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V. Artemyev 1, Q. Ma2,3, R. W. Ebert 4,5, X.-J. Zhang6,1, F. Allegrini4,5 +1Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, USA +2Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, USA +3Center for Space Physics, Boston University, Boston, MA, USA +4Southwest Research Institute, San Antonio, TX, USA +5Department of Physics and Astronomy, University of Texas at San Antonio, San Antonio, TX, USA +6Department of Physics, University of Texas at Dallas, Richardson, TX, USA +Key Points: +• We report Juno observations of thin anisotropic current sheets in the Jovian +magnetodisk +• The contribution of electron streams to the current sheet stress balance is esti- +mated +• We show force-free current sheet configuration supported by strong electron +field-aligned currents +Corresponding author: Anton Artemyev, aartemyev@igpp.ucla.edu +–1– +arXiv:2301.03731v1 [physics.space-ph] 10 Jan 2023 + +manuscript submitted to JGR: Space Physics +Abstract +Current sheets are an essential element of the planetary magnetotails, where strong +plasma currents self-consistently support magnetic field gradients. The current sheet +configuration is determined by plasma populations that contribute to the current den- +sity. +The most commonly investigated configuration is supported by diamagnetic +cross-field currents of hot ions, typical for the magnetospheres of magnetized plan- +ets. +In this study, we examine a new type of the current sheet configuration sup- +ported by field-aligned currents from electron streams in the Jovian magnetodisk. +Such bi-directional streams increase the electron thermal anisotropy close to the fire- +hose instability threshold and lead to strong magnetic field shear. The current sheet +configuration supported by electron streams is nearly force-free, with |B| ≈ const +across the sheet. Using Juno plasma and magnetic field measurements, we investigate +the internal structure of such current sheets and discuss possible mechanisms for their +formation. +1 Introduction +Current sheets are observed in all planetary magnetotails, the night-side regions +of stretched magnetic field lines. The configuration of magnetotails depends on char- +acteristics of the planetary magnetic field interaction with solar wind (Bagenal, 1992; +Bagenal & Murdin, 2000; Jackman et al., 2014; Khurana & Liu, 2018), but all mag- +netotails contain current sheets, spatially localized regions of strong plasma currents. +Instabilities of such current sheets, either internally or externally driven, can result in +the magnetic reconnection that further transforms the magnetic energy to the plasma +heating and charged particle acceleration (e.g., Birn et al., 2012; Gonzalez & Parker, +2016; Sitnov et al., 2019). Among all the current sheets in space plasmas, the one +in Earth’s magnetotail has been most intensively investigated, where strong diamag- +netic currents, predominantly carried by hot protons (with a small fraction of oxygen +ions), support the magnetic field configuration and pressure balance self-consistently +(see Schematic in Fig. 1(a) and Birn, Schindler, and Hesse (2004); Sitnov and Merkin +(2016); Zelenyi et al. (2011); Artemyev and Zelenyi (2013); Lu et al. (2016)). The +current sheet in Earth’s magnetotail is characterized by large plasma β ∼ 100 (β is +the ratio of plasma and magnetic field pressures), which leads to the dominant role +of cross-field currents, j⊥ ≫ j∥, in the current sheet configuration. The relative con- +tribution of ions and electrons to j⊥ depends on the polarization electric fields, E⊥ +(Schindler & Birn, 2002; Schindler et al., 2012); thin (ion-kinetic-scale) current sheets +are, therefore, mostly electron dominated (Hesse et al., 1998; Runov et al., 2006; Arte- +myev et al., 2009; Lu et al., 2019) due to the strong polarization electric field within +(Zelenyi et al., 2010; Lu et al., 2016). However, in the Earth’s magnetotail, there are +two interesting exceptions that j∥ can be appreciably large. First, in the near-Earth +magnetotail, during the current sheet thinning (formation of thin current sheets during +the substrom growth phase, see Birn, Dorelli, et al. (2004); Petrukovich et al. (2007); +Hsieh and Otto (2015)) the spatial scale (thickness) of the current sheet can become +smaller than the thermal proton gyroradius. In such sub-ion scale thin current sheets +the proton pressure cannot be redistributed within sub-gyroradius scale. To establish +the pressure balance, the intensified field-aligned electron currents form strong mag- +netic field shear, contributing to a (partially) force-free current sheet configuration +with j⊥ ≤ j∥ (see Schematic in Fig. 1(b) and Nakamura et al. (2008); Artemyev et al. +(2013); Artemyev, Angelopoulos, Liu, and Runov (2017)). Second, a similar, partially +force-free current sheet configuration has been observed in the distant (lunar orbit) +magnetotail (Xu et al., 2018) where plasma β can be as low as ∼ 1, and plasma pres- +sure is not sufficient to establish the pressure balance. This second force-free current +sheet with low β is also typical in Mars (Artemyev, Angelopoulos, Halekas, et al., 2017; +DiBraccio et al., 2015) and Venus (Rong et al., 2015) magnetotails occupied by cold +planetary plasmas. +–2– + +manuscript submitted to JGR: Space Physics +Figure 1. +Schematic of current sheet configurations. (a) 2D configuration of the magnetic +field line in the current sheet with plasma pressure P gradients (the dashed line parallel to the +x-axis illustrates its projection to the xy plane). The main current density flows along the y-axis, +transversely to magnetic field lines. (b) 3D configuration of the magnetic field line in the current +sheet with B2 +y/8π gradients playing the role of P gradients (see discussion on the P → B2 +y transi- +tion in Syrovatskii, 1981; Lukin et al., 2018). The dashed curve illustrates the field line projection +to the xy plane. The main current density flows along B, i.e., parallel to the magnetic field line. +–3– + +(a) +Z +jyBz +V-PA +X +个jvBx +jy-i~vp +y +jx-j +(b) +B +Z +jyBz +jxBy个 +X +jz By +jy-jinmanuscript submitted to JGR: Space Physics +Between the two mechanisms for the force-free (or partially force-free) current +sheet formation, sub-ion scale thickness versus low plasma β, the first one is more +interesting, because such sub-ion current sheets with strong field-aligned electron cur- +rents can be favorable to kinetically-driven magnetic field reconnection and current +filamentation (Drake & Lee, 1977; Zelenyi & Taktakishvili, 1987; Wilson et al., 2016). +Investigations of such current sheet configurations, however, are quite limited, because +these current sheets are rather transient (dynamical) in the Earth’s magnetotail (see +discussions in Nakamura et al., 2008). An alternative plasma environment to investi- +gate these specific current sheets would require hot heavy ions, curved magnetic field +lines, and fast electron plasma flows. The best possible, accessible system is the Jovian +magnetodisk, filled by sulfur and oxygen ions (Thomas et al., 2004; Krupp et al., 2004; +Mauk et al., 2004; Kim et al., 2020a) with various charge states (e.g., Selesnick & Co- +hen, 2009; Clark et al., 2016; Allen et al., 2019; Kim et al., 2020b) and conjugate to the +Jovian aroural region, a powerful source of field-aligned electron streams (e.g., Mauk et +al., 2017b, 2017a). Therefore, We will use the recently available plasma and magnetic +field measurements (Bagenal et al., 2017) from Juno in the Jovian magnetodisk to +systematically examine sub-ion scale, force-free (or partially force-free) current sheets. +In this study, we focus on 18 events of Juno current sheet crossings during the +first 30 orbits, i.e., those with a strong magnetic shear (field-aligned currents), electron +field-aligned streams, and different combinations of proton and heavy ion contributions +to the pressure. We estimate the current sheet spatial scale (thickness) and current +density during its flapping motion. The following of paper includes three sections: de- +scription of Juno instruments and data analysis techniques in Sect. 2, detailed analysis +of 9 current sheet crossings in Sect. 3, and discussion on the results in Sect. 4. +2 Data analysis technique and instruments +We use data from the Juno magnetometer (MAG) in 2017-2018, with 1s time +resolution (Connerney, Benn, et al., 2017; Connerney, Adriani, et al., 2017). We focus +on measurements at r > 25RJ radial distances in the Jovian magnetodisk. Figure +2 shows a typical one-day magnetic field measurements in the magnetodisk by Juno: +quasi-periodic crossings of zeros of the radial magnetic field component Br = 0 (current +sheet) are due to flapping oscillations of the magnetodisk. For each such crossing we +transform the magnetic field into local coordinate systems (Sonnerup & Cahill, 1968): +Bl is the most varying magnetic field component, Bn is the less varying component, +and Bm is transverse to Bl and Bn. We keep only those current sheets with a Bm peak +at Bl = 0 and with available plasma measurements by the Jovian Auroral Distributions +Experiment (JADE) instrument. The times of selected crossings are given in table 1, +along with their radial distances. Overview plots of plasma and magnetic field profiles +during each current sheet crossing are provided in the Supporting Information. In the +main text, we mostly discuss six force-free current sheets, in comparison with three +current sheets supported by plasma pressure gradients (non-force-free sheets), but our +conclusions are supported by the entire dataset. +Jovian Auroral Distributions Experiment (JADE) measures (see McComas, Alexan- +der, et al., 2017; McComas, Szalay, et al., 2017; Kim et al., 2020a, 2020b) electron +distributions from below ∼ 0.1 to 100 keV (at a 1 s cadence) and ions from ∼ 13 eV to +∼ 50 keV (including ion composition, at a 1 s cadence). We have averaged JADE data +over the spin period (30 s) to obtain a complete pitch angle coverage from 0◦ to 180◦. +These energy ranges cover the main (thermal) plasma populations in Jupiter’s magne- +todisk. We use the following data products from JADE: electron pitch-angle/energy +distributions averaged over time interval of the current sheet crossing, electron omni- +directional energy spectra (energy flux) Fe(t, E), electron pressure pe(t) and density +ne(t) profiles, electron pressure anisotropy Ae = pe,∥/pe,⊥ averaged over the current +sheet crossing interval, omnidirectional proton and heavy ion energy spectrum (energy +–4– + +manuscript submitted to JGR: Space Physics +Br +Bφ +Bθ +B, nT +−20 +−10 +0 +10 +20 +20:0020:1520:3020:4521:0021:1521:30 +B, nT +−20 +−10 +0 +10 +20 +10:30 +10:40 +10:50 +11:00 +11:10 +B, nT +−30 +−20 +−10 +0 +10 +20 +30 +00:00 +02:00 +04:00 +06:00 +08:00 +10:00 +12:00 +14:00 +16:00 +18:00 +20:00 +22:00 +00:00 +Figure 2. +Top panel shows one-day measurements by Juno MAG: radial Br, azimuthal Bϕ, +and north-south Bθ components. Bottom panels show the expanded view of two current sheet +crossings from the top panel. +flux) Fp(t, E) and Fi(t, E), and proton and heavy ion densities np(t), ni(t). Note that +field view of the JADE electron detector may not cover the entire [0, 180◦] pitch-angle +range at the current sheet boundaries, so we average electron measurements over each +current sheet crossing interval to obtain a more complete pitch-angle distribution. The +rotation of the background magnetic field direction across the current sheet ensures a +wide coverage of the pitch-angle range (see pitch-angle distributions below), which is +needed for the estimate of Ae. In this study, we use the heavy ion (or ”i”) to denote +integrated quantities of those ion populations with mass-to-charge ratio larger than +five. We also estimate proton and heavy ion pressures pp(t), pi(t) as moments of the +omnidirectional energy spectrum, i.e., we assume that thermal proton and ion speeds +well exceed their bulk flow speed (this is a reasonable assumption, see Kane et al., +1999; Frank et al., 2002; Waldrop et al., 2005; Kim et al., 2020b). +Using plasma and magnetic field measurements, we estimate βe,p,i = 8πpe,p,i/B2 +profiles and electron fire-hose parameter Λe ≈ βe(Ae − 1)/2 = 4π(p∥e − p⊥e)/B2. +To show that this parameter controls the contribution of the electron anisotropy to +the current density, we illustrate the case for a simple quasi-1D current sheet with +∂Bl/∂rn = 4πjm/c, ∂Bm/∂rn = 4πjl/c, and Bn = const ̸= 0. The electron current +due to cross-field drifts in such current sheet is (Shkarofsky et al., 1966) +j⊥e = −ecn[E × B] +B2 +− c[∇p⊥e × B] +B2 ++ cΛe +4π +[B × (B∇) B] +B2 +(1) +where E⊥ is the transverse component of the polarization electric field. This equation +should be supplemented by the field-aligned stress balance equation +enE∥ = −∇∥p∥e + Λe +4π (B∇) B +–5– + +manuscript submitted to JGR: Space Physics +# +date +time +radial +comments +distance +1 +2017 doy 027 +09:00-09:20 +65RJ +ff CS, heavy ions +2 +2017 doy 080 +07:10-07:40 +61RJ +ff CS, heavy ions +3 +2017 doy 080 +17:10-17:50 +61RJ +non-ff CS, heavy ions +4 +2017 doy 128 +08:00-08:30 +86RJ +ff CS, heavy ions +5 +2017 doy 128 +16:30-17:30 +86RJ +ff CS, heavy ions +6 +2017 doy 133 +05:10-06:10 +61RJ +non-ff CS, heavy ions +7 +2017 doy 181 +06:50-07:20 +86RJ +ff CS, heavy ions +8 +2017 doy 181 +09:45-10:20 +86RJ +ff CS, heavy ions +9 +2018 doy 031 +09:30-10:40 +50RJ +non-ff CS, heavy ions +10 +2018 doy 034 +13:30-14:30 +50RJ +non-ff CS, heavy ions +11 +2018 doy 085 +21:00-21:50 +62RJ +non-ff CS, protons +12 +2018 doy 086 +10:15-11:15 +55RJ +ff CS, heavy ions & protons +13 +2018 doy 088 +01:20-02:30 +38RJ +ff CS, protons +14 +2018 doy 088 +11:30-12:30 +38RJ +ff CS, protons +15 +2018 doy 088 +21:30-22:30 +38RJ +ff CS, heavy ions +16 +2018 doy 141 +10:30-11:15 +38RJ +ff CS, heavy ions & protons +17 +2018 doy 141 +19:45-21:00 +38RJ +ff CS, heavy ions & protons +18 +2018 doy 142 +20:00-20:45 +26RJ +non-ff CS, heavy ions & protons +Table 1. +List of current sheet crossings. In the comments column, ff CS and non-ff CS stand +for the force-free and non force-free current sheets, respectively. The dominant ion type (heavy +ions or protons) is also indicated in the comments column. +where E∥ is the field-aligned component of the polarization electric field and ∇∥ = +(B/B)∇. +For force-free current sheets with ∇npe⊥ = ∇npe∥ = ∇nB = 0, the current +density equation can be rewritten as +j⊥e = cΛe +4π +� +−el +B2 +n +B2 ∇nBm + em +B2 +n +B2 ∇nBl + en +Bn +B2 (Bl∇nBm − Bm∇nBl) +� +whereas the parallel current density can be obtained from the divergence free condition: +j∥e = − 1 +B (Bl∇nBm − Bm∇nBl) +Thus, the total electron current is +je += +j⊥e + j∥e +B +B = 4πcΛe +� +−el +� +∇nBm − Bm∇nB +B +� ++ em +� +∇nBl − Bm∇nB +B +�� +(2) += +cΛe +4π (−el∇nBm + em∇nBl) = Λej +where j = (c/4π)∇ × B. +3 Current sheet examples +Figure 3 shows six typical examples of thin current sheets with an almost constant +magnetic field magnitude across the sheet, |B| ≈ const. Such a constant magnetic field +implies the dominant role of field-aligned currents in the current sheet configuration: +if j = C · B, then j × B = 0 and there is no pressure variation across the sheet (note +–6– + +manuscript submitted to JGR: Space Physics +that typical crossings of the magnetodisk current sheet show strong variations of the +plasma pressure (density) across the sheet, see, e.g., Huscher et al., 2021). Panels +(a) show that |B| = const is due to peak of Bm component that compensates the +drop of B2 +l around the neutral plane (where Bl = 0). As expected for the force-free +current sheet, there are no appreciable variations in the ion fluxes (protons or heavy +ions) across the sheet (see panel (c)). Electron fluxes may show some variations (see +panel (b)), but variations of the electron thermal pressure are insufficient to cause any +significant variations of the magnetic field pressure (see panel (d); note that a variation +of 10−2 · cm−3keV corresponds to ≈ 2nT variation of the magnetic field). +Let us explain the absence of ion pressure variations during the observed current +sheet crossings. As shown in Fig. 3, the time-scale of current sheet crossings varies from +T ∼ 5 min to 30 min; taking into account the flapping speed of ∼ ωJR tan θ, this time- +scale can be converted to a spatial scale L ≈ 1000km·(r/50RJ)·(T/60s) ∈ [5, 30]·103km +(here ωJ is the Jupiter rotational frequency, r is the radial distance of the current sheet, +and θ ≈ 9.5◦ is the tilt angle of the magnetodisk with respect to the planetary equator, +see Connerney et al. (1998); Khurana and Schwarzl (2005)). Despite that we used the +upper limit for the flapping speed (see discussion and observations in Hill, 1979; Kim +et al., 2020b), this spatial scale is much smaller than the typical thicknesses of Jovian +magnetodisk current sheets, L ∼ 2RJ ≈ 105km (Connerney et al., 2020; Liu et al., +2021; Khurana et al., 2022). More importantly, this spatial scale is comparable to +(or smaller than) the hot proton or heavy ion gyroradius: for the equatorial field of +a typical current sheet, ∼ 5nT, ∼ 30keV protons and sulfur ions have gyroradii of +∼ 5000km and ∼ 25000km, respectively. +Thus, these current sheets are likely on +sub-ion scale, within which ions cannot redistribute their pressure. To establish the +pressure balance in such current sheets, the field-aligned electron currents create a +local Bm peak. +To estimate the electron contribution to the field-aligned currents, we use Eq. (2) +and the measured electron pitch-angle, energy distributions. Figure 4(b) shows that +all six current sheets are characterized by field-aligned bi-directional electron streams. +These streams occupy ∼ 30◦ in pitch angles around the parallel and anti-parallel (with +respect to the background magnetic field) directions, in the energy range below ∼ 10 +keV. Such field-aligned streams may be generated by reconnection further downtail +(e.g., Kronberg et al., 2012) or originate from the aurora acceleration region (Mauk et +al., 2017b, 2020; Elliott et al., 2020; Allegrini et al., 2020). In the Earth’s magneto- +sphere, similar field-aligned streams are observed in the near-Earth magnetotail (Hada +et al., 1981; Walsh et al., 2013; Artemyev, Angelopoulos, Liu, & Runov, 2017), but +their energies are well below ∼ 1keV, in agreement with the capability of the Earth’s +aurora acceleration (Ergun et al., 2004; Chaston et al., 2007; Watt & Rankin, 2009). +More effective aurora acceleration in the Jupiter magnetosphere may produce ∼ 10 +keV beams (Kollmann et al., 2018; Saur et al., 2018; Damiano et al., 2019; Lysak +et al., 2021), which are likely further expanded in the pitch-angle space by various +scattering mechanisms and form the electron streams observed in the plasma sheet +(see discussion in Zhang et al., 2020). In the presence of a large electron βe ∼ 1, such +field-aligned streams create a strong pressure anisotropy with the fire-hose parameter +Λe reaching one (see Fig. 4(c)). Thus, Eq. (2) shows that for this large Λe, electrons +will carry almost all the current to support Bl and Bm variations across the sheet. +We explain the formation of force-free (partial force-free) current sheets with +strong field-aligned electron currents (shown in Fig. +4) as a need to balance the +magnetic field pressure decrease ∼ B2 +l on a sub-ion scale. This explanation implies +that similar electron currents should be observed in current sheets on larger scales, +where they will not create Bm peaks, but rather contribute to the cross-field current +density, in agreement with Eq. (1) (see discussion in Artemyev et al., 2016). Figure +5 shows such current sheets with a significant ion pressure contribution to the stress +–7– + +manuscript submitted to JGR: Space Physics +(e) +10−3 +10−2 +10−1 +2018 doy 086 +10:00 +10:15 +10:30 +10:45 +11:00 +11:15 +11:30 +(d) +10−3 +10−2 +(c) +102 +103 +104 +(b) +102 +103 +104 +105 +(a) +−5 +0 +5 +10 +(e) +10−2 +10−1 +2018 doy 088 +11:30 +11:40 +11:50 +12:00 +12:10 +12:20 +12:30 +p +e +i +(d) +10−3 +10−2 +10−1 +1/cm2/s/sr +105 +106 +(c) +102 +103 +104 +1/cm2/s/sr +103 +104 +105 +106 +107 +(b) +102 +103 +104 +105 +|B| +Bl +Bm +Bn +(a) +−15 +−10 +−5 +0 +5 +10 +15 +(e) +p, eV⋅cm-3 +10−3 +10−2 +10−1 +2017 doy 181 +09:50 +09:55 +10:00 +10:05 +10:10 +(d) +n, cm-3 +10−3 +10−2 +(c) +energy, eV +103 +104 +(b) +energy, eV +102 +103 +104 +105 +(a) +B, nT +−4 +−2 +0 +2 +4 +6 +(e) +10−3 +10−2 +10−1 +2017 doy 128 +16:30 +16:40 +16:50 +17:00 +17:10 +17:20 +17:30 +i +e +p +(d) +10−3 +10−2 +1/cm2/s/sr +105 +106 +(c) +103 +104 +1/cm2/s/sr +103 +104 +105 +106 +(b) +102 +103 +104 +105 +(e) +10−3 +10−2 +10−1 +100 +2017 doy 080 +07:16 07:18 07:20 07:22 07:24 07:26 07:28 07:30 +(d) +10−3 +10−2 +10−1 (c) +103 +104 +(b) +102 +103 +104 +105 +(e) +p, eV⋅cm-3 +10−3 +10−2 +10−1 +100 +2017 doy 027 +09:00 +09:05 +09:10 +09:15 +09:20 +(d) +n, cm-3 +10−3 +10−2 +(c) +energy, eV +103 +104 +(b) +energy, eV +102 +103 +104 +105 +|B| +Bl +Bm +Bn +(a) +−4 +−2 +0 +2 +4 +(a) +−7.5 +−5 +−2.5 +0 +2.5 +5 +7.5 +(a) +B, nT +−10 +−5 +0 +5 +10 +Figure 3. +Six examples of force-free current sheets observed by Juno in different radial dis- +tances (see table 1). (a) Magnetic field components in the local (MVA) coordinate system and +the magnetic field magnitude. (b, c) Omnidirectional spectra of electrons and dominant ion +species (blue for protons and red for heavy ions). (d,e) Densities and pressures of electrons and +dominant ion species. +–8– + +manuscript submitted to JGR: Space Physics +(d) +10−2 +10−1 +100 +2018 doy 086 +10:00 +10:15 +10:30 +10:45 +11:00 +11:15 +11:30 +(b) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(a) +10−2 +10−1 +100 +Bl, nT +−10 +−5 +0 +5 +10 +(d) +10−2 +10−1 +100 +2018 doy 088 +11:30 +11:40 +11:50 +12:00 +12:10 +12:20 +12:30 +1/cm2/s/sr +103 +104 +105 +106 +107 +(b) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +p +e +i +(a) +10−1 +100 +(d) +Λ +10−1 +100 +2017 doy 181 +09:50 +09:55 +10:00 +10:05 +10:10 +(b) +energy, eV +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(a) +β +10−2 +10−1 +100 +101 +Bl, nT +−10 +−5 +0 +5 +10 +(d) +10−2 +10−1 +100 +2017 128 +16:30 +16:40 +16:50 +17:00 +17:10 +17:20 +17:30 +1/cm2/s/sr +103 +104 +105 +106 +(b) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(d) +10−2 +10−1 +100 +2017 080 +07:16 07:18 07:20 07:22 07:24 07:26 07:28 07:30 +(b) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(c) +Λ +10−2 +10−1 +100 +2017 027 +09:00 +09:05 +09:10 +09:15 +09:20 +(b) +energy, eV +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +i +e +p +(a) +10−1 +100 +101 +(a) +10−2 +10−1 +100 +101 +102 +(a) +β +10−2 +10−1 +100 +101 +Figure 4. +Six examples of force-free current sheets from Fig. 3. (a) Electron and ion betas. +(b) Electron pitch-angle, energy distribution averaged over the entire event. (c) Electron fire-hose +parameter and Bl field to illustrate the current sheet center. +–9– + +manuscript submitted to JGR: Space Physics +balance and strong field-aligned electron anisotropy. There is almost no Bm peak in +the current sheet center (where Bl ∼ 0, see panel (a)), whereas ion fluxes and pressures +exhibit peaks (panels (c,d)). Temporal scales of current sheet crossings from Fig. 5 +are about 20-30min, corresponding to a spatial scale larger than the typical proton +and heavy ion gyroradius in these current sheets. Note the two current sheets on 2017 +doy 080 (shown in Fig. 3 and 5) exhibit different characteristics: the force-free current +sheet in Fig. 3 was crossed within a couple of minutes and shows no pi variations, +whereas the one in Fig. 5 was crossed within ∼ 15 min and shows strong pi variations. +In current sheets from Fig. 5, electron pitch-angle distributions contain strong +field-aligned streams with characteristics very similar to those in the force-free cur- +rent sheets (compare Figs. 4(b) and 5(g)). However, contrary to force-free current +sheets, these field-aligned streams mostly contribute to anisotropic cross-field electron +currents, see Eq. (1). Indeed, small magnetic field pressure at the current sheet center +(where Bl ∼ 0) increases βe and leads to a large electron fire-hose parameter Λe ≈ 1 +even for a moderate anisotropy Ae (see Fig. 5(f,h)). +Comparison of Figs. 3, 4, and 5 suggests the following mechanism for the forma- +tion of the force-free current sheet. Certain external conditions (e.g., electron acceler- +ation in the aurora region, see Mauk et al. (2017b); Saur et al. (2018); Damiano et al. +(2019); Lysak et al. (2021)) generate field-aligned electron streams bouncing within the +current sheet in the Jovian magnetodisk. These streams contribute to the field-aligned +electron anisotropy, Ae > 1, and fire-hose parameter Λe > 0. In typical thick current +sheets such anisotropy supports the cross-field electron currents (see Eq.(1)) and may +create a thin, sub-ion scale current sheet embedded into a thick, ion scale current sheet +(e.g., Zelenyi et al., 2004; Mingalev et al., 2018; Kamaletdinov et al., 2020; Zelenyi +et al., 2022). Indeed, the magnetic field profiles in Fig. 5 exhibit stronger gradients +around Bl ∼ 0. If external drivers result in the current sheet thinning, the current +sheet may reach sub-ion spatial scale where ions cannot redistribute their pressure +and maintain the stress balance. In this case, the electron currents form Bm peaks to +balance the B2 +l drop at the current sheet center, and self-consistently evolve from the +cross-field currents to field-aligned currents (see Schematic in Fig. 1). +4 Discussion +In this study, we investigate force-free (and partially force-free) current sheets, +where field-aligned electron streams support the pressure anisotropy and parallel cur- +rents, leading to the formation of the Bm peak at the current sheet center, Bl = 0. +Let us discuss the difference of the stress balance in such current sheets from that +in more typical thick current sheets. In current sheets, the 2D stress balance in the +equatorial plane (balance along the radial direction) is maintained by a combination +of the centrifugal force, plasma pressure force, and magnetic field line tension force +(Hill & Carbary, 1978; Cheng, 1983; Zimbardo, 1989): +1 +c jϕBθ + minω2 +Jr + ∇r ˆp = 0 +(3) +where ∇r ˆp is the radial component of the plasma pressure tensor gradient. In the +local coordinate system, l ≈ er and m ≈ eφ. Based on the vertical stress balance, +8πp = max B2 +l − max B2 +m, we may estimate the current density as: +jm ≈ c +4π +max Bl +Bn +∇r max Bl ≈ 5.5nA +m2 · (r/RJ)−2 ≈ 12pA +m2 · +� +r +30RJ +�−2 +(4) +where max Bl ≈ 50nT·(r/RJ)−1 and max Bl/Bn ≈ 20 are the empirical relations (see, +e.g., Artemyev et al., 2014; Liu et al., 2021). The corresponding current sheet thickness +L = c max Bl/4πjm ≈ 1.5RJ · (r/30RJ)−1 should be larger than 1RJ at r > 30RJ, +which is consistent with the thickness estimates for typical thick current sheets (e.g., +–10– + +manuscript submitted to JGR: Space Physics +Bl, nT +−20 +−10 +0 +10 +20 +(h) +Λ +10−4 +10−3 +10−2 +10−1 +100 +101 +2018 doy 034 +13:30 +13:40 +13:50 +14:00 +14:10 +14:20 +(h) +Λ +10−3 +10−2 +10−1 +100 +2017 doy 133 +05:20 +05:30 +05:40 +05:50 +06:00 +06:10 +(h) +Λ +10−2 +10−1 +100 +2017 doy 080 +17:10 +17:20 +17:30 +17:40 +17:50 +(g) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(f) +10−2 +10−1 +100 +101 +102 +2017 doy 133 +05:20 +05:30 +05:40 +05:50 +06:00 +06:10 +1/cm2/s/sr +103 +104 +105 +106 +107 +(g) +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(f) +10−2 +10−1 +100 +101 +102 +2018 doy 034 +13:30 +13:40 +13:50 +14:00 +14:10 +14:20 +(g) +energy, eV +102 +103 +104 +105 +pitch-angle, ○ +0 +30 +60 +90 +120 +150 +180 +(f) +β +10−2 +10−1 +100 +101 +102 +2017 doy 080 +17:10 +17:20 +17:30 +17:40 +17:50 +(e) +10−3 +10−2 +10−1 +100 +i +e +(d) +10−3 +10−2 +10−1 +1/cm2/s/sr +105 +106 +(c), ions +103 +104 +1/cm2/s/sr +103 +104 +105 +106 +107 +(b) +102 +103 +104 +105 +(e) +10−3 +10−2 +10−1 +100 +(d) +10−4 +10−3 +10−2 +10−1 (c), ions +103 +104 +(b) +102 +103 +104 +105 +(e) +p, eV⋅cm-3 +10−3 +10−2 +10−1 +100 +(d) +n, cm-3 +10−3 +10−2 +(c), ions +energy, eV +103 +104 +(b) +energy, eV +102 +103 +104 +105 +|B| +Bl +Bm +Bn +(a) +−20 +−10 +0 +10 +20 +(a) +−7.5 +−5 +−2.5 +0 +2.5 +5 +7.5 +(a) +B, nT +−10 +−5 +0 +5 +10 +Figure 5. +Three examples of current sheets with large field-aligned electron currents and +plasma pressure variations observed by Juno in different radial distances (see table 1). (a) Mag- +netic field components in the local (MVA) coordinate system and the magnetic field magnitude. +(b, c) Omnidirectional spectra of electrons and dominant ion species. (d, e) Densities and pres- +sures of electrons and dominant ion species. (f) Electron and ion beta. (g) Electron pitch-angle +and energy distribution averaged over the entire event. (h) Electron fire-hose parameter and Bl +field to show the current sheet center. +–11– + +manuscript submitted to JGR: Space Physics +Khurana et al., 2004; Liu et al., 2021). Such current sheets will be crossed during an +interval of ∆t = L/ωjr tan θ ≈ 3hours · (r/30RJ)−2; for much thinner current sheets +as in our dataset, the traversal timescale will be less than 10 min (see Fig. 3). The +stress balance in such thin current sheets cannot be maintained by centrifugal force +and radial gradient of the plasma pressure. Instead, the electron pressure anisotropy +contributes to the stress balance (Rich et al., 1972): +∇r ˆp = ∇rp⊥ + ∇θ +p∥e − p⊥e +B2 +BrBθ ≈ ∇θ +�Λe +4π BrBθ +� +(5) +This equation shows that the thin current sheet configuration resembles a classical +rotational discontinuity, with no variations of the Alfven speed because of the pressure +anisotropy (Hudson, 1970): +∆vA = +� +B2 +4πnmi +(1 − Λe) ∼ 0 +(6) +This condition allows for a balance of the 1D current sheet (with thickness L much +smaller than the spatial scale of the radial gradient of the plasma density) without +fast plasma flows typical for rotational discontinuities in anisotropic plasma, where +cross-sheet change of the plasma flow velocity equals to ∆vA (see discussions of the +anisotropy contribution to the force-free current sheet configurations in Vasko et al., +2014; Artemyev, Angelopoulos, Vasko, & Zelenyi, 2020). Formation of such 1D current +sheets around the fire-hose marginally stability threshold has been predicted theoreti- +cally (e.g., Francfort & Pellat, 1976; Cowley, 1978; Cowley & Pellat, 1979), but never +have been detected under quiet geomagnetic conditions in the Earth’s magnetotail (see +discussion in Sitnov et al., 2006; Artemyev & Zelenyi, 2013). Observations of these +current sheets in the Jovian magnetotail confirm the theoretical predictions, which can +further lead to improved current sheet models (see discussion on development of the +next generation of current sheet models in Sitnov & Merkin, 2016; Y. D. Yoon et al., +2021; Zelenyi et al., 2022). +It is worth to note that these current sheets are electromagnetically disconnected +from the Jovian ionosphere, because the local Alfven speed is zero, vA = (B/√4πnmi)· +√1 − Λe = 0, and there are no field-aligned perturbations propagating from the current +sheet to the ionosphere. Such local destruction of magnetosphere-ionosphere coupling +is an interesting phenomenon that we do not observe in the Earth’s magnetotail, where +Λe is much more moderate (Artemyev, Angelopoulos, Vasko, Petrukovich, et al., 2020). +Theoretical investigations of these force-free current sheets in the Jovian mag- +netodisk is a real challenge for plasma kinetics, because these current sheets share +properties of 2D plasma equilibria (with the tension force ∼ (4π/c) · jmBn) and prop- +erties of rotational discontinuities (with j = C ·B and Bn ̸= 0). All existing 2D kinetic +current sheet models operate with the plasma pressure gradients, ∇p = c−1j × B, +and do not include field-aligned currents (see, e.g., P. H. Yoon & Lui, 2005; Vasko +et al., 2013; Zelenyi et al., 2011; Sitnov et al., 2006; Sitnov & Merkin, 2016, and +references therein). Existing models of force-free current sheets mostly assume 1D +tangential discontinuities with Bn = 0 (see, e.g., Harrison & Neukirch, 2009; Panov +et al., 2011; Neukirch, Vasko, et al., 2020; Neukirch, Wilson, & Allanson, 2020, and +references therein). Construction of the kinetic model for 1D rotational discontinuities +requires assumptions of an additional system symmetry (e.g., Sonnerup & Su, 1967; +Artemyev, 2011; Mingalev et al., 2012; Vasko et al., 2014), whereas kinetic models +of 2D rotational discontinuities have not yet been constructed. Although fluid mod- +els of 2D rotational discontinuities (with j = C · B, jnBm = jmBn, and Bn ̸= 0) +can be constructed (e.g., Cowley, 1978; Hilmer & Voigt, 1987; Hau & Voigt, 1992; +Lukin et al., 2018), their kinetic realization has not been demonstrated. Therefore, +further theoretical investigations are needed to explain Juno observations in the Jovian +magnetodisk. +–12– + +manuscript submitted to JGR: Space Physics +5 Conclusions +We have investigated thin (thickness is smaller than or about the hot ion gyro- +radius) current sheets observed by Juno in the Jovian magnetodisk and characterized +by strong field-aligned electron streams. We have demonstrated that electron streams +support the strong field-aligned anisotropy, which may increase the fire-hose parameter +up to the instability threshold, Λe = 4π(pe∥ − pe⊥)/B2 → 1. In such thin, marginally +stable current sheets, almost all current is field-aligned, and the current sheet configu- +ration is force-free, with j×B ≈ 0. This is a new type of strongly anisotropic, force-free +current sheets, which has not been reported in the quiet-time Earth’s magnetopshere +or solar wind. Further numerical investigations of such current sheet formation and +dynamics will reveal their potential role in the particle acceleration (e.g., via magnetic +reconnection). +Acknowledgments +This work is supported by Grants 80NSSC19K1593 (A.V.A.) under Juno Participat- +ing Scientist program, and by subcontract 699046X to UCLA under prime contract +ZZM06AA75C (X.J.Z. and Q.M.). +Open Research +Data processing was done using SPEDAS V4.1, see Angelopoulos et al. (2019). All +the adopted data are available in the archive https://doi.org/10.5281/zenodo +.7470240 +References +Allegrini, F., Mauk, B., Clark, G., Gladstone, G. R., Hue, V., Kurth, W. S., . . . Wil- +son, R. J. +(2020, April). +Energy Flux and Characteristic Energy of Electrons +Over Jupiter’s Main Auroral Emission. Journal of Geophysical Research (Space +Physics), 125(4), e27693. doi: 10.1029/2019JA027693 +Allen, R. C., Paranicas, C. P., Bagenal, F., Vines, S. K., Hamilton, D. C., Allegrini, +F., . . . Wilson, R. J. 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4Southwest Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' San Antonio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' USA 5Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' University of Texas at San Antonio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' San Antonio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' USA 6Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' University of Texas at Dallas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Richardson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' USA Key Points: We report Juno observations of thin anisotropic current sheets in the Jovian magnetodisk The contribution of electron streams to the current sheet stress balance is esti- mated We show force-free current sheet configuration supported by strong electron field-aligned currents Corresponding author: Anton Artemyev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' aartemyev@igpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='edu –1– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='03731v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='space-ph] 10 Jan 2023 manuscript submitted to JGR: Space Physics Abstract Current sheets are an essential element of the planetary magnetotails, where strong plasma currents self-consistently support magnetic field gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The current sheet configuration is determined by plasma populations that contribute to the current den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The most commonly investigated configuration is supported by diamagnetic cross-field currents of hot ions, typical for the magnetospheres of magnetized plan- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In this study, we examine a new type of the current sheet configuration sup- ported by field-aligned currents from electron streams in the Jovian magnetodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Such bi-directional streams increase the electron thermal anisotropy close to the fire- hose instability threshold and lead to strong magnetic field shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The current sheet configuration supported by electron streams is nearly force-free, with |B| ≈ const across the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Using Juno plasma and magnetic field measurements, we investigate the internal structure of such current sheets and discuss possible mechanisms for their formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1 Introduction Current sheets are observed in all planetary magnetotails, the night-side regions of stretched magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The configuration of magnetotails depends on char- acteristics of the planetary magnetic field interaction with solar wind (Bagenal, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Bagenal & Murdin, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Jackman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Khurana & Liu, 2018), but all mag- netotails contain current sheets, spatially localized regions of strong plasma currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Instabilities of such current sheets, either internally or externally driven, can result in the magnetic reconnection that further transforms the magnetic energy to the plasma heating and charged particle acceleration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Birn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Gonzalez & Parker, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Sitnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Among all the current sheets in space plasmas, the one in Earth’s magnetotail has been most intensively investigated, where strong diamag- netic currents, predominantly carried by hot protons (with a small fraction of oxygen ions), support the magnetic field configuration and pressure balance self-consistently (see Schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1(a) and Birn, Schindler, and Hesse (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Sitnov and Merkin (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev and Zelenyi (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The current sheet in Earth’s magnetotail is characterized by large plasma β ∼ 100 (β is the ratio of plasma and magnetic field pressures), which leads to the dominant role of cross-field currents, j⊥ ≫ j∥, in the current sheet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The relative con- tribution of ions and electrons to j⊥ depends on the polarization electric fields, E⊥ (Schindler & Birn, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Schindler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' thin (ion-kinetic-scale) current sheets are, therefore, mostly electron dominated (Hesse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Runov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Arte- myev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2019) due to the strong polarization electric field within (Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' However, in the Earth’s magnetotail, there are two interesting exceptions that j∥ can be appreciably large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' First, in the near-Earth magnetotail, during the current sheet thinning (formation of thin current sheets during the substrom growth phase, see Birn, Dorelli, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Petrukovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Hsieh and Otto (2015)) the spatial scale (thickness) of the current sheet can become smaller than the thermal proton gyroradius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In such sub-ion scale thin current sheets the proton pressure cannot be redistributed within sub-gyroradius scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' To establish the pressure balance, the intensified field-aligned electron currents form strong mag- netic field shear, contributing to a (partially) force-free current sheet configuration with j⊥ ≤ j∥ (see Schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1(b) and Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev, Angelopoulos, Liu, and Runov (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Second, a similar, partially force-free current sheet configuration has been observed in the distant (lunar orbit) magnetotail (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018) where plasma β can be as low as ∼ 1, and plasma pres- sure is not sufficient to establish the pressure balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' This second force-free current sheet with low β is also typical in Mars (Artemyev, Angelopoulos, Halekas, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' DiBraccio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2015) and Venus (Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2015) magnetotails occupied by cold planetary plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –2– manuscript submitted to JGR: Space Physics Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Schematic of current sheet configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (a) 2D configuration of the magnetic field line in the current sheet with plasma pressure P gradients (the dashed line parallel to the x-axis illustrates its projection to the xy plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The main current density flows along the y-axis, transversely to magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (b) 3D configuration of the magnetic field line in the current sheet with B2 y/8π gradients playing the role of P gradients (see discussion on the P → B2 y transi- tion in Syrovatskii, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lukin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The dashed curve illustrates the field line projection to the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The main current density flows along B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', parallel to the magnetic field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –3– (a) Z jyBz V-PA X 个jvBx jy-i~vp y jx-j (b) B Z jyBz jxBy个 X jz By jy-jinmanuscript submitted to JGR: Space Physics Between the two mechanisms for the force-free (or partially force-free) current sheet formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' sub-ion scale thickness versus low plasma β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' the first one is more interesting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' because such sub-ion current sheets with strong field-aligned electron cur- rents can be favorable to kinetically-driven magnetic field reconnection and current filamentation (Drake & Lee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zelenyi & Taktakishvili, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Investigations of such current sheet configurations, however, are quite limited, because these current sheets are rather transient (dynamical) in the Earth’s magnetotail (see discussions in Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' An alternative plasma environment to investi- gate these specific current sheets would require hot heavy ions, curved magnetic field lines, and fast electron plasma flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The best possible, accessible system is the Jovian magnetodisk, filled by sulfur and oxygen ions (Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Krupp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020a) with various charge states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Selesnick & Co- hen, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020b) and conjugate to the Jovian aroural region, a powerful source of field-aligned electron streams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017b, 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Therefore, We will use the recently available plasma and magnetic field measurements (Bagenal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017) from Juno in the Jovian magnetodisk to systematically examine sub-ion scale, force-free (or partially force-free) current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In this study, we focus on 18 events of Juno current sheet crossings during the first 30 orbits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', those with a strong magnetic shear (field-aligned currents), electron field-aligned streams, and different combinations of proton and heavy ion contributions to the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We estimate the current sheet spatial scale (thickness) and current density during its flapping motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The following of paper includes three sections: de- scription of Juno instruments and data analysis techniques in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 2, detailed analysis of 9 current sheet crossings in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3, and discussion on the results in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 2 Data analysis technique and instruments We use data from the Juno magnetometer (MAG) in 2017-2018, with 1s time resolution (Connerney, Benn, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Connerney, Adriani, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We focus on measurements at r > 25RJ radial distances in the Jovian magnetodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Figure 2 shows a typical one-day magnetic field measurements in the magnetodisk by Juno: quasi-periodic crossings of zeros of the radial magnetic field component Br = 0 (current sheet) are due to flapping oscillations of the magnetodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' For each such crossing we transform the magnetic field into local coordinate systems (Sonnerup & Cahill, 1968): Bl is the most varying magnetic field component, Bn is the less varying component, and Bm is transverse to Bl and Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We keep only those current sheets with a Bm peak at Bl = 0 and with available plasma measurements by the Jovian Auroral Distributions Experiment (JADE) instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The times of selected crossings are given in table 1, along with their radial distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Overview plots of plasma and magnetic field profiles during each current sheet crossing are provided in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In the main text, we mostly discuss six force-free current sheets, in comparison with three current sheets supported by plasma pressure gradients (non-force-free sheets), but our conclusions are supported by the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Jovian Auroral Distributions Experiment (JADE) measures (see McComas, Alexan- der, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' McComas, Szalay, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020a, 2020b) electron distributions from below ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='1 to 100 keV (at a 1 s cadence) and ions from ∼ 13 eV to ∼ 50 keV (including ion composition, at a 1 s cadence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We have averaged JADE data over the spin period (30 s) to obtain a complete pitch angle coverage from 0◦ to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' These energy ranges cover the main (thermal) plasma populations in Jupiter’s magne- todisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We use the following data products from JADE: electron pitch-angle/energy distributions averaged over time interval of the current sheet crossing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' electron omni- directional energy spectra (energy flux) Fe(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' electron pressure pe(t) and density ne(t) profiles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' electron pressure anisotropy Ae = pe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='∥/pe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='⊥ averaged over the current sheet crossing interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' omnidirectional proton and heavy ion energy spectrum (energy –4– manuscript submitted to JGR: Space Physics Br Bφ Bθ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −20 −10 0 10 20 20:0020:1520:3020:4521:0021:1521:30 B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −20 −10 0 10 20 10:30 10:40 10:50 11:00 11:10 B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −30 −20 −10 0 10 20 30 00:00 02:00 04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00 22:00 00:00 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Top panel shows one-day measurements by Juno MAG: radial Br, azimuthal Bϕ, and north-south Bθ components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Bottom panels show the expanded view of two current sheet crossings from the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' flux) Fp(t, E) and Fi(t, E), and proton and heavy ion densities np(t), ni(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Note that field view of the JADE electron detector may not cover the entire [0, 180◦] pitch-angle range at the current sheet boundaries, so we average electron measurements over each current sheet crossing interval to obtain a more complete pitch-angle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The rotation of the background magnetic field direction across the current sheet ensures a wide coverage of the pitch-angle range (see pitch-angle distributions below), which is needed for the estimate of Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In this study, we use the heavy ion (or ”i”) to denote integrated quantities of those ion populations with mass-to-charge ratio larger than five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We also estimate proton and heavy ion pressures pp(t), pi(t) as moments of the omnidirectional energy spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', we assume that thermal proton and ion speeds well exceed their bulk flow speed (this is a reasonable assumption, see Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Waldrop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Using plasma and magnetic field measurements, we estimate βe,p,i = 8πpe,p,i/B2 profiles and electron fire-hose parameter Λe ≈ βe(Ae − 1)/2 = 4π(p∥e − p⊥e)/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' To show that this parameter controls the contribution of the electron anisotropy to the current density, we illustrate the case for a simple quasi-1D current sheet with ∂Bl/∂rn = 4πjm/c, ∂Bm/∂rn = 4πjl/c, and Bn = const ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The electron current due to cross-field drifts in such current sheet is (Shkarofsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 1966) j⊥e = −ecn[E × B] B2 − c[∇p⊥e × B] B2 + cΛe 4π [B × (B∇) B] B2 (1) where E⊥ is the transverse component of the polarization electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' This equation should be supplemented by the field-aligned stress balance equation enE∥ = −∇∥p∥e + Λe 4π (B∇) B –5– manuscript submitted to JGR: Space Physics # date time radial comments distance 1 2017 doy 027 09:00-09:20 65RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 2 2017 doy 080 07:10-07:40 61RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 3 2017 doy 080 17:10-17:50 61RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 4 2017 doy 128 08:00-08:30 86RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 5 2017 doy 128 16:30-17:30 86RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 6 2017 doy 133 05:10-06:10 61RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 7 2017 doy 181 06:50-07:20 86RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 8 2017 doy 181 09:45-10:20 86RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 9 2018 doy 031 09:30-10:40 50RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 10 2018 doy 034 13:30-14:30 50RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 11 2018 doy 085 21:00-21:50 62RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' protons 12 2018 doy 086 10:15-11:15 55RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions & protons 13 2018 doy 088 01:20-02:30 38RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' protons 14 2018 doy 088 11:30-12:30 38RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' protons 15 2018 doy 088 21:30-22:30 38RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions 16 2018 doy 141 10:30-11:15 38RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions & protons 17 2018 doy 141 19:45-21:00 38RJ ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions & protons 18 2018 doy 142 20:00-20:45 26RJ non-ff CS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' heavy ions & protons Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' List of current sheet crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In the comments column, ff CS and non-ff CS stand for the force-free and non force-free current sheets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The dominant ion type (heavy ions or protons) is also indicated in the comments column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' where E∥ is the field-aligned component of the polarization electric field and ∇∥ = (B/B)∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' For force-free current sheets with ∇npe⊥ = ∇npe∥ = ∇nB = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' the current density equation can be rewritten as j⊥e = cΛe 4π � −el B2 n B2 ∇nBm + em B2 n B2 ∇nBl + en Bn B2 (Bl∇nBm − Bm∇nBl) � whereas the parallel current density can be obtained from the divergence free condition: j∥e = − 1 B (Bl∇nBm − Bm∇nBl) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' the total electron current is je = j⊥e + j∥e B B = 4πcΛe � −el � ∇nBm − Bm∇nB B � + em � ∇nBl − Bm∇nB B �� (2) = cΛe 4π (−el∇nBm + em∇nBl) = Λej where j = (c/4π)∇ × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3 Current sheet examples Figure 3 shows six typical examples of thin current sheets with an almost constant magnetic field magnitude across the sheet, |B| ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Such a constant magnetic field implies the dominant role of field-aligned currents in the current sheet configuration: if j = C · B, then j × B = 0 and there is no pressure variation across the sheet (note –6– manuscript submitted to JGR: Space Physics that typical crossings of the magnetodisk current sheet show strong variations of the plasma pressure (density) across the sheet, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Huscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Panels (a) show that |B| = const is due to peak of Bm component that compensates the drop of B2 l around the neutral plane (where Bl = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' As expected for the force-free current sheet, there are no appreciable variations in the ion fluxes (protons or heavy ions) across the sheet (see panel (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Electron fluxes may show some variations (see panel (b)), but variations of the electron thermal pressure are insufficient to cause any significant variations of the magnetic field pressure (see panel (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' note that a variation of 10−2 · cm−3keV corresponds to ≈ 2nT variation of the magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Let us explain the absence of ion pressure variations during the observed current sheet crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3, the time-scale of current sheet crossings varies from T ∼ 5 min to 30 min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' taking into account the flapping speed of ∼ ωJR tan θ, this time- scale can be converted to a spatial scale L ≈ 1000km·(r/50RJ)·(T/60s) ∈ [5, 30]·103km (here ωJ is the Jupiter rotational frequency, r is the radial distance of the current sheet, and θ ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5◦ is the tilt angle of the magnetodisk with respect to the planetary equator, see Connerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Khurana and Schwarzl (2005)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Despite that we used the upper limit for the flapping speed (see discussion and observations in Hill, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020b), this spatial scale is much smaller than the typical thicknesses of Jovian magnetodisk current sheets, L ∼ 2RJ ≈ 105km (Connerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' More importantly, this spatial scale is comparable to (or smaller than) the hot proton or heavy ion gyroradius: for the equatorial field of a typical current sheet, ∼ 5nT, ∼ 30keV protons and sulfur ions have gyroradii of ∼ 5000km and ∼ 25000km, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Thus, these current sheets are likely on sub-ion scale, within which ions cannot redistribute their pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' To establish the pressure balance in such current sheets, the field-aligned electron currents create a local Bm peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' To estimate the electron contribution to the field-aligned currents, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2) and the measured electron pitch-angle, energy distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Figure 4(b) shows that all six current sheets are characterized by field-aligned bi-directional electron streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' These streams occupy ∼ 30◦ in pitch angles around the parallel and anti-parallel (with respect to the background magnetic field) directions, in the energy range below ∼ 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Such field-aligned streams may be generated by reconnection further downtail (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Kronberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2012) or originate from the aurora acceleration region (Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2017b, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Elliott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Allegrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In the Earth’s magneto- sphere, similar field-aligned streams are observed in the near-Earth magnetotail (Hada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Walsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev, Angelopoulos, Liu, & Runov, 2017), but their energies are well below ∼ 1keV, in agreement with the capability of the Earth’s aurora acceleration (Ergun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Chaston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Watt & Rankin, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' More effective aurora acceleration in the Jupiter magnetosphere may produce ∼ 10 keV beams (Kollmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Damiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lysak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021), which are likely further expanded in the pitch-angle space by various scattering mechanisms and form the electron streams observed in the plasma sheet (see discussion in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In the presence of a large electron βe ∼ 1, such field-aligned streams create a strong pressure anisotropy with the fire-hose parameter Λe reaching one (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2) shows that for this large Λe, electrons will carry almost all the current to support Bl and Bm variations across the sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We explain the formation of force-free (partial force-free) current sheets with strong field-aligned electron currents (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 4) as a need to balance the magnetic field pressure decrease ∼ B2 l on a sub-ion scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' This explanation implies that similar electron currents should be observed in current sheets on larger scales, where they will not create Bm peaks, but rather contribute to the cross-field current density, in agreement with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (1) (see discussion in Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Figure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 shows such current sheets with a significant ion pressure contribution to the stress ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='–7– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='manuscript submitted to JGR: Space Physics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='10−3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV⋅cm-3 10−3 10−2 10−1 2017 doy 181 09:50 09:55 10:00 10:05 10:10 (d) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' cm-3 10−3 10−2 (c) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 103 104 (b) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 (a) B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −4 −2 0 2 4 6 (e) 10−3 10−2 10−1 2017 doy 128 16:30 16:40 16:50 17:00 17:10 17:20 17:30 i e p (d) 10−3 10−2 1/cm2/s/sr 105 106 (c) 103 104 1/cm2/s/sr 103 104 105 106 (b) 102 103 104 105 (e) 10−3 10−2 10−1 100 2017 doy 080 07:16 07:18 07:20 07:22 07:24 07:26 07:28 07:30 (d) 10−3 10−2 10−1 (c) 103 104 (b) 102 103 104 105 (e) p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV⋅cm-3 10−3 10−2 10−1 100 2017 doy 027 09:00 09:05 09:10 09:15 09:20 (d) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' cm-3 10−3 10−2 (c) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 103 104 (b) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 |B| Bl Bm Bn (a) −4 −2 0 2 4 (a) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 −5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 (a) B, nT −10 −5 0 5 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Six examples of force-free current sheets observed by Juno in different radial dis- tances (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (a) Magnetic field components in the local (MVA) coordinate system and the magnetic field magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (b, c) Omnidirectional spectra of electrons and dominant ion species (blue for protons and red for heavy ions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (d,e) Densities and pressures of electrons and dominant ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –8– manuscript submitted to JGR: Space Physics (d) 10−2 10−1 100 2018 doy 086 10:00 10:15 10:30 10:45 11:00 11:15 11:30 (b) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (a) 10−2 10−1 100 Bl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −10 −5 0 5 10 (d) 10−2 10−1 100 2018 doy 088 11:30 11:40 11:50 12:00 12:10 12:20 12:30 1/cm2/s/sr 103 104 105 106 107 (b) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 p e i (a) 10−1 100 (d) Λ 10−1 100 2017 doy 181 09:50 09:55 10:00 10:05 10:10 (b) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (a) β 10−2 10−1 100 101 Bl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −10 −5 0 5 10 (d) 10−2 10−1 100 2017 128 16:30 16:40 16:50 17:00 17:10 17:20 17:30 1/cm2/s/sr 103 104 105 106 (b) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (d) 10−2 10−1 100 2017 080 07:16 07:18 07:20 07:22 07:24 07:26 07:28 07:30 (b) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (c) Λ 10−2 10−1 100 2017 027 09:00 09:05 09:10 09:15 09:20 (b) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 i e p (a) 10−1 100 101 (a) 10−2 10−1 100 101 102 (a) β 10−2 10−1 100 101 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Six examples of force-free current sheets from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (a) Electron and ion betas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (b) Electron pitch-angle, energy distribution averaged over the entire event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (c) Electron fire-hose parameter and Bl field to illustrate the current sheet center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –9– manuscript submitted to JGR: Space Physics balance and strong field-aligned electron anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' There is almost no Bm peak in the current sheet center (where Bl ∼ 0, see panel (a)), whereas ion fluxes and pressures exhibit peaks (panels (c,d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Temporal scales of current sheet crossings from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 5 are about 20-30min, corresponding to a spatial scale larger than the typical proton and heavy ion gyroradius in these current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Note the two current sheets on 2017 doy 080 (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3 and 5) exhibit different characteristics: the force-free current sheet in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3 was crossed within a couple of minutes and shows no pi variations, whereas the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 5 was crossed within ∼ 15 min and shows strong pi variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In current sheets from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 5, electron pitch-angle distributions contain strong field-aligned streams with characteristics very similar to those in the force-free cur- rent sheets (compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 4(b) and 5(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' However, contrary to force-free current sheets, these field-aligned streams mostly contribute to anisotropic cross-field electron currents, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Indeed, small magnetic field pressure at the current sheet center (where Bl ∼ 0) increases βe and leads to a large electron fire-hose parameter Λe ≈ 1 even for a moderate anisotropy Ae (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 5(f,h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Comparison of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3, 4, and 5 suggests the following mechanism for the forma- tion of the force-free current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Certain external conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', electron acceler- ation in the aurora region, see Mauk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2017b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Damiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lysak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2021)) generate field-aligned electron streams bouncing within the current sheet in the Jovian magnetodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' These streams contribute to the field-aligned electron anisotropy, Ae > 1, and fire-hose parameter Λe > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In typical thick current sheets such anisotropy supports the cross-field electron currents (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (1)) and may create a thin, sub-ion scale current sheet embedded into a thick, ion scale current sheet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Mingalev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Kamaletdinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Indeed, the magnetic field profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 5 exhibit stronger gradients around Bl ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' If external drivers result in the current sheet thinning, the current sheet may reach sub-ion spatial scale where ions cannot redistribute their pressure and maintain the stress balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In this case, the electron currents form Bm peaks to balance the B2 l drop at the current sheet center, and self-consistently evolve from the cross-field currents to field-aligned currents (see Schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 4 Discussion In this study, we investigate force-free (and partially force-free) current sheets, where field-aligned electron streams support the pressure anisotropy and parallel cur- rents, leading to the formation of the Bm peak at the current sheet center, Bl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Let us discuss the difference of the stress balance in such current sheets from that in more typical thick current sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In current sheets, the 2D stress balance in the equatorial plane (balance along the radial direction) is maintained by a combination of the centrifugal force, plasma pressure force, and magnetic field line tension force (Hill & Carbary, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Cheng, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zimbardo, 1989): 1 c jϕBθ + minω2 Jr + ∇r ˆp = 0 (3) where ∇r ˆp is the radial component of the plasma pressure tensor gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In the local coordinate system, l ≈ er and m ≈ eφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Based on the vertical stress balance, 8πp = max B2 l − max B2 m, we may estimate the current density as: jm ≈ c 4π max Bl Bn ∇r max Bl ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5nA m2 · (r/RJ)−2 ≈ 12pA m2 · � r 30RJ �−2 (4) where max Bl ≈ 50nT·(r/RJ)−1 and max Bl/Bn ≈ 20 are the empirical relations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Artemyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The corresponding current sheet thickness L = c max Bl/4πjm ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5RJ · (r/30RJ)−1 should be larger than 1RJ at r > 30RJ, which is consistent with the thickness estimates for typical thick current sheets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –10– manuscript submitted to JGR: Space Physics Bl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' nT −20 −10 0 10 20 (h) Λ 10−4 10−3 10−2 10−1 100 101 2018 doy 034 13:30 13:40 13:50 14:00 14:10 14:20 (h) Λ 10−3 10−2 10−1 100 2017 doy 133 05:20 05:30 05:40 05:50 06:00 06:10 (h) Λ 10−2 10−1 100 2017 doy 080 17:10 17:20 17:30 17:40 17:50 (g) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (f) 10−2 10−1 100 101 102 2017 doy 133 05:20 05:30 05:40 05:50 06:00 06:10 1/cm2/s/sr 103 104 105 106 107 (g) 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (f) 10−2 10−1 100 101 102 2018 doy 034 13:30 13:40 13:50 14:00 14:10 14:20 (g) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 pitch-angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ○ 0 30 60 90 120 150 180 (f) β 10−2 10−1 100 101 102 2017 doy 080 17:10 17:20 17:30 17:40 17:50 (e) 10−3 10−2 10−1 100 i e (d) 10−3 10−2 10−1 1/cm2/s/sr 105 106 (c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ions 103 104 1/cm2/s/sr 103 104 105 106 107 (b) 102 103 104 105 (e) 10−3 10−2 10−1 100 (d) 10−4 10−3 10−2 10−1 (c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ions 103 104 (b) 102 103 104 105 (e) p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV⋅cm-3 10−3 10−2 10−1 100 (d) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' cm-3 10−3 10−2 (c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ions energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 103 104 (b) energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' eV 102 103 104 105 |B| Bl Bm Bn (a) −20 −10 0 10 20 (a) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 −5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5 (a) B, nT −10 −5 0 5 10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Three examples of current sheets with large field-aligned electron currents and plasma pressure variations observed by Juno in different radial distances (see table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (a) Mag- netic field components in the local (MVA) coordinate system and the magnetic field magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (b, c) Omnidirectional spectra of electrons and dominant ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (d, e) Densities and pres- sures of electrons and dominant ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (f) Electron and ion beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (g) Electron pitch-angle and energy distribution averaged over the entire event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (h) Electron fire-hose parameter and Bl field to show the current sheet center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –11– manuscript submitted to JGR: Space Physics Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Such current sheets will be crossed during an interval of ∆t = L/ωjr tan θ ≈ 3hours · (r/30RJ)−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' for much thinner current sheets as in our dataset, the traversal timescale will be less than 10 min (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' The stress balance in such thin current sheets cannot be maintained by centrifugal force and radial gradient of the plasma pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Instead, the electron pressure anisotropy contributes to the stress balance (Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1972): ∇r ˆp = ∇rp⊥ + ∇θ p∥e − p⊥e B2 BrBθ ≈ ∇θ �Λe 4π BrBθ � (5) This equation shows that the thin current sheet configuration resembles a classical rotational discontinuity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' with no variations of the Alfven speed because of the pressure anisotropy (Hudson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' 1970): ∆vA = � B2 4πnmi (1 − Λe) ∼ 0 (6) This condition allows for a balance of the 1D current sheet (with thickness L much smaller than the spatial scale of the radial gradient of the plasma density) without fast plasma flows typical for rotational discontinuities in anisotropic plasma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' where cross-sheet change of the plasma flow velocity equals to ∆vA (see discussions of the anisotropy contribution to the force-free current sheet configurations in Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev, Angelopoulos, Vasko, & Zelenyi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Formation of such 1D current sheets around the fire-hose marginally stability threshold has been predicted theoreti- cally (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Francfort & Pellat, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Cowley, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Cowley & Pellat, 1979), but never have been detected under quiet geomagnetic conditions in the Earth’s magnetotail (see discussion in Sitnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev & Zelenyi, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Observations of these current sheets in the Jovian magnetotail confirm the theoretical predictions, which can further lead to improved current sheet models (see discussion on development of the next generation of current sheet models in Sitnov & Merkin, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' It is worth to note that these current sheets are electromagnetically disconnected from the Jovian ionosphere, because the local Alfven speed is zero, vA = (B/√4πnmi)· √1 − Λe = 0, and there are no field-aligned perturbations propagating from the current sheet to the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Such local destruction of magnetosphere-ionosphere coupling is an interesting phenomenon that we do not observe in the Earth’s magnetotail, where Λe is much more moderate (Artemyev, Angelopoulos, Vasko, Petrukovich, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Theoretical investigations of these force-free current sheets in the Jovian mag- netodisk is a real challenge for plasma kinetics, because these current sheets share properties of 2D plasma equilibria (with the tension force ∼ (4π/c) · jmBn) and prop- erties of rotational discontinuities (with j = C ·B and Bn ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' All existing 2D kinetic current sheet models operate with the plasma pressure gradients, ∇p = c−1j × B, and do not include field-aligned currents (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Yoon & Lui, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Zelenyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Sitnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Sitnov & Merkin, 2016, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Existing models of force-free current sheets mostly assume 1D tangential discontinuities with Bn = 0 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Harrison & Neukirch, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Panov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Neukirch, Vasko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Neukirch, Wilson, & Allanson, 2020, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Construction of the kinetic model for 1D rotational discontinuities requires assumptions of an additional system symmetry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Sonnerup & Su, 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Artemyev, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Mingalev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Vasko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2014), whereas kinetic models of 2D rotational discontinuities have not yet been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Although fluid mod- els of 2D rotational discontinuities (with j = C · B, jnBm = jmBn, and Bn ̸= 0) can be constructed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Cowley, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Hilmer & Voigt, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Hau & Voigt, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lukin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 2018), their kinetic realization has not been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Therefore, further theoretical investigations are needed to explain Juno observations in the Jovian magnetodisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –12– manuscript submitted to JGR: Space Physics 5 Conclusions We have investigated thin (thickness is smaller than or about the hot ion gyro- radius) current sheets observed by Juno in the Jovian magnetodisk and characterized by strong field-aligned electron streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' We have demonstrated that electron streams support the strong field-aligned anisotropy, which may increase the fire-hose parameter up to the instability threshold, Λe = 4π(pe∥ − pe⊥)/B2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' In such thin, marginally stable current sheets, almost all current is field-aligned, and the current sheet configu- ration is force-free, with j×B ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' This is a new type of strongly anisotropic, force-free current sheets, which has not been reported in the quiet-time Earth’s magnetopshere or solar wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Further numerical investigations of such current sheet formation and dynamics will reveal their potential role in the particle acceleration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', via magnetic reconnection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Acknowledgments This work is supported by Grants 80NSSC19K1593 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=') under Juno Participat- ing Scientist program, and by subcontract 699046X to UCLA under prime contract ZZM06AA75C (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Open Research Data processing was done using SPEDAS V4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='1, see Angelopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' All the adopted data are available in the archive https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='5281/zenodo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='7470240 References Allegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', Mauk, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (2020, January).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' –13– manuscript submitted to JGR: Space Physics Ion Nongyrotropy in Solar Wind Discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 889(1), L23.' metadata={'source': 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Magnetodisk as Observed by Juno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Journal of Geophysical Research (Space Physics), 125(8), e27957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='1029/2020JA027957 Zimbardo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' (1989, July).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' A self-consistent picture of Jupiter’s nightside magneto- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=', 94, 8707-8719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} +page_content='1029/JA094iA07p08707 –20–' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQfNAZ6/content/2301.03731v1.pdf'} diff --git a/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/2301.13238v1.pdf.txt b/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/2301.13238v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..09f0e4e9e566c49be4ca52db36278658efce2713 --- /dev/null +++ b/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/2301.13238v1.pdf.txt @@ -0,0 +1,2751 @@ +Prepared for submission to JHEP +IFIC/23-03 +Asymmetries in Extended Dark Sectors: +A Cogenesis Scenario +Juan Herrero-García , Giacomo Landini and Drona Vatsyayan +Departamento de Física Teórica, Universidad de Valencia and IFIC, Universidad de Valencia- +CSIC, C/ Catedrático José Beltrán, 2 | E-46980 Paterna, Spain +E-mail: juan.herrero@ific.uv.es, giacomo.landini@ific.uv.es, +drona.vatsyayan@ific.uv.es +Abstract: The observed dark matter relic abundance may be explained by different mech- +anisms, such as thermal freeze-out/freeze-in, with one or more symmetric/asymmetric com- +ponents. In this work we investigate the role played by asymmetries in determining the +yield and nature of dark matter in scenarios with more than one dark matter particle. In +particular, we show that the energy density of a particle may come from an asymmetry, +even if the particle is asymptotically symmetric by nature. To illustrate the different effects +of asymmetries we adopt a model with two dark matter components. We embed it in a co- +genesis scenario that is also able to reproduce neutrino masses and the baryon asymmetry. +The framework predicts a monochromatic neutrino line for some of the scenarios. +arXiv:2301.13238v1 [hep-ph] 30 Jan 2023 + +Contents +1 +Introduction +2 +2 +General framework +3 +3 +A model for neutrino masses, the baryon asymmetry, and dark matter +7 +3.1 +The scalar sector and spontaneous symmetry breaking +8 +3.2 +The gauge sector +9 +3.3 +The fermionic sector +10 +4 +Dark matter components +11 +4.1 +Asymmetric dark matter via cogenesis +12 +4.2 +Contribution of ψ +13 +4.2.1 +Production of ψ from χ decays +13 +4.2.2 +Constraints +16 +4.3 +Contribution of S +17 +4.3.1 +Production of S from freeze-out +17 +4.3.2 +Production of S from χ decays +18 +4.3.3 +Constraints +19 +5 +Dark matter relic abundance +20 +5.1 +Scenario 1: ψ-LD-A + S-FO-A +22 +5.2 +Scenario 2: ψ-LD-A + S-FOLD-PA +22 +5.3 +Mixed Scenario 1-2: ψ-LD-A + S-FOLD-A +23 +5.4 +Scenario 3: ψ-FILD-PA + S-FO-A +24 +5.5 +Scenario 4: ψ-FILD-PA + S-FOLD-PA +24 +5.6 +Mixed Scenario 3-4: ψ-FILD-PA + S-FOLD-A +24 +5.7 +Scenario 5: ψ-FI-S + S-FO-A +25 +5.8 +Scenario 6: S-FOLD-S +25 +5.9 +Summary of the scenarios +26 +6 +Phenomenological signals +28 +7 +A low-energy variant: The inverse seesaw +31 +8 +Conclusions +33 +A Decays of ϕ to SM particles +34 +B Constraints on massless A′ +µ +35 +C Implications of fermion mixing +35 +– 1 – + +D Contributions to operator O6 +36 +1 +Introduction +The nature of Dark Matter (DM), that makes up roughly a quarter of the energy density of +our universe, along with the origin of neutrino masses and the baryon asymmetry (BAU), +are among the most important open problems that the Standard Model (SM) fails to ex- +plain, with overwhelming experimental evidence. Several extensions of the SM have been +proposed, where either a single particle or several stable particles, i.e., multi-component DM +[1–9], make up the observed DM relic abundance, ΩDMh2 ∼ 0.1 [10]. The most popular +mechanisms to reproduce this relic abundance include thermal freeze-out (FO) of weakly in- +teracting massive particles (WIMPs) [11–16] and freeze-in (FI) of feebly interacting massive +particles (FIMPs) [17–19]. +WIMPs are initially in thermal equilibrium with the SM and undergo annihilations +until they freeze-out when the annihilation rate drops below the Hubble expansion rate; +therefore, the DM abundance is inversely proportional to the annihilation rate. On the +other hand, FIMPs have a negligible initial abundance and are produced mainly by tiny +interactions with particles in the thermal bath so that they never thermalise, and they +freeze-in once the mother particle decouples from the bath; thus, the DM abundance is +directly proportional to the production rate. +Depending on the interactions, a further +contribution may come from late decays (LD) of the mother particle, the size of which is +model-dependent. +Most of these models assume that the new states are symmetric in nature, i.e., the +abundance of the DM particle is the same as that of the antiparticle. However, there exist +a wide variety of models that propose that the DM abundance is rather set by an initial +asymmetry in the dark sector, in analogy to the visible sector (where ηB = 0.88 × 10−11), +motivated by the closeness of baryonic and DM energy densities, ρDM ∼ 5ρB [20]. The +asymmetry can be first generated in the visible sector and then transferred to dark sector +(or vice versa) [21, 22], or an asymmetry can be generated simultaneously in both the +sectors (cogenesis). Such asymmetric dark matter (ADM) models (see Refs. [23, 24] for a +review) aim to explain the observed baryon asymmetry and DM abundance in a common +framework. +An intermediate scenario between the two extremes involves the asymmetric freeze-out +of a species, where the DM particle and its antiparticle freeze-out with different number +densities, depending on the initial asymmetry, and the dark matter is partially asymmetric +[25]. Hence, a further distinction can be made regarding the nature of DM, whether it is +symmetric, asymmetric or partially asymmetric. Therefore, the presence of an asymmetry +can have significant implications for the mechanisms discussed above in reproducing the +DM abundance. +The possibility that asymmetric single-component WIMP DM is produced in a cogen- +esis scenario has been studied in Ref. [26]. Another model has been presented in Ref. [27], +– 2 – + +with the possibility to restore the symmetric nature of DM through late decays of an extra +particle. Finally, in Ref. [28] a model of symmetric multi-component DM, which combines +freeze-out and freeze-in, accompanied by the generation of the baryon asymmetry, has been +proposed. +In this work, we aim to generalize this picture, starting from the concrete cogenesis +scenario realized in Ref. [27], by considering an extended dark sector in which we can re- +alize multicomponent DM, so that one DM component, or even both, can be asymmetric. +Furthermore, we combine freeze-out and freeze-in production, including the possibility of +asymmetric freeze-in [29–32]. To this end, we propose a model in which the dark sector is +connected to the visible sector via a mediator particle. The asymmetries in both sectors +are generated via cogenesis, which yields a relation between neutrino masses, the baryon +asymmetry and the DM relic abundance. We then investigate the role of the dark sec- +tor asymmetry in determining the relic abundance of one/several particles. For example, +depending on the model parameters, the DM may either be single-component or multi- +component, with either all symmetric or asymmetric components or a mixture of both. +Similarly, the production mechanisms may be freeze-out, freeze-in or some via freeze-out +and the others via freeze-in. +In particular, we explore the possibility of asymmetric freeze-in, which has been over- +looked in the literature. Unlike the case of asymmetric freeze-out, it is not so straightforward +and requires the presence of a richer dark sector. Moreover, we show that in certain sce- +narios the particle abundance may be set by an asymmetry even if its nature is symmetric. +We find that some of them predict the existence of an observable neutrino line. +The paper is structured as follows. In Section 2, we elaborate on the framework of +generating an asymmetry and the possibility of asymmetric freeze-in. We discuss the com- +plete model, which also generates neutrino masses and the baryon asymmetry, in Section 3. +The different DM candidates are discussed in Section 4. The contribution to the DM relic +abundance is studied in Section 5. In section 6, we discuss the main phenomenological +signatures, with special emphasis on the prediction of a monochromatic neutrino line. In +Section 7, we discuss a low scale variant of the model (an inverse seesaw). Finally, we give +our conclusions in Section 8. We show further details in some appendices. +2 +General framework +In order to generate a dark sector asymmetry, we focus on the cogenesis of both DM and +baryon asymmetries and adopt the two-sector thermal leptogenesis mechanism of Ref. [27]. +In this case, the asymmetries are produced from out-of-equilibrium CP-violating decays of +right handed neutrinos (RHNs), which are well-motivated to reproduce neutrino masses. +This requires the leptons and some of the dark sector particles to be charged under a +lepton symmetry that is broken by Majorana masses of the RHNs, thus satisfying all the +conditions for the dynamical generation of an asymmetry [33]. The RHNs are initially in +thermal equilibrium, and once the temperature drops below the mass of the lightest RHN, +MN1, the washout and other interactions leading to transfer of asymmetries between the two +sectors become inefficient. Subsequently, the asymmetries get frozen in the two sectors. The +– 3 – + +leptonic asymmetry is partially converted into the baryonic one via sphalaeron processes. +The dark asymmetry (ηi ≡ Y + +i +− Y − +i , where we introduce the yield as the number density +upon entropy density, Yi = ni/s) is carried by a dark fermion (we denote it by χ) and +once its symmetric component gets annihilated away, the asymmetric one sets the relic +abundance, ΩDM ∝ ηχmχ. This framework, therefore, connects neutrino mass generation +with the baryon asymmetry and the DM relic abundance. +See Refs. [34–41] for other +extensions of seesaw framework that address the three issues under the same umbrella. +In our work, we go a step further and enlarge the dark sector so that the different +possibilities discussed in the introduction are feasible. For this purpose, the particle χ in +our set-up is not the dark matter but rather decays to another stable fermion (say ψ), which +may constitute all or part of the DM abundance. The schematic framework is shown in +Fig. 1. Therefore, the asymmetry in χ can be transferred to ψ via its decays. Indeed, the +N +L +H +χ +S +ψ +φ +yν +yS +yφ +Figure 1. Schematic framework of cogenesis and DM production in the model at T < MN1 via the +indicated Yukawa interactions yi. Here, L and H are the SM lepton and Higgs doublet, respectively, +whereas S and φ are complex scalars belonging to the dark sector. +set-up offers a richer phenomenology as it is now possible to accommodate multi-component +DM (in this case, it could be ψ and S), as well as different dynamics thanks to the presence +and size of the different interactions. In principle, we can have four cases of equilibration +between different sectors, as illustrated also in Ref. [42]. In Fig. 2, we show the four cases +of entropy transfer between the various sectors: the SM + NR (green), χ, S, φ (orange) and +ψ (gray).1 +The first case (EE) involves equilibration among all sectors. It is the multi-component +freeze-out scenario that has been widely considered (see for instance Ref. [1]), and we will +not consider it here in the following. Cases 2 and 3, FF and FE, respectively, correspond +to scenarios where χ is not in equilibrium with the SM + NR sector, which implies that +yS ≪ 10−7. In the considered framework where the asymmetries are produced from the +decays of heavy right-handed neutrinos, such a small Yukawa coupling would then be unable +to generate a dark asymmetry comparable to the visible one, and therefore DM would have +to be symmetric. Therefore, we are interested in the last case (EF), where: +i) the orange sector is in equilibrium with the SM + NR, +1In the scenarios considered below, φ, even if it comes from χ decays (see Fig. 1), is included in the +middle blob. +– 4 – + +EE: +χ, S +φ +SM ++NR +ψ +Equilibrium +Equilibrium +FF: +χ, S +φ +SM ++NR +ψ +Freeze-in +Freeze-in +FE: +χ, S +φ +SM ++NR +ψ +Equilibrium +Freeze-in +EF: +χ, S +φ +SM ++NR +ψ +Equilibrium +Freeze-in +Figure 2. The four scenarios for entropy transfer between the SM sector and the dark sectors +formed by χ, S and ψ, φ. Similar figure in Ref. [42]. +ii) the dark sector asymmetry may be generated via co-genesis, of size comparable to the +baryonic one, and +iii) the ψ sector is not in equilibrium and is produced via freeze-in. +Whether the abundance produced by freeze-in is asymmetric depends on the value +of Yukawa yφ, because the ψ population can be symmetric even if the mother particle χ +carries an asymmetry. This can be understood as follows. Let us define xi ≡ mi/T, for a +species of mass mi, where T is the temperature. The asymmetry freezes out at xi ∼ 20, +whereas the freeze-in from early decays takes place around the mass of the mother particle, +i.e., xi ∼ 1, when both the mother particle and its antiparticle are in equilibrium. So, if +the production from early decays is greater than the asymmetric yield after freeze-out, i.e., +Yψ = Y ¯ψ > Yχ ≈ ηD, then the daughter particle will be symmetric in nature, because the +production from late-decays (that take place at xi ≫ 1) will be sub-dominant and negligible. +However, if the production from early decays is smaller than the asymmetry, Yψ < ηD, then +the late decays that are active much later after the asymmetry has frozen-in will produce +more ψ than ¯ψ, hence generating an asymmetry in ψ. Realising this last scenario therefore +yields an example of asymmetric freeze-in, which up to our knowledge has not been studied +in the literature2. +As we will see, in certain scenarios, the late decays of an asymmetric particle may +populate the symmetric component of a species. Hence, the late decays play an important +role in determining the final nature of a species. +In order to distinguish between the +scenarios, we use the notation of Ref. [25] to define the asymmetric ratio for a particle +species as +ri ≡ Y − +i /Y + +i +with +0 < ri ≤ 1 , +(2.1) +where +(−) denotes the particle (antiparticle). Here, the upper (lower) limit in r signi- +fies that the species is completely symmetric (asymmetric) and ΩDM ∝ Y + +i ++ Y − +i . The +2In Ref. [43], the contributions from early and late decays have been compared for a symmetric DM +model. +– 5 – + +asymptotic asymmetric ratio of a species i with mass mi can be written as [25] +r(∞) +i +≃ exp +� +− +� πg∗ +45xf +MPl ⟨σv⟩i ηD mi +� +, +(2.2) +where ⟨σv⟩i is the thermally-averaged annihilation cross section, MPl ≃ 1.2×1019 GeV, mp +is the mass of the proton and xf ≡ mi/T∗ ∼ 20 is the mass over freeze-out temperature +ratio. In the following, we use r(∞) +i +≡ ri to alleviate notation. As shown in Ref. [1], different +regimes appear: +• For ri < 10−2, the behaviour of DM is highly asymmetric (A). +• In the range, 10−2 < ri < 0.9, DM behaves as partially asymmetric (PA). +• For ri > 0.9, DM is highly symmetric (S). +In the rest of the paper we adopt these ranges to define the nature of DM. Further classi- +fication can be made on the basis of the dominant production mechanism that determines +the asymptotic nature of the dark matter, be it freeze-out (FO), early decays from freeze-in +(FI), or late decays (LD). This opens up a plethora of possibilities, as illustrated in Fig. 3. +DM +FO +FI +LD +S +PA +A +S +PA +A +A +PA +S +Figure 3. Possible production and final nature of dark matter components in a model. +Scenario +Symmetric +Partially Asymmetric +Asymmetric +(r > 0.9) +(10−2 < r < 0.9) +(r < 10−2) +Freeze-out +FO-S +FO-PA +FO-A +Freeze-in +FI-S +FI-PA +FI-A +Late decays +LD-S +LD-PA +LD-A +Table 1. +Classification schemes corresponding to the behaviour and dominant production mech- +anism of DM in a model. +We characterise the multiple combinations for the nature of one-component DM as per +the name outlined in Table. 1. When multiple components are present, we prefix the name +of the component to the scheme, for example, the scenario X-FI-S+Y -FO-A corresponds to +the case of two DM components, X and Y , where X is symmetric and produced via freeze- +in whereas Y is asymmetric and freezes out. In this work, we aim to focus exclusively +– 6 – + +on scenarios where the asymmetry is directly involved in reproducing the relic abundance. +For this goal, we propose a complete model that displays the different roles played by the +asymmetries. +3 +A model for neutrino masses, the baryon asymmetry, and dark matter +Our initial hypothesis for the construction of a model that explains dark matter, neutrino +masses and the baryon asymmetry in a common framework is that the symmetry baryon +minus lepton number, B−L (under which all SM quarks have charge 1/3 and all SM leptons +have charge -1), plays a key role. This is one of the best-motivated symmetries beyond the +SM: it is accidental and anomaly-free in the SM, and when gauged it requires the presence +of three sterile neutrinos (which in turn naturally generate active neutrino masses) and is +easily embedded in GUTs. When considering a complex asymmetric dark sector, however, +this symmetry is not enough, and we require extra U(1)s to forbid Majorana masses and +some interaction terms, as well as to annihilate the symmetric components. +Therefore, we augment the SM gauge group by 3 new gauge U(1) symmetries: U(1)B−L +, and the dark product U(1)D ⊗ U(1)X. We add three right-handed neutrinos NR (RHNs) +to cancel the gauge anomalies of the first one. We also extend the particle content by the +addition of two Dirac dark fermions, ψ0 and χ0, and 3 scalars σ, S and φ. The gauge bosons +associated with the three new groups are Z0 +B−L, Z0 +D, A +′0, with gauge couplings gB−L, gD and +gX, respectively. The quantum numbers of the new fields are summarised in Table 2. Note +that all the new fields are SM singlets. The new part of the Lagrangian of the model can +thus be written as +Lnew = L0 +χψ + L0 +kin + Lint − V (σ, S, φ, H) , +(3.1) +where L0 +kin includes the kinetic terms of the gauge bosons and the scalars, V (σ, S, φ, H) is +the most general scalar potential that one can write given the symmetries of the model, +where H is the SM Higgs doublet field, and +L0 +χψ = ¯χ0(i /D − m0 +χ)χ0 + ¯ψ0(i /D − m0 +ψ)ψ0, +Lint = −yαi +ν ¯Lα ˜HN i +R − yij +σ σNic +R Nj +R − yi +SS ¯Ni +Rχ0 − yφφ ¯ψ0χ0 + H.c. . +(3.2) +Here m0 +χ,ψ are bare mass terms of the dark fermions, while the indices α = e, µ, τ and +i = 1, 2, 3 run over the generations of leptons and right handed neutrinos, respectively. We +use ˜H = iσ2H∗. Note that yν is a 3 × 3 general complex matrix, yσ is a 3 × 3 complex +symmetric matrix, yS is 3-component vector and yφ is a complex number. However, several +phases are unphysical. Four phases of yσ may be removed by rephasing σ and Ni +R. Two +phases of yS may be removed by rephasing S and χ0. yφ can be taken real by rephasing φ +or ψ. +In accordance with the framework discussed above, we take +yφ ≪ , 1 +gX ≪ 1 , +(3.3) +so that ψ0 cannot thermalise with the SM bath. We further assume a vanishing initial +abundance nψ = 0, consistent with an inflationary epoch. On the other hand, the particles +– 7 – + +Field +Spin +U(1)B−L +U(1)D +U(1)X +Ni +R +1/2 +-1 +0 +0 +σ +0 ++2 +0 +0 +χ0 +1/2 +-1 +1 +0 +ψ0 +1/2 +0 +0 ++1 +S +0 +0 +-1 +0 +φ +0 ++1 +-1 ++1 +Table 2. Particle content of the model and their respective charge assignments under Lorentz and +the U(1) groups. The first two states correspond to the sterile neutrino sector, and the last four to +the dark sector. +χ0, σ, S, φ and Ni +R reach thermal equilibrium with the SM thermal bath through sizeable +new gauge (gB−L and gD) and scalar interactions. +Note that the Lagrangian also contains the kinetic mixing between the U(1) gauge +factors. An unavoidable contribution to kinetic mixing among all U(1)s arises at one-loop +level with φ running in the loop. Also, χ contributes in the case of U(1)B−L − U(1)D +mixing. +Hence, the kinetic mixing −(κ/2)Zµν +B−LZD,µν is naturally of the order of κ ≳ +gDgB−L/(16π2) ∼ 10−3 gB−L gD. The kinetic mixing of U(1)X with the other U(1) factors +is ≳ gXgi/(16π2). Since gX ≪ 1 by assumption, this contribution can be safely ignored. +Finally, a kinetic mixing among ZB−L and the SM U(1)Y gauge boson is generated through +a loop of SM quarks and leptons, of the order ≳ gY gB−L/(16π2) � +i=q,l Yi(B − L)i. Notice +that, given the conservation of U(1)em, the photon does not couple to the B − L current. +In any case, the bounds on the kinetic mixing are not relevant for the range of parameters +that we consider in the paper. +Finally, let us remark that the necessary ingredients for the model to work could also +be achieved with a global U(1)B−L [16, 44], explicitly violated by right-handed neutrino +masses, since: +i) χ is Dirac in nature because of the gauge U(1)D, +ii) it has sizeable interactions with the SM (with NR, yS) to thermalise, and +iii) it can undergo efficient annihilations due to the U(1)D gauge group. +However, such scenario is not as theoretically appealing as the gauged B − L version that +we consider, which demands the existence of 3 right-handed neutrinos. +In the following subsections, we discuss the scalar, gauge, fermionic and dark sectors +in detail. +3.1 +The scalar sector and spontaneous symmetry breaking +The details of the full scalar potential V (σ, S, φ, H) are quite involved as one can write +quadratic, quartic and mixed quartic terms for each combination of the scalar fields σ, S, φ +and the SM Higgs doublet H. However, without entering into the details of the scalar +potential, we can safely assume that there is a region of the parameter space in which σ +– 8 – + +takes a large vev, ⟨σ⟩ = vB−L ≳ 1011 GeV, which breaks the U(1)B−L symmetry by 2 units +and generates Majorana masses for the sterile neutrinos as well as a mass for the U(1)B−L +gauge boson, ZB−L. The value vB−L ≳ 1011 GeV is chosen so that the lightest RHN mass +(MN1) safely obeys the equivalent Davidson-Ibarra lower bound [45] to achieve thermal +Leptogenesis in the model [27], but any larger value will not change the following analysis +and conclusions. Therefore, an asymmetry can be generated once the inverse decays of N1 +go out of equilibrium [46, 47]. Note that we take MZB−L > MN1 and mσ > MN1, so that +the heavy ZB−L gauge boson and the radial component of σ are therefore naturally very +heavy and decay fast into quark and leptons. +The other scalar, φ, takes a much smaller vev than that of σ, i.e., ⟨φ⟩ = vφ ≪ vB−L, +which breaks U(1)D ⊗ U(1)X → U(1)X+D. After symmetry breaking, we can write φ(x) = +vφ+ϕ(x)/ +√ +2. The overall symmetry breaking pattern of the model can then be represented +as +U(1)B−L ⊗ U(1)D ⊗ U(1)X +⟨σ⟩ +−→ U(1)D ⊗ U(1)X +⟨φ⟩ +−→ U(1)X+D . +(3.4) +The only fields that are charged under the unbroken U(1)X+D symmetry are the fermions +χ0, ψ0 with charge +1 and the scalar S with charge -1. We discuss in Section 4 the conse- +quences of this for the DM stability. +A phenomenologically relevant parameter is the mixing between the SM Higgs boson +h (coming from H = (vEW + h)/ +√ +2) and the scalar ϕ, which can be characterised by the +mixing angle θ. This is directly related to the mixed quartic term λHφφ†φH†H in the scalar +potential. In the rest of the paper, we assume that this mixing angle is small (sin θ ≃ θ ≪ 1) +so we can safely trade h and ϕ for the mass eigenstates.3 Finally, the scalar S does not +obtain a vev, ⟨S⟩ = 0, and may therefore be a DM candidate in the model because of its +charge under U(1)X+D (see discussion at the beginning of Section 4). +We are interested in the regime +MN3, MN2 ≫ MN1 ≫ m0 +χ ≫ m0 +ψ, mS > mφ , +(3.5) +so that the decay channels shown in Fig. 1 are kinematically open. We consider values of +m0 +ψ and mS of similar order of magnitude, in the GeV ballpark, because we are interested +in scenarios where both may contribute significantly to the DM relic abundance. +3.2 +The gauge sector +The symmetry breaking pattern outlined in Eq. 3.4 leads to one massless (two massive) +gauge boson(s), which correspond to the unbroken (broken) generator(s). The masses are +given by +� +� +� +� +� +� +� +m2 +A′ = 0 , +m2 +ZD = 2g2 +Dv2 +φ +� +1 + O +� +g2 +X, (vφ/vB−L)2 , κ2�� +, +m2 +ZB−L = 8g2 +B−Lv2 +B−L +� +1 + O +� +g2 +X, (vφ/vB−L)2 , κ2�� +, +(3.6) +3In Appendix A, we briefly discuss the mixing when studying the decays of ϕ to SM particles. We show +that even a very small mixing angle allows efficient decays. +– 9 – + +where A′, ZD, ZB−L are the mass eigenstates. Since gX ≪ 1, κ ≪ 1 and vφ ≪ vB−L, their +contribution to the masses can be neglected in the following analysis. +The tiny values of the parameters also suppress the mixing between the gauge bosons so +that the mass eigenstates mostly coincide with the original eigenstates. The mixing among +them at the leading order in the small expansion parameters (omitting Lorentz indices) can +be expressed as +� +� +� +� +� +� +� +A +′0 ≃ A +′ − gX/gD ZD , +Z0 +D ≃ ZD + gX/gD A +′ − +� +(gD/4gB−L)(v2 +φ/v2 +B−L) + κ +� +ZB−L , +Z0 +B−L ≃ ZB−L + (gD/4gB−L)(v2 +φ/v2 +B−L)(1 + κgB−L/gD)Z +′ +D − (gX/4gB−L)(v2 +φ/v2 +B−L)A′ . +The massless A′ +µ is decoupled from all the other fields (as gX ≪ 1) and does not play any +role in the following discussion (see Appendix B for a discussion on the bounds on a massless +dark gauge boson). The mass/kinetic mixing among ZD and ZB−L induces an interaction +of the type Zµ +DJB−L +µ +, where JB−L +µ +is the B − L current,4 which may lead to decays of ZD +into SM quarks and leptons. However, the decay width is suppressed by (vφ/vB−L)4 and it +is subleading with respect to other decay channels: +• If mZD > 2mχ, the gauge boson decays into ¯χχ or S†S at tree level, or into ϕϕ at +one-loop level through a loop of S. This last process depends on the coupling between +ϕ and S. As we will show in Appendix A, the particles ϕ must have a fast decay to +SM fermions, so this gives a lower bound on their mass, and therefore on the mass of +ZD. +• If 2mS < mZD < 2mχ, the gauge boson decays mainly as ZD → S†S. The decays +have width ∼ O(g2 +DmZD/100) and are very fast in the relevant temperature regime +(T < vφ).5 +• If 2mS > mZD > 2mϕ, the gauge boson can only decay into 2ϕ, see the first point. +In the rest of the paper we focus on the second scenario, namely 2mS < mZD < 2mχ. +3.3 +The fermionic sector +We first discuss the generation of tiny neutrino masses via the type-I see-saw mechanism +[49–54]. The breaking of U(1)B−L gives masses to the heavy RHNs, MNi ∼ yi +σ vB−L ≳ 1011 +GeV. In the following we use MN1 ≲ vB−L, i.e., we take y1 +σ ≲ 1. The masses for the active +neutrinos are given by the seesaw expression, +mν = −mD M−1 +N mT +D , +(3.7) +with mD = yνvEW/ +√ +2. For the dark fermions, once φ obtains a vev, a mixing is induced +between χ0 and ψ0 due to the Yukawa coupling yφ, see Eq. (3.2). We define the fermion +mixing parameter as +ϵf ≡ +yφvφ +m0χ − m0 +ψ +, +(3.8) +4Note that in the absence of the mass mixing there would not be such an interaction even for κ ̸= 0 [48]. +5We ignore the possibility mZD ≳ 2mS in which the phase space of the decay closes. +– 10 – + +where yφ ≪ 1 and typically vφ < m0 +χ. Thus, the mixing parameter is highly suppressed, +ϵf ≪ 1 (we remind that we are assuming hierarchical masses, i.e., m0 +χ ≫ m0 +ψ). Diagonalising +the fermionic sector at leading order in ϵf leads to the following masses for the the mass- +eigenstates χ and ψ, +� +mψ = m0 +ψ − ϵ2 +f(m0 +χ − m0 +ψ) + O(ϵ3 +f) , +mχ = m0 +χ + ϵ2 +f(m0 +χ − m0 +ψ) + O(ϵ3 +f) . +(3.9) +The fields in the mass basis are related to the original fields as +� +ψ = (1 − ϵ2 +f/2) ψ0 − ϵf χ0 + O(ϵ3 +f) , +χ = (1 − ϵ2 +f/2) χ0 + ϵf ψ0 + O(ϵ3 +f) . +(3.10) +As the mixing is very suppressed, ϵf ≪ 1, in the following we drop the subscripts 0, trading +the original masses/fields for the physical ones. +Finally, let us mention that no Majorana masses for χ or ψ are generated due to the +preserved gauge U(1)X+D, due to the fact that S does not take a vev. Higher dimensional +operators may be written at dimension 6 and 8 for χ and ψ, respectively, +Oχ = χcχσSS , +(3.11) +Oψ = ψcψσSSφ†φ† . +(3.12) +However, as S does not take a vev, these operators do not generate Majorana masses. +4 +Dark matter components +Recall that the only fields that are charged under the remnant U(1)X+D symmetry are χ, ψ +(with charge +1) and S (with charge −1). As mχ ≫ mψ, mS, only ψ, S may be stable. The +X + D charge is preserved in decays of the type ψ → S† + P or the opposite S → ¯ψ + P, +where P is some uncharged state. Therefore, in principle the lightest among ψ and S is +stable and would be the only DM candidate. +However, the decay ψ → S† + P (or the opposite) is suppressed by the masses of the +right-handed neutrino N1 and χ in the propagators. At low energies, E ≪ mχ ≪ MN1, +one can integrate out both the particles and study the decay in terms of higher-dimensional +operators. As we analyse below in Section 6, P corresponds to active neutrinos in the +model, i.e., P = ν, yielding a monochromatic neutrino line from ψ decays [55–59]. In that +section we show that in a broad region of the parameter space, the decays are suppressed +on cosmological timescales and obey current limits, and therefore both particles (ψ and S) +contribute significantly to the DM relic abundance. The scenario in which the decay is fast +enough and there is only 1 DM candidate has been studied extensively in the literature, so +we will not discuss it here. Hence, we focus on the two-DM scenario. The conservation at +low energies of the U(1)X+D symmetry yields the constraint +0 = QX+D = ηψQψ + ηSQS +=⇒ ηψ = ηS , +(4.1) +where in the last step we used that the U(1)X+D charges are given by Qψ = −QS = 1. +First, we discuss the generation of the asymmetry and then the DM production. +– 11 – + +4.1 +Asymmetric dark matter via cogenesis +In the following, we assume a hierarchical scenario MN2,3 ≫ MN1, which in turn implies +y2,3 +σ +≫ y1 +σ. The simultaneous generation of lepton and dark sector asymmetries takes place +at a high scale (T ∼ MN1) via the decays of the lightest RHN, N1, into the two channels +(the asymmetry generated in the decays of N2,3 are washed out by N1 interactions), as +shown in Fig. 1: +1. N1 → LH generates a lepton asymmetry, ηL, which is later reprocessed into a baryon +asymmetry ηB by sphalerons. This is the case of Type-I thermal leptogenesis, well +studied in the literature (see Ref. [60] for a review), but with extra contributions, see +below. +2. N1 → χS generates an asymmetry in the dark sector, ηD, analogous to the lepton +asymmetry. +Here, we focus on the regime where the asymmetry generated by N1 decays into S is not +washed out, and is comparable to the asymmetry in χ, i.e., ηχ ∼ ηS ∼ ηD. In order to +generate an asymmetry, CP needs to violated. The CP asymmetry generated in the decays +of N1 can be written as +εL = +� +α +ΓN1→LαH − ΓN1→¯LαH† +ΓN1 +, +εχ = ΓN1→χS − ΓN1→¯χS† +ΓN1 +, +(4.2) +where ΓN1 = ((yνy† +ν)11 + |y1 +S|2)MN1/(8π) is the total tree-level decay width of N1 and y1 +S +is the relevant Yukawa coupling for N1 → χ + S. In the following we refer to it as yS. +However, due to CPT invariance, no asymmetry can be generated at the tree level. +CP violation arises via the interference of tree-level and one-loop level decay (vertex and +self-energy corrections) amplitudes, which depends on the imaginary part of the product +of the Yukawas involved. +Therefore, for this asymmetry to be non-zero, we require at +least two distinct phases that come from the Yukawas yν and yS, so the couplings to the +heavier neutrinos N2,3 are important. Further, having an imaginary part in the internal +loop contribution demands that the would-be decay products can be produced on-shell (i.e., +the optical theorem), which is easily satisfied as we take MN1 ≫ mχ,S,L,H. +In this cogenesis scenario, it can be seen that both εL and εχ depend on yν and yS, +as the dark sector particles (χ, S) are involved in the one-loop self-energy correction for +N1 → LH and vice versa. Thus, the ratio of the decay asymmetries depends on the ratio +of the couplings yν and yS and may be correlated with the branching ratio of N1 decay in +each sector (BrL and Brχ). This relates neutrino mass generation to the baryon and DM +abundances. +In Ref. [27] it has been shown that the dark asymmetry ηD can be quite different from +the visible one ηB (contrary to the ADM models that predict ηB ∼ ηD) due to different +branching ratios, washout effects and transfer between the sectors via inverse decays and +2 ↔ 2 scatterings. The asymptotic asymmetries for the two sectors can be written as +η∞ +L = εL ξL Y eq +N1(T ≪ MN1) ≃ 2.6 × 10−10 , +η∞ +χ = εχ ξχ Y eq +N1(T ≪ MN1) , +(4.3) +– 12 – + +where ξL (ξχ) is the leptonic (dark) efficiency parameter characterising the effects of washout +and transfer interactions, and Y eq +N1(T ≪ MN1) = 135ζ(3)/4π4g∗ is the initial equilibrium +N1 yield, with g∗ ≃ 106.75, the number of relativistic degrees of freedom. The numerical +value of η∞ +L is selected to match the observed baryon asymmetry, i.e., ηB = (28/79) ηL, +generated via sphaleron processes. Note that the presence of N − ZB−L interactions in the +model may modify the usual picture; however, the correct value of the asymmetry may be +generated [46, 61]. +Typically, producing the DM relic abundance in ADM models constrains the DM mass, +depending on the ratio ηD/ηB. However, since χ is not the DM candidate in our set-up, +it is not constrained to be of the order of few GeVs (for ηD ∼ ηB) and thus may be +much larger, ∼ O(TeV). In the analysis below, we consider dark asymmetries in the range +ηD ∼ (0.1 − 1) ηB, which may be achieved if BrL ∼ Brχ and the Yukawa couplings of both +sectors have similar hierarchies. Thus, we work with DM masses in the GeV ballpark. We +refer the reader to Fig. 6 of Ref [27], where it is shown the order of the asymmetries that +can be obtained for a given value of MN1 and different branching ratios. +A key feature of any ADM model is the presence of an interaction that efficiently +depletes the symmetric component. In our set-up, in order for the DM component ψ to +be asymmetric, we require that the symmetric population of χ is annihilated and only the +asymmetric component survives before it decays to ψ. For example, this can take place +via annihilations of the form ¯χχ → ZDZD. In the non-relativistic limit, s ≃ 4m2 +χ, the +thermally-averaged cross section (considering s-wave annihilations) for mχ > mZD is given +by +σv(¯χχ → ZDZD) = +g4 +D +16πm2χ +� +1 − +m2 +ZD +m2χ +�3/2 � +1 − +m2 +ZD +2m2χ +�−2 +. +(4.4) +For mχ > mS, an extra contribution comes from annihilations of the form ¯χχ → ZD → S†S, +which opens up the region of parameter space where mχ < mZD. It should be noted that +there is another contribution to annihilations from the channel ¯χχ → ZB−L → ¯qq (¯ll), +where q (l) is a SM quark (lepton). However, given the large mass of ZB−L, this turns out +to be negligible. In the left plot of Fig. 4 we show the mass ranges of ZD and χ where the +latter is asymmetric, i.e., rχ < 10−2, see Eq. 2.2. One observes how, for larger values of the +asymmetry, a larger region of the parameter space has rχ < 10−2. Note also the presence +of the resonance for mZD ≃ 2mχ. +4.2 +Contribution of ψ +4.2.1 +Production of ψ from χ decays +First, we study the dynamics of ψ, neglecting the contribution to the relic abundance +from S. We assume that the ¯χχ annihilation processes studied in the previous section are +efficient enough so that rχ < 10−2 and the freeze-out abundance of χ is determined by its +asymmetry. The production of ψ is driven by decays of χ: χ → ψφ for T > vφ and χ → ψϕ +for T < vφ. As the h − ϕ mixing is small (θ ≪ 1), it is safe to trade ϕ, h for the mass +eigenstates. The production via decays can be divided into two types: +– 13 – + +Figure 4. The red regions indicate the regions of mχ, mZD (left) and mS, mϕ (right) where the +annihilations of χ, S respectively are strong enough to result in a fractional asymmetry rχ (S) < 10−2. +Left: The star represents the benchmark point {mχ, mZD} = {3.5 TeV, 500 GeV}. In the gray +region the annihilation channel ¯χχ → ZDZD is closed. Right: In the gray region, annihilations to +ϕ are closed. +1. While χ is in thermal equilibrium with the SM bath (T > T∗, T∗ ≃ mχ/20 being the +freeze-out temperature of χ,), the decays are symmetric, i.e., ψ and ¯ψ are produced +in equal amounts from the decays of χ and ¯χ, that are symmetric during this period, +i.e., Y + +χ ≃ Y − +χ = Y eq +χ +≫ ηD. This is the usual freeze-in contribution. We also refer +to these processes as early decays. We denote the abundance of ψ particles produced +by freeze-in by YFI. This symmetric production peaks around T ∼ mχ > vφ, so that +the channel is χ → ψφ. +2. Once χ freezes out (T < T∗), the population of χ becomes asymmetric, i.e., Y + +χ ≃ +ηD ≫ Y − +χ ≃ rχY + +χ . Such an asymmetry is then subsequently transferred to ψ via +decays. This is the asymmetric freeze-in contribution of ψ from late decays (LD). +Since the decays occur late, i.e TD ≪ vφ, in this case the channel is χ → ψϕ. The +populations are given by +Y + +LD = +ηD +1 − rχ +≃ ηD , +Y − +LD = rχY + +LD ≃ ηD rχ , +(4.5) +where in the last step we used rχ < 10−2. +Therefore, the asymptotic abundance of ψ and ¯ψ can be written as +Yψ ≡ Y + +ψ = YFI +2 + Y + +LD ≃ YFI +2 + ηD, +(4.6) +Y ¯ψ ≡ Y − +ψ = YFI +2 + Y − +LD ≃ YFI +2 + ηD rχ . +where in the last step we used Eq. (4.5). The three interesting cases for the asymmetric +ratio are +rψ ≃ +� +� +� +� +� +� +� +rχ +if ηDrχ ≫ YFI , +YFI/(2ηD) +if ηDrχ ≪ YFI ≪ ηD , +1 +if YFI ≫ ηD . +(4.7) +– 14 – + +105 +9p= 0.5 +104 +103 +ND = NB +102 +ND = 0.1 NB +101 +102 +103 +104 +105 +101 +mx [GeV]101 +np = 0.1 nB, mzp = 500 GeV +ms +mp [GeV] +100 +Λsp = 0.05 +Λs = 0.01 +Λss = 0.001 +10-1 +101 +102 +100 +ms [GeV]We remind that, according to our definition in Section 2, DM is asymmetric if rψ < 10−2, +partially asymmetric if 10−2 < rψ < 0.9 and symmetric if rψ > 0.9. Hence, in order for +ψ to be asymmetric, not only we need to require that rχ ≪ 1 but also the symmetric +freeze-in contribution to the abundance, YFI/2, should be suppressed, i.e., YFI ≪ ηD. In +the opposite regime, where the freeze-in production from early decays is dominant, the final +ψ abundance is always symmetric. The decay width is given by +Γχ→ψϕ ≃ +y2 +φmχ +32π ∆2(fψ, fϕ) , +(4.8) +where fψ ≡ mψ/mχ and fϕ ≡ mϕ/mχ, and ∆(fψ, fϕ) is the phase space suppression factor, +∆2(fψ, fϕ) = +� +1 − f2 +ψ − f2 +ϕ + 2fψ +�2 �� +1 − f2 +ψ − f2 +ϕ +�2 − 4f2 +ψf2 +ϕ +� +, +(4.9) +with 0 ≤ ∆(fψ, fϕ) ≤ 1. We have neglected the small corrections due to fermion mixing ϵf. +There is an analogous expression for ϕ → φ. Here onwards, we omit the arguments of the +function ∆ and we consider values of the parameters such that mχ ≫ mψ, mϕ. Within this +approximation ∆ ≃ 1. Furthermore, in this limit the width is practically independent of +mϕ or mφ, so that in the following we do not differentiate among decays into ψϕ and ψφ. +Notice that the Yukawa yφ also leads to annihilation processes (such as ¯χχ → ¯χψ) which +would contribute to ψ production. However, as long as ∆ ≃ 1, these are subleading with +respect to decays. +The freeze-in contribution from early decays can be computed numerically by solving +the Boltzmann equations for χ and ψ. However, a very good analytic estimate is given by +YFI ≃ 135 +8π4 +� +45 +πg3∗ +Γχ→ψϕ MPl +m2χ +≃ 6 × 10−6 y2 +φ ∆2 MPl +mχ +, +(4.10) +where we used that the production peaks around the mass of the heaviest particle involved +in the decay, i.e., at T ≃ mχ. The late decays of χ would instead peak at temperature TD +at which Γχ→ψϕ/H|T=TD ≃ 1, +TD ≈ yφ ∆ +� +mχMPl +32π +≈ 10 MeV ∆ +� +yφ +10−12 +� � +mχ +3.5 TeV +�1/2 +. +(4.11) +In order to get asymmetric DM, we impose the condition YFI ≲ 10−2ηD (here we are +assuming values of {gD, mχ, mZD} such that rχ ≲ 10−2) which implies +yφ ∆ ≲ 6.4 × 10−12 +�ηD +ηB +�1/2 � +mχ +3.5 TeV +�1/2 +. +(4.12) +In Fig. 5, we show the parameter space in the plane yφ versus mψ where the contribution +of ψ is dominant. We fix mχ = 3.5 TeV. We highlight the regions where it is symmetric, +partially-asymmetric and asymmetric, subject to the constraints discussed below. +– 15 – + +Figure 5. Parameter space for the scenario in which the DM is composed solely of ψ and we assume +that S is light enough in each point of the plot so that its contribution to the DM abundance can be +safely neglected. We distinguish three different regions depending on the nature of DM (symmetric, +asymmetric or partially asymmetric), labeled by the value of rψ. In the darker gray region DM is +produced through freeze-in and is symmetric. In the white region it is produced via late decays +and is asymmetric. The DM relic abundance is reproduced along the blue solid (dashed) line which +corresponds to ηD/ηB = 1 (0.1), whereas the horizontal dashed lines indicate the shift in the regions +for ηD = 0.1ηB. +4.2.2 +Constraints +The gauge coupling gX is constrained by long-range force experiments, gX ≲ 10−8, see +Appendix B. In the rest of the paper, we take gX sufficiently small so that gauge interactions +can not drive thermalisation of ψ and therefore gX never plays a role. On the other side ψ +could thermalise through processes involving the Yukawa coupling yφ. All these processes +involve at least one χ particle (or the conjugate). The most important ones are the decays +(and inverse decays) χ → ψϕ. The condition for non-thermalisation is Γχ→ψϕ/H|T≃mχ ≪ +1. Indeed, for T > mχ, the interaction rate/Hubble ratio grows when the temperature +decreases, while for T < mχ decay and inverse decays become inefficient. Therefore, we +evaluate the ratio at its maximal value, i.e., T ≃ mχ, which gives +yφ ∆ ≪ 30 +� mχ +MPl += 5 × 10−7 +� +mχ +3.5 TeV . +(4.13) +Notice that taking the limit ∆ ≪ 1 does not help in pushing towards higher values of the +Yukawa. Indeed, if ∆ is small, the decay channel closes and annihilations become more +relevant. Annihilations processes are independent from ∆ and give a condition analogous +to Eq. (4.13) with ∆ = 1. However, as we are considering ∆ ≃ 1, decays are more important +than annihilations. +In the case in which ψ is produced mainly by late decays, we require that these are +peaked much after the freeze-out of χ, i.e., TD ≪ T∗, which ensures that the population of +– 16 – + +— nD=NB +10-8 +mx= 3.5 TeV, △ = 1 +np=0.1 nB +Symmetric DM +10-9 +Freeze-in +rμ > 0.9 +10-10 +Partially asymmetric DM +0.9 >r ≥10-2 +10-11 +10-12 +Late decays +Asymmetric DM +rμ<10-2 +10-13 +BBN +100 +101 +10-1 +my [GeV]ψ is asymmetric. This gives a stronger condition, +yφ ∆ ≪ 0.5 +� mχ +MPl += 8.5 × 10−9 +� +mχ +3.5 TeV . +(4.14) +If ψ is produced by early decays, peaked around Tprod ≃ mχ, it beheaves as cold DM as +long as mψ ≳ keV. However, if the production from late decays is significant, we must take +into account an additional constraint. Indeed, as we have a heavy particle (χ) decaying late +into a lighter stable one (ψ), we must check that the DM free streaming length is smaller +than 0.1 Mpc, which gives [62] +mψ > 3.5 keV ⟨p/T⟩prod +� +10 +g∗(TD) +�1/3 +≳ 1.7 keV mχ ∆ +TD +� +10 +g∗(TD) +�1/3 +, +(4.15) +where in the last line we used that the typical DM momentum at production is ⟨p⟩ ∼ +mχ∆/2, while ⟨Tprod⟩ = TD. Using Eqs. (4.11) and (4.15), this translates into a lower +bound on the coupling, +yφ ≳ 5.6 × 10−14 � +mχ +3.5 TeV +�1/2 �5 GeV +mψ +� � +10 +g∗(TD) +�1/3 +. +(4.16) +This bound only applies if i) ψ reproduces the DM relic abundance, and ii) the production +from late decays is dominant, i.e., Y + +LD ≫ YFI, corresponding to the vertical blue lines of +Fig. 5. We can use BBN bounds to constrain the lifetime of χ. If χ did not decay, its +abundance today, Ωχh2, would be mχ/mψ times the would-be DM (ψ) abundance. +In +particular, for mχ ∼ O(TeV) and would-be abundance ηD ∼ 10−11, this leads to τχ ≲ +(0.1 − 1) s [63] for +yφ∆ ≳ 10−13 +�3.5 TeV +mχ +�1/2 +. +(4.17) +Here we assumed that the decay of ϕ into SM radiation (mainly hadrons) occurs instanta- +neously right after χ decays; otherwise, the bound should read τχ + τϕ ≲ (0.1 − 1) s. This +is discussed in Appendix A. The effects of fermion mixing are discussed in Appendix C. +4.3 +Contribution of S +4.3.1 +Production of S from freeze-out +The scalar S is produced by N1 decays at high temperature and shares the same asymmetry +of χ, i.e, ηD. Once produced, it thermalises with the SM bath through scalar and gauge +interactions and undergoes annihilations. In the following, we focus mainly on masses of S +in the 1−50 GeV range, so that its contribution to the relic abundance may be of the same +order as that of ψ. In this range of masses, annihilations of S into ZD may be kinematically +forbidden. However, if mϕ < mS (possible by tuning the coefficients of the scalar potential, +e.g., mϕ/vφ ≲ 10−3 and mϕ ∼ O(GeV) or lighter), then the annihilations S†S → ϕϕ are +– 17 – + +allowed. Next we consider the minimal option of just the scalar portal λSφ|S|2|φ|2. The +non-relativistic cross section for S†S → ϕϕ induced by the operator is given by 6 +σv(S†S → ϕϕ) ≃ +λ2 +Sφ +32πm2 +S +� +1 − m2 +ϕ +m2 +S +�1/2 � +1 − m2 +ϕ/2m2 +S − 2λSφv2 +φ/m2 +S +1 − m2ϕ/2m2 +S +�2 +. +(4.18) +Even for moderately small values of the coupling, the cross section is significant and annihi- +lations are strong enough to destroy the symmetric population of S. Hence, the symmetric +population of S annihilates around T (S) +∗ +∼ mS/20, leaving only the asymmetric population, +with the abundance fixed by the dark asymmetry Y + +S ∼ ηD, Y − +S ∼ rSYS ≪ Y + +S , completely +analogous to the computation of χ annihilations. In Fig. 4 (right panel) we show the re- +gion of mϕ and mS where S is asymmetric for different values of the coupling λSφ fixing +ηD = 0.1ηB and mZD ∼ 500 GeV. +In principle, other annihilation channels for S are possible, depending on the scalar +potential parameters, such as S†S → h → ¯ff (which is more suppressed for mh > mS), +allowing for a larger set of possibilities. For simplicity, we focus only on annihilations into +ϕϕ, which involve only one coupling λSφ and need not be very large, O(10−2). +4.3.2 +Production of S from χ decays +Integrating out the lightest of the heavy sterile neutrinos, N1, induces the effective interac- +tions +yνyS +¯L ˜HSχ +MN1 +, +y2 +S +χ2S2 +MN1 +. +(4.19) +The former leads to χ → S†νL and χ → S†νLh decays, which compete with χ → ψϕ, +eventually populating the S† sector, whereas, the latter induces χχ → S†S† annihilations. +Focusing on decays, we can use the estimate (see also Ref. [59]) +Γ(χ → S†ν) ≈ |yS|2mχ +32π +� mν +MN1 +� � +1 − m2 +S +m2χ +�2 +. +(4.20) +These decays are peaked around T (S) +D +defined as Γχ→S†ν/H|T=T (S) +D += 1 (analogous to the +decays into ψ computed earlier) and given by +T (S) +D +≃ |yS| +� +mνmχMPl +32πMN1 +≃ MeV +� |yS| +10−3 +� � +mν +0.05eV +mχ +3.5TeV +1011GeV +MN1 +�1/2 +. +(4.21) +Since for the values of the parameters T (S) +D +< T∗, the only important decay is χ → S†νL, +while the conjugate process is irrelevant as the population of ¯χ after freeze-out is negligible +(recall that we are in the region of the parameter space in which rχ < 10−2).7 +6We neglect the diagram containing the self-interaction ϕ3, which enters if the φ quartic coupling is not +extremely small. +7Also in this case we can distinguish between the late decays, peaked at T (S) +D , and the early decays at +T > T∗. The latter produce a symmetric population of S and S† from the decays of χ and ¯χ (the analogue +of the freeze-in population of ψ, peaked at T ≃ mχ). However, this symmetric population thermalises and +undergoes annihilations leaving no imprint in the final abundance. +– 18 – + +We can parametrise the dominant decay channel of χ by defining the ratio of branching +ratios +R ≡ Br(χ → S†ν) +Br(χ → ψϕ) ∼ |yS|2 +y2 +φ +mν +MN1 +, +(4.22) +where in the last step we used Eqs. (4.8) and (4.20). The decay of χ into ψ is the dominant +channel, i.e., R ≪ 1, for +|yS| ≪ +� +yφ +2 × 10−11 +� � +MN1 +1011GeV +�1/2 �0.05eV +mν +�1/2 +. +(4.23) +Notice that the contribution of the 3-body decays χ → νhS† should be similar to the 2- +body ones, as the decay rate gets suppressed by the phase space factor while at the same +time it has an enhancement (mχ/v)2 ∼ 200 (eventually it can become dangerous for mχ ≫ +TeV). For a generic value of R, the probability to decay into ψ is 1/(1 + R) while into S† +is R/(1 + R). Therefore the abundances of ψ, ¯ψ in Eq. (4.6) get divided by (1 + R). +Concerning S, we can distinguish two possibilities: +1. If T (S) +D +< T (S) +∗ +< T∗, at the time where decays into S† peak, the latter has already de- +coupled from the thermal bath with an asymmetric abundance YS = ηD and therefore +the S and S† population can not annihilate. Then the abundances of S, S† are +YS ≡ Y + +S = ηD, +YS† ≡ Y − +S = +R +1 + R ηD , +(4.24) +where we assume that rS < 10−2 and we have ignored it for simplicity. This corre- +sponds to +|yS| ≲ 0.1 +� mS +GeV +� � +MN1 +1011GeV +�1/2 �0.05eV +mν +�1/2 �3.5TeV +mχ +�1/2 +. +(4.25) +2. If T (S) +∗ +< T (S) +D +< T∗, the decays produce a population of S†, while S†S annihilations +are still efficient. As a result there is a partial washout of ηS, which gets reduced to +ηD/(1 + R). Therefore, at T < T (S) +∗ +a population of S decouples with abundance +Y + +S = +1 +1 + R ηD , +Y − +S ≪ Y + +S , +(4.26) +where we assume rS < 10−2. +Notice that for R ≪ 1 Eqs. (4.24) and (4.26) are +equivalent as the decays are irrelevant. +4.3.3 +Constraints +We can constrain the value of |yS| for the case in which R > 1. +As for this choice χ +decays mostly into S†, we must impose that the decay occurs before BBN, in analogy with +Section 4.2.2. Using the decay rate in Eq. (4.20) we find that the BBN bound translates +into the constraint +|yS| ≳ 10−3 +�0.05 eV +mν +MN1 +1011 GeV +3.5 TeV +mχ +�1/2 +. +(4.27) +– 19 – + +For R < 1 we can choose smaller yS while BBN constrains the value of yφ, see Eq. (4.17). +We also find that for R > 1 the model is characterised by an interesting feature: +the decays χ → S†νL, which occur before BBN and neutrino decoupling (around T ∼ +O(10 MeV) for the choice of parameters we adopted and |yS| ∼ 10−2), generate also an +asymmetric population of neutrinos, +∆ην = +R +(1 + R) ηD , +(4.28) +which is maximal (≈ ηD) for large R. Therefore, the neutrino population is more asym- +metric than in the standard case, as this new contribution sums up to the usual leptonic +asymmetry generated earlier on by leptogenesis. Note, however, that since these decay +processes occur below the scale of electroweak symmetry breaking, this leptonic asymmetry +is not transferred to the baryonic one. +Finally, if R > 1 (and T (S) +D +< T (S) +∗ +), the S† population that arises from late decays +of χ may be warm. In such a case, if a significant fraction of the DM was made by this +S† population, constraints from free streaming length (see Eq. 4.15 with mψ → mS and +TD → T (S) +D ) would give |yS| ≳ 5 MeV/mS. +However, even in the cases in which the +contribution of ψ DM is negligible, the DM is composed by a mixture of S (produced by +freeze-out, always cold) and S†, where Y + +S +≥ Y − +S +(the equality applies if R ≫ 1). This +corresponds to a mixture of cold/warm DM, which in general is less constrained than the +full warm DM case [64]. +In the following, we discuss the scenarios arising due to the different dynamics of the +dark components in the model. It can be checked that Eq. 4.1 is always fulfilled. +5 +Dark matter relic abundance +The precise values of some of the parameters do not change qualitatively the results, as- +suming that we are always in the red regions of Fig. 4, i.e., that annihilations are efficient. +However, for definiteness, in the following we fix some of the parameters of the model, as +shown in Table 3. +Parameter +Benchmark Value +mχ +3.5 TeV +mZD +500 GeV +gD +0.5 +MN1 +1011 GeV +mν +0.05 eV +Table 3. Benchmark values of the model parameters used in the analysis. +Notice that with this choice, rχ ≪ 10−2, i.e., ¯χχ annihilations efficiently erase the +symmetric population. We also assume that rS ≪ 10−2. At the same time, depending on +the values of the Yukawa couplings yφ and |yS|, and the masses of the DM particles mψ +and mS, it is possible to realise the different scenarios of Table 1 while satisfying all the +– 20 – + +constraints. In Table 4 we outline the different scenarios, as well as provide expressions for +the relic abundance of the DM components. Next we analyse them one by one. The reader +may want to skip the following discussion and go directly to the summary of scenarios in +Section 5.9. +Scenario +Model +ψ +S +Asymmetric +Asymmetric +1 +ψ-LD-A +LD χ → ψϕ +FO S†S → ϕϕ +S-FO-A +Y + +ψ = ηD +Y + +S = ηD +Y − +ψ ≪ Y + +ψ +Y − +S ≪ Y + +S +Asymmetric +Partially asymmetric +2 +ψ-LD-A +LD χ → ψϕ +FO S†S → ϕϕ + LD χ → S†νL +S-FOLD-PA +Y + +ψ = ηD/(1 + R) +Y + +S = ηD +Y − +ψ ≪ Y + +ψ +Y − +S = ηDR/(1 + R) +Asymmetric +Asymmetric +Mixed +ψ-LD-A +LD χ → ψϕ +FO S†S → ϕϕ + LD χ → S†νL +1-2 +S-FOLD-A +Y + +ψ = ηD/(1 + R) +Y + +S = ηD/(1 + R) +Y − +ψ ≪ Y + +ψ +Y − +S ≪ Y + +S +Partially asymmetric +Asymmetric +3 +ψ-FILD-PA +FI + LD χ → ψϕ +FO S†S → ϕϕ +S-FO-A +Y + +ψ = YFI/2 + ηD +Y + +S = ηD +Y − +ψ = YFI/2 +Y − +S ≪ Y + +S +Partially Asymmetric +Partially Asymmetric +4 +ψ-FILD-PA +FI +LD χ → ψϕ +FO S†S → ϕϕ + LD χ → S†νL +S-FOLD-PA +Y + +ψ = (YFI/2 + ηD)/(1 + R) +Y + +S = ηD +Y − +ψ = YFI/(2(1 + R)) +Y − +S = ηDR/(1 + R) +Partially Asymmetric +Asymmetric +Mixed +ψ-FILD-PA +FI +LD χ → ψϕ +FO S†S → ϕϕ + LD χ → S†νL +3-4 +S-FOLD-A +Y + +ψ = (YFI/2 + ηD)/(1 + R) +Y + +S = ηD/(1 + R) +Y − +ψ = YFI/(2(1 + R)) +Y − +S ≪ Y + +S +Symmetric +Asymmetric +5 +ψ-FI-S +FI χ → ψϕ +FO S†S → ϕϕ +S-FO-A +Y + +ψ = YFI/2 + ηD ≃ YFI/2 +Y + +S = ηD +Y − +ψ = YFI/2 +Y − +S ≪ Y + +S +Symmetric +6 +S-FOLD-S +Negligible production +FO S†S → ϕϕ + LD χ → S†νL +Y + +S = ηD +Y − +S = ηD +Table 4. +Classification of scenarios in our model on the basis of dominant production mechanism +and asymptotic nature of both dark matter components, ψ and S. +– 21 – + +5.1 +Scenario 1: ψ-LD-A + S-FO-A +In this scenario, both components are asymmetric. For this to happen, we demand +• yφ ≲ 6.4×10−12� +ηD/ηB: the freeze-in population of ψ from early decays is negligible, +while the late decays are dominant. Hence, ψ is asymmetric with abundance Y + +ψ = +ηψ = ηD. +• R ≪ 1, which means |yS| ≪ yφ/(2 × 10−11): S freezes-out once all the symmetric +population has annihilated. The population of S† produced by the late decays of χ +is negligible. The abundance is determined by its asymmetry, i.e Y + +S = ηS = ηD. +Therefore, the model has two asymmetric components with individual abundance ηD, +where the abundance of one (ψ) is set by late decays and that of the other one (S) by +freeze-out. The relative abundance of the two DM species and the total abundance are +given by +Ωψ +ΩS += mψ +mS +, +ΩDM +ΩB +≃ 5 = ηD(mψ + mS) +mpηB +. +(5.1) +In Fig. 6 (black lines), we show the region of the parameter space in which the correct +DM relic abundance is reproduced in the plane mS versus mψ, for two values of the dark +asymmetry ηD. Ωψ/ΩS increases when the curves are followed clockwise. We can see that +for ηD ≃ 0.1 (1) ηB, dashed (solid) black line, we have mψ ≃ mS ≃ 30 (3) GeV if both +species contribute similarly. Alternatively, we can push the mass of S down to GeV in +such a way that its contribution to DM abundance is negligible and reproduce the relic +abundance for mψ ≃ 50 (5) GeV for ηD ≃ 0.1 (1) ηB, or vice versa. +5.2 +Scenario 2: ψ-LD-A + S-FOLD-PA +Similar to the previous scenario, here we also take yφ ≲ 6.4 × 10−12� +ηD/ηB, so that +the dominant production of ψ comes from late decays, leading to its asymmetric nature. +However, here we take R ∼ O(1) (i.e., |yS| ∼ 0.5 × 1011yφ), which makes it qualitatively +distinct. We also assume that |yS| is small enough so that Eq. (4.25) is satisfied, i.e., the +decays of χ to S† occur when the latter has already decoupled from the thermal bath, +T (S) +D +< T (S) +∗ +. +Due to R being order one, χ partially decays into ψ and partially into S†. The probabil- +ity to decay into ψ is 1/(1+R) whereas that of into S† is R/(1+R). Therefore, ψ is highly +asymmetric with abundance Y + +ψ = ηψ ∼ ηD/(1 + R), whereas S becomes partially asym- +metric because a population of S† is produced by χ → S†ν decays. As T (S) +D +< T (S) +∗ +, S†S +annihilations are not active. Therefore ηS reduces to ηS = ηD −R ηD/(1+R) = ηD/(1+R), +while the total abundance is determined by the sum Y + +S +Y − +S = (1+2R)ηD/(1+R). There- +fore, the contribution to the DM abundance becomes +Ωψ +ΩS +≈ mψ +mS +1 +(1 + 2R) , +ΩDM +ΩB +≃ 5 = ηD(mψ + (1 + 2R)mS) +(1 + R)mpηB +. +(5.2) +– 22 – + +Figure 6. +Values of mψ and mS for which the correct relic abundance can be reproduced for +Scenario 1: ψ-LD-A + S-FO-A (black lines) and Scenario 2: ψ-LD-A + S-LD-PA (red lines). We +show ηD ≃ (0.1)ηB using solid (dashed) lines. The relative abundance of ψ with respect to S in- +creases when the curves are followed clockwise. The gray dashed line corresponds to mψ = mS. The +parameters are fixed as per Table 3, so that ¯χχ → ZDZD annihilations are efficient. Furthermore, +we fix yφ = 10−12. For Scenario 1 (2): |yS| = 10−3 (5 × 10−2), so that R ≃ 0.0005 (1.25). In both +cases YFI ≃ 10−4ηB. +In the limit R = 0, all χ decay into ψ and we recover Eq. (5.1). If R = 1 the relative +ψ/S abundance is mψ/(3mS). The factor 3 comes from the fact that while we only have +ψ and no ¯ψ, we have both S and S† with Y + +ψ = Y − +S += ηD/2 and Y + +S += ηD. The limit +R ≫ 1 is discussed later (Scenario 6). As discussed earlier, this case leads to an enhanced +background of an asymmetric neutrino population. The allowed parameter space is shown +in Fig. 6 by red lines. In this case, the shape of the curves is not symmetric as in Scenario +1; in the limiting case where ψ (S) dominates the abundance one needs mψ ≃ 100 GeV +(mS ≃ 30 GeV) for ηD ≃ 0.1 ηB, and masses roughly one order of magnitude smaller for +ηD ≃ ηB, as expected. +5.3 +Mixed Scenario 1-2: ψ-LD-A + S-FOLD-A +We consider the same range for yφ and R but we assume that |yS| is large enough to violate +Eq. (4.25), i.e., χ decays to S† while S†S annihilations are still efficient, T (S) +D +> T (S) +∗ +. For ψ +we find the same results of the previous section. The S† population produced by χ decays +partially washes-out ηS, leaving an asymmetric population of S with abundance Y + +S += +ηD/(1 + R), while the S† population gets erased by S†S annihilations. The contribution to +the DM abundance is now +Ωψ +ΩS +≃ mψ +mS +, +ΩDM +ΩB +≃ 5 = ηD(mψ + mS) +(1 + R)mpηB +. +(5.3) +Notice that this scenario is a mixture between Scenario 1 (the nature of the DM particles +is the same, both asymmetric, and the abundances are the same ones rescaled by (1 + R)) +– 23 – + +102 +Scenario 1 +Scenario 2 +Np= 0.1 NB +ms [GeV] +101 +ND= NB +101 +100 +102 +my [GeV]and Scenario 2 (the ranges of yφ and R are identical and the same processes determine the +final abundance). Therefore we denote it as Mixed Scenario 1-2. +5.4 +Scenario 3: ψ-FILD-PA + S-FO-A +For larger values of the Yukawa, 6.4 × 10−12� +ηD/ηB ≲ yφ ≲ 1.9 × 10−10� +ηD/ηB, the +freeze-in contribution to ψ production from early decays grows and becomes comparable to +the contribution from late decays. So ψ is partially asymmetric with Y + +ψ ≃ YFI/2 + ηD and +Y − +ψ ≃ YFI/2. We take R ≪ 1, so that all χ decay into ψ, partially while being in thermal +equilibrium (freeze-in) and partially at a later time (late decays). The ψ abundance is +Y + +ψ + Y − +ψ = YFI + ηD. +On the other hand, the abundance of S is determined by thermal freeze-out, once +the symmetric population is annihilated away. Thus, S freezes-out with an asymmetry +Y + +S = ηD. In this case the relative abundance between the two DM component and the +total DM abundance are given by +Ωψ +ΩS += mψ(ηD + YFI) +mSηD +, +ΩDM +ΩB += mψ(ηD + YFI) + ηDmS +ηBmp +. +(5.4) +5.5 +Scenario 4: ψ-FILD-PA + S-FOLD-PA +If R ∼ O(1) for the values of Yukawa yφ considered in the previous scenario, then χ decays +half of the time into S† and the other half into ψ. +Again, we assume that Eq. (4.25) +is satisfied. +The population of ψ, ¯ψ is produced partially by freeze-in and partially by +late decays, whereas the decays into S† washout the asymmetry in S, making it partially +asymmetric. Therefore, in this scenario, the late decays determine the asymptotic nature +of both DM components, i.e., it is the late decays that finally make the DM to be partially +asymmetric. The corresponding expressions for the DM abundance are8 +Ωψ +ΩS += mψ +mS +ηD + YFI +ηD(1 + 2R) , +ΩDM +ΩB += mψ(ηD + YFI) + ηD(1 + 2R)mS +ηB(1 + R)mp +. +(5.5) +5.6 +Mixed Scenario 3-4: ψ-FILD-PA + S-FOLD-A +Here we consider the same range of yφ and R as in the previous scenario but a larger |yS|, +which violates Eq. (4.25). The discussion for ψ is the same as in Scenario 4. In analogy +with the Mixed Scenario 1-2, S†S annihilations erase the S† from the thermal bath, leaving +an asymmetric population of S which survives the annihilations, so that +Ωψ +ΩS +≃ mψ +mS +YFI + ηD +ηD +, +ΩDM +ΩB +≃ 5 = mψ(YFI + ηD) + mSηD +(1 + R)mpηB +. +(5.6) +This scenario is a mixture between Scenarios 3 and 4. +8Notice that the previous equations are quite general, as Scenario 1, 2, 3, 5 and 6 can be seen as particular +cases of them. +– 24 – + +5.7 +Scenario 5: ψ-FI-S + S-FO-A +For even larger Yukawas, in the range 1.9 × 10−10� +ηD/ηB ≲ yφ ≲ 5 × 10−7, the ψ sec- +tor is mainly populated during freeze-in, while late decays only produce a sub-dominant +component, i.e., YFI ≫ ηD. Therefore, the ψ population is (almost) symmetric, Y + +ψ += +YFI/2 + ηD ≃ YFI. Concurrently, R is typically small because of the larger value of yφ and +therefore S freezes with an asymmetric abundance, Y + +S = ηD. +Therefore, DM is mostly symmetric in ψ and asymmetric in S, produced by freeze-in +and freeze-out, respectively. The relative abundance of the two species and the total DM +abundance are +Ωψ +ΩS += mψ +mS +YFI +ηD +, +ΩDM +ΩB +≃ 5 = mψYFI + mSηD +mpηB +. +(5.7) +In Fig. 7 we show the mass ranges of ψ and S for which the DM relic abundance can be +reproduced for different values of ηD. Notice that if S is subdominant, the mass of ψ is fixed, +irrespective of the value of asymmetry. This is due to the fact that the DM mass is fixed by +the freeze-in contribution in Eq. (4.10), which is (almost) independent of mψ. On the other +hand, it depends on the mass of χ. For the values used in the figure (0.1ηB < ηD < ηB, +etc.), we see that DM could be mainly composed by light symmetric ψ of mass around 500 +MeV, mainly by asymmetric GeV-ish S, or by a combination of them. +Figure 7. +Values of mψ and mS for which the correct relic abundance can be reproduced +for Scenario 5: ψ-FI-S + S-FO-A. The gray dashed line corresponds to mψ = mS. We fix the +parameters |yS| = 2 × 10−4 and yφ = 2 × 10−10. With this choice, R ≃ 5 × 10−10 and YFI ≃ 10ηB. +As we have underlined above, for these values of yφ, R is typically small. However, +if we lower the vB−L scale it is possible to reach R > 1 even in this scenario. We briefly +discuss this possibility at the end of Section 7. +5.8 +Scenario 6: S-FOLD-S +Finally, when R ≳ O(10), for whatever value of yφ compatible with it, the majority of +the χ population decays into S† after freeze-out of S (but before BBN if 10−3 < |yS| < +– 25 – + +102 +Scenario 5 +Np =0.1 NB +10 +ms [GeV] +ND= NB +10- +100 +10-1 +my [GeV]0.1(mS/GeV)) and the asymmetry is completely washed-out. However, the populations of +S and S† survive independently, as S†S annihilations are not active. Hence, the DM relic +is almost completely made up by the symmetric population of S and S†, i.e., Y + +S = Y − +S , +while there is a negligible abundance of ψ produced from early or late decays. Therefore, +the scenario leads to practically only 1 DM component with abundance completely fixed +by the asymmetry: Y + +S + Y − +S = 2ηD, leading to the prediction +mS ≃ 2.5 GeV +�ηB +ηD +� +. +(5.8) +This scenario is phenomenologically interesting because S†S annihilations get enhanced +with respect to the usual freeze-out case (indeed the abundance is set by the asymmetry +and not by the annihilation cross section), leading to enhanced indirect detection signals. +Additionally, if the S† population arises from late decays of χ there can be a mixture of +cold/warm DM where S particles, coming from the thermal plasma, represent the cold +component, while their anti-particles, coming from late decays, the warm one, with a pos- +sible impact on structure formation [27]. Furthermore, the additional contribution to the +asymmetric background neutrino population is maximal in this case, ∆Yν ≈ ηD. +Notice that for larger |yS| the decays take place while S, S† are still in equilibrium so +that S†S annihilations washout the asymmetry and one recovers the standard scenario of +symmetric freeze-out in which the S abundance is determined by the annihilation cross +section σvS†S instead of the asymmetry. Depending on mS and σvS†S it may be possible to +reproduce the correct relic abundance. We do not consider this possibility in the following +discussion. +5.9 +Summary of the scenarios +In the scenarios discussed above, we considered the parameter space where +1. Both ψ and S are stable. +2. ¯χχ → ZDZD annihilations are efficient enough so that rχ < 10−2, i.e., the late decays +of χ are always asymmetric. +3. S†S → ϕϕ annihilations are efficient enough so that rS < 10−2, i.e., only the asym- +metric population of S survives the annihilations. +As long as yφ lies in the range 10−13 ≲ yφ ≲ 5 × 10−7, we can summarise all the +scenarios with the following expressions for the individual and total relic abundance: +Ωψ +ΩS += mψ(ηD + YFI) +ηDmSf(R) +, +ΩDM +ΩB += mψ(ηD + YFI) + ηDmSf(R) +ηB(1 + R)mp +, +(5.9) +where +f(R) ≡ +� +1 + 2R +if T (S) +D +< T (S) +∗ +1 +if T (S) +D +> T (S) +∗ +. +(5.10) +– 26 – + +Figure 8. Contours of DM relic abundance in the mψ - mS plane corresponding to ηD = ηB for +the different scenarios discussed above. We fix the values of the yukawa couplings (yφ, |yS|) for +each scenario: Scenario 1: (10−12, 10−3), Scenario 2: (10−12, 5 × 10−2), Scenario 3: (10−10, 10−3), +Scenario 4: (6.4 × 10−12, 10−1), Scenario 5: (2 × 10−10, 2 × 10−4), Scenario 6: (10−13, 5 × 10−2). +The gray dashed line corresponds to mψ = mS. +In Fig. 8 we show contours of correct relic abundance in the mS versus mψ plane for +the different scenarios. +We take ηD = ηB and for each scenario we fix an appropriate +value for the Yukawa couplings |yS| and yφ (Scenarios 1, 2 and 5 are the same already +shown in Figs. 6 and 7, respectively). DM could be mainly composed by ψ, with mψ in +between hundreds of MeV and tens of GeV, mainly by the scalar S with mS ∼ GeV, or by +a combination of them (both GeV-ish). For fixed R, ψ DM is heavier when is asymmetric +(Scenario 1) and lighter when symmetric (Scenario 5). For fixed rψ, ψ gets heavier as R +gets larger. On the contrary, S DM gets lighter while R grows. The minimal value of +mS is fixed by Eq. (5.8), corresponding to Scenario 6. Notice the presence of a four-fold +degeneracy between scenarios 1, 2, 4, 6. This corresponds to the case mS = mψ, in which +the mass of both the DM particles is fixed by Eq. (5.8) (the relation is exact for Scenarios +1, 2 and 6, while for Scenario 4 it is approximately valid if YFI < ηD). Choosing another +value for ηD leads to a rescaling of DM masses by a factor ηB/ηD. +The main results of the paper are provided in Table 5 and Fig. 9. In Table 5, we +show the requirements on yφ and R for each scenario and summarise the contribution to +the relative and the total DM relic abundance. In Fig. 9 we show the allowed parameter +space in the plane |yS| versus yφ. We fix the parameters as: mχ = 3.5 TeV, mZD = 500 +GeV, gD = 0.5, MN1 = 1011 GeV, mν = 0.05 eV and ηD = ηB. We restrict our analysis to +the region 10−13 ≲ yφ ≲ 5 × 10−7 (no thermalisation of ψ and BBN bound on yφ fulfilled +for R < 1). The Yukawa |yS| is large enough to generate a sizeable dark asymmetry (and +respects the BBN bound in the region in which R > 1, i.e., |yS| ≳ 10−3). The gray dot- +dashed (dotted) line corresponds to R = 1 for MN1 = 109 (104) GeV. Clearly, as N1 gets +lighter, χ preferably decays into S†. Notice that in the plot the masses of the DM particles +are not fixed, but at every point there are always some values of the latter for which the +correct DM relic abundance is reproduced. +– 27 – + +101 +ND = NB +Scenario +1 +ms [GeV] +3 +4 +5 +6 +100 +10-1 +101 +my [GeV]Sc. +10−10yφ/ +� +ηD/ηB +R +T (S) +D /T (S) +∗ +ΩDM/ΩB +Ωψ/ΩS +1 +≤ 0.064 +≪ 1 +Any +ηD +ηB +mψ+mS +mp +mψ +mS +2 +≤ 0.064 +O(1) +< 1 +ηD +ηB +mψ+(1+2R)mS +(1+R)mp +mψ +mS(1+2R) +1-2 +≤ 0.064 +O(1) +> 1 +ηD +ηB +mψ+mS +(1+R)mp +mψ +mS +3 +0.064 − 1.9 +≪ 1 +Any +mψ(ηD+YFI)+ηDmS +ηBmp +mψ(ηD+YFI) +mSηD +4 +0.064 − 1.9 +O(1) +< 1 +mψ(ηD+YFI)+ηD(1+2R)mS +ηB(1+R)mp +mψ(ηD+YFI) +mSηD(1+2R) +3-4 +0.064 − 1.9 +O(1) +> 1 +mψ(ηD+YFI)+ηDmS +ηB(1+R)mp +mψ(ηD+YFI) +mSηD +5 +≳ 1.9 +≪ 1 +Any +ηD +ηB +mψ(YFI/ηD)+mS +mp +mψYFI +mSηD +6 +yφ ≲ 5 × 10−7 +≳ O(10) +< 1 +ηD +ηB +2mS +mp +≪ 1 +Table 5. +Contribution to the DM relic abundance in different scenarios depending on the values +of yφ (in units of +� +ηD/ηB) and R for the values of parameters provided in Table 3. We take +yφ < 5 × 10−7 to avoid thermalisation of ψ. In Scenarios 2, 4 and 6 the Yukawa satisfies |yS| ≲ +0.1(mS/GeV). In the Mixed Scenarios 1-2 and 3-4 we have |yS| > 0.1(mS/GeV). Scenarios 1, 3 +and 5 give the same result independently of this condition. +6 +Phenomenological signals +In principle, the models considered in this work may be difficult to test and disentangle in +their current version, because of several reasons: +i) Freeze-in scenarios invoke very small couplings, gX ≪ 1, yφ ≪ 1; +ii) The considered scale of B−L breaking is very large, vB−L ≫ vEW, so collider searches +are not an option; +iii) Asymmetric DM yields suppressed indirect detection signals in general. Moreover, in +our scenarios, the symmetric component is typically erased into the dark sector, via +¯χχ → ZDZD and S†S → ϕϕ, see Fig. 4. +iv) The mixing of ϕ with the Higgs, generated by λHφH†Hφ†φ, was taken to be very +small. +However, there are a few distinctive signals of S, through the usual Higgs portal, +λHSH†HS†S: +• Direct detection in the case in which the DM is mainly composed of S, which currently +sets the limits λHS ≲ 0.01 [65]. +• Higgs invisible decays, for mS < mh/2, which currently sets the limits λHS ≲ 0.01 +[65, 66]. In this case, it could be that S was produced via Higgs decays, but it did +not constitute a dominant part of the DM. +• Similarly, indirect detection signals from annihilations are present if the final abun- +dance is composed of S and partially-asymmetric (Scenarios 2 and 4) or symmetric +(Scenario 6). Scenario 6 is interesting, since it requires larger than usual annihilation +– 28 – + +Figure 9. Parameter space of the scenarios discussed above. We fix mχ = 3.5 TeV, mZD = 500 +GeV, gD = 0.5, MN1 = 1011 GeV, mν = 0.05 eV and ηD = ηB and 10−13 ≲ yφ ≲ 5 × 10−7 (no +thermalisation and BBN bound). The Yukawa |yS| is large enough to generate a sizeable dark +asymmetry (and respects the BBN bound). +Notice that in every point the masses of the DM +particles, mψ and mS, are not fixed, and it is not possible to get the correct relic abundance for +arbitrary values of mψ and mS, but only for particular values, i.e., along the curves in Fig. 8. +The black diamond (red circle) is the benchmark point in Fig. 6 for scenario 1 (2); the purple star +corresponds to the benchmark of Scenario 5 in Fig. 7. Scenario 4 is not visible in the plot but would +appear in the intersection between 2 (red region) and 3 (blue region). Depending on the value of mS +a portion of region 2 (3) could convert into the Mixed Scenario 1-2 (3-4). However, for mS ≳ GeV, +this requires |yS| ≳ 0.1. The gray dot-dashed (dotted) line corresponds to R = 1 for MN1 = 109 +(104) GeV. Clearly, as N1 gets lighter, χ preferably decays into S†. Above the red (blue) [purple] +line the heavier between S and ψ, for fixed mass max(mS, mψ) = 3 GeV, has a lifetime, re-scaled +by its relative number density, of 1023 (1024) [1025] s. Therefore, the region above the red line is +excluded, see Section 6. +rates, and is very predictive, with a mass of 2.5 GeV, see Eq. (5.8). In this case, +on-shell S-annihilations into muons, pions and electrons are possible. Typically, such +light thermal DM is severely constrained by its energy injection in the CMB. There- +fore, it may be interesting to further investigate for which values of λHS is Scenario +6 allowed. +Last but not least, there is a very interesting phenomenological signal of our model: the +presence of a monochromatic flux of neutrinos coming from the late decay of the heaviest +of the two DM components. Let us assume mS > mψ (the opposite case mψ > mS is +completely analogous). +At low energies, E ≪ mχ ≪ MN1, the decay S → ¯ψ + νL is +generated by the dimension-6 operator +O6 = ¯L ˜HSφ†ψ . +(6.1) +– 29 – + +10-1 +6 +ND = NB + Scenario +Scenario 2 +10-2 +R +Ts(n1 DM/ns) [s] +1023 +0.1 +S +Scenario 3 +1024 +10-3 +— - 10 +1025 +Scenario 1 +Scenario5 +rμ > 0.9 +rμ<10-2 +10-4 +10-12 +10-10 +10-9 +10-13 +10-11 +ysThis operator arises by first integrating the right-handed neutrino field NR, which generates +the interaction given in Eq. (4.19), and then at the lower scale the fermion χ, giving rise to +the interaction 9 +ySyφyν +MN1mχ +O6 . +(6.2) +Once H and φ acquire vevs, the decay S → ¯ψ + νL is generated. The decay width reads +Γ(S → ¯ψ + νL) ≈ +|yS|2y2 +φmS +32π +� vφ +mχ +�2 � mν +MN1 +� � +1 − +m2 +ψ +m2 +S +� +. +(6.3) +To guarantee that both S and ψ are cosmologically stable and contribute to the DM +abundance, the lifetime of the heavier particle needs to be larger than the age of the +Universe, τS > tU ∼ 4 × 1017 s. +However, there are stronger constraints if the decay +products include an active neutrino, τS ≳ 1023 s (for a GeV-ish single-component DM) +[55, 56]. If S decays at late times, it leads to a very distinctive signature: a neutrino line +peaked at mS/2 ∼ O(GeV). Therefore, depending on the relative abundance of ψ and +S, some region of the parameter space could be excluded, see Refs. [57–59]. +To quan- +tify this, we compare the experimental bound with the re-scaled lifetime of the particle, +τ re−sc +S += τS (n1DM/nS) > 1023 s, where n1DM (nS) is the single-component DM (S) number +density. This constraint gives a condition on the parameters, +yφ|yS| ≲ 3 × 10−12 +�3 GeV +mS +�1/2 +�700GeV +vφ +� � +mχ +3.5TeV +� �0.05eV +mν +�1/2 � +MN1 +1011GeV +�1/2 � +ΩDM +ΩS(yφ, |yS|, mS, ηD) +�1/2 +, +(6.4) +valid if mS > mψ. In general the condition is not linear in the parameters. If ΩS/ΩDM ≪ 1, +the constraint is extremely weak: DM is made only by ψ and there are no decays. On the +contrary, if ΩS/ΩDM ∼ O(1), the condition becomes linear in the Yukawas. In the opposite +regime, mψ > mS, there is an equivalent condition to that of Eq. (6.4), with mS → mψ +and ΩS → Ωψ. If the two states are almost degenerate mS ≃ mψ there is a strong phase +space suppression so that the constraint gets weaker. +Notice that, even taking into account that the lifetime is re-scaled by the relative +number density, this constraint is much stronger than the one coming from τS > tU, so that +it automatically guarantees the stability of both S and ψ. +The neutrino line emerging from S → ¯ψνL (ψ → S†νL) decay is the main smoking +gun of our scenarios. As an illustrative example, in Fig. 9 we show three different lines +in red, blue and purple corresponding respectively to τ re−sc +S += 1023, 1024, 1025 s, for fixed +max(mS, mψ) = 3 GeV. For this choice of mass, as we can see in Fig. 8, ΩS/ΩDM ∼ +Ωψ/ΩDM ∼ O(1); therefore, Eq. (6.4) gives a simple linear constraint on the Yukawas and +the DM masses. The corresponding region above the red line is excluded. In the region +9Operator O6 may also be generated by a UV completion of our model at scales above vB−L. This +produces additional constraints, weaker than Eq. (6.4). We discuss them in Appendix D. +– 30 – + +between the red and the purple lines the signal is close to the experimental sensitivity and +could lead to the observation of a neutrino line in existing or near-future neutrino telescopes +(see Refs. [67–72]). From the figure we deduce that Scenarios 3 and 5 are therefore the most +testable ones. Notice that, for different choices of max(mS, mψ), the picture can be more +complicated because Eq. (6.4) becomes non-linear. By performing a numerical scan over the +relevant Yukawa and DM mass ranges, we have checked that, as expected, there is always +an excluded region in the upper right corner of Fig. 8, which corresponds to yφ ≳ 10−10, +|yS| ≳ 0.03. +7 +A low-energy variant: The inverse seesaw +So far, we have assumed that active neutrinos get masses through the Type-I seesaw at +a very high scale. In this framework, the lepton and dark asymmetries are generated in +a similar way as in high-scale leptogenesis. +In the following, we discuss the possibility +to embed our models into a low-scale leptogenesis scenario, trying to lower the scale of +B − L of the scenarios considered in this work. This is an alternative path, which may +yield interesting phenomenological signals. In this case, there can be DM interactions with +the SM, mediated by ZB−L. Apart from direct and indirect detection signals, χ may be +produced at colliders via ¯qq → ZB−L → ¯χχ and then decay, χ → ψϕ or χ → S†ν, yielding +missing energy.10 For R ≲ 1, if the mixing of ϕ with the Higgs is in the correct range, +displaced vertices may be produced from ϕ → SM SM. Searches for long-lived particles are +very active, with lots of experiments running or designed for the following years [73–75]. +It has been shown that leptogenesis at a low scale is possible, for example via resonant +leptogenesis, with O(10 TeV) right-handed neutrino masses [76–78]. Another possibility is +to adopt an inverse see-saw (ISS) mechanism to give mass to neutrinos [79, 80]. Next we +consider this option within a B − L set-up, see also Refs. [81–85] and the review [86]. We +can modify our model by replacing the scalar σ with two new fields: a scalar σ′ and three +copies of a fermion SL, with the quantum numbers outlined in Table 6. +Field +Spin +U(1)B−L +U(1)D +U(1)X +SL +1/2 +0 +0 +0 +σ′ +0 ++1 +0 +0 +Table 6. Quantum numbers of the new states in the inverse seesaw variant. +The Lagrangian is the same as Eq. (3.1) (excluding the terms involving σ), with the +addition of +LISS = SLi/∂SL − σ′SLyσ′NR − 1 +2SLµSc +L + H.c. , +(7.1) +where µ is a 3 × 3 complex symmetric matrix which can be taken to be real and diagonal +without loss of generality, and yσ′ is a 3 × 3 complex matrix. If the new scalar takes a vev, +⟨σ′⟩ = vB−L, B − L is spontaneously broken and SL and NR form a pseudo-Dirac pair, +with mass MD = yσ′ vB−L. In this case, neutrinos get a mass through the inverse see-saw +10Notice that there is no ZB−LS†S vertex in the Lagrangian. +– 31 – + +mechanism. We focus on the range MD > mD ≫ µ. The mass of the active neutrinos is +given by +mν ≃ mD M−1 +D µ (M−1 +D )T mT +D , +(7.2) +so that light neutrino masses may be reproduced with small values of MD ∼ vB−L ∼ O +(TeV) for small enough values of µ. +Both low-scale variants result into a massive gauge boson, ZB−L, with a mass in the +5 − 10 TeV range, allowed by experimental constraints [87]. In this case, the larger B − L +gauge interactions allow to erase the symmetric χ population. Therefore, the U(1)D gauge +interactions are not needed, and in the limit gD → 0, U(1)D acts as a global symmetry.11 +Notice that the presence of a global U(1)D is still crucial to stabilise the DM particles and +forbid dangerous operators, such as ¯ψφNR, which would generate ψ−NR mixing. However, +U(1)D could also be replaced by a Z2 symmetry, under which all fields but χ, S and φ are +even. In such a case, the symmetry stabilising the DM particles is a Z′ +2, coming from the +combination of the broken U(1)X and the original Z2. +Figure 10. Parameter space for the low-scale variant with the gauge coupling fixed at 0.5 and +ηD = ηB. In the red region the fractional asymmetry rχ < 10−2. The gray region is excluded +because ZB−L is too light whereas in the purple region ZB−L is lighter than χ. Since we expect +mZB−L ≳ mN1, in the purple region N1 → χS decays are forbidden. +The star indicates the +benchmark point {mχ, mZB−L} = {3.5 TeV, 10 TeV}. +Thanks to the low scale, the annihilations ¯χχ → ZB−L → ¯qq (¯ll) are strong enough to +erase the symmetric population of χ and make the model more testable. The cross section +11In the limit gD → 0, the unbroken U(1)X+D symmetry is now global. The would-be Goldstone boson +of the broken U(1)X−D is now eaten by A′, which becomes massive. +A′ is light because its mass is +proportional to the gauge coupling gX, mA′ ∼ gXvφ ≪ vφ (recall that gX is tiny by assumption to avoid ψ +thermalisation), and it does not thermalise because it has only gauge interactions driven by gX. A′ couples +to the B − L current proportionally to (v2 +φ/v2 +B−L)gX. The interactions of a massive gauge boson are less +constrained than those of a massless one. However, as now vφ ≲ vB−L, one obtains a constraint of the order +of gX ≲ 10−12. +– 32 – + +ND = nB, 9p = 0.5 +105 +104 +103 +X > 7-8zu +102 +101 +102 +103 +104 +101 +mx [GeV]for the ¯χχ → ZB−L → ¯ff process (assuming we are far enough from the resonance, i.e., +4m2 +χ − m2 +ZB−L ≫ ΓZB−LmZB−L), reads +σv(¯χχ → ¯ff) = NC(f)g4 +B−Lq2 +B−L(f) +2π +� +1 − +m2 +f +m2χ +2m2 +χ + m2 +f +(4m2χ − m2 +ZB−L)2 , +(7.3) +where NC(f) = 3 (1) and qB−L(f) = 1/3 (−1) for quarks (leptons). +The different scenarios for DM are equivalent to the ones studied in the previous sec- +tions. However, the decay χ → S†νL is now enhanced (with respect to the previous case) +by the larger value of (mν/MN1). Therefore, we need smaller values of |yS| to realise a +scenario with R ≪ 1. However, too small |yS| may be problematic for the generation of the +dark asymmetry. A precise lower bound is not well established and could be computed in +future works. Assuming |yS| ≳ 10−4, the most natural scenario is that the decays of χ into +S† occur while the latter are still in equilibrium, at T (S) +D +> T (S) +∗ +, (partially) washing-out the +asymmetry ηS. Therefore, if yφ is small enough such that R > 10, χ mostly decay into S† +and ηS gets completely erased. Subsequently S undergoes a standard symmetric freeze-out +and its abundance is determined by the annihilation cross section σvS†S. +If yφ is larger, other scenarios can be realised, such as Scenarios 3, Mixed 3-4 or 5. +Eventually, if both yφ > 1.9 × 10−10� +ηD/ηB and R > 1, a possibility not realised when +vB−L ≫ TeV, a new scenario appears: ψ is symmetric and is produced mainly by freeze-in +with abundance Y + +ψ ≃ Y − +ψ = YFI/(2(1 + R)) while the decays into S† partially washout +ηS, leaving an asymmetric abundance of S, Y + +S = ηD/(1 + R) ≫ Y − +S . This is basically a +generalisation of Scenario 5 with all the abundances rescaled by a 1+R factor. Notice that, +in any case, in order to have multicomponent DM, S must be light enough to satisfy the +neutrino constraints, corresponding to the decay S → ψνL discussed in Section 6, which +get stronger the smaller the vB−L scale. Interestingly, in this case regions of smaller yφ and +|yS| could be probed. +8 +Conclusions +Most probably, the dark sector is very rich, with a plethora of theoretical possibilities re- +garding the number of stable particles, their nature, and their production mechanisms. In +this work we classify and analyse the different options by adopting a cogenesis model that +simultaneously explains neutrino masses, the baryon asymmetry and the DM relic abun- +dance. While neutrino masses and the baryon asymmetry are produced via the standard +Type-I seesaw mechanism and leptogenesis (with some extra contributions), respectively, +we find that in such a framework there is a variety of viable scenarios for explaining the +nature and abundance of dark matter. +Once the decays of right handed neutrinos into the visible and dark sectors generate +the asymmetries, some dark sector particles undergo asymmetric freeze-out and others are +produced via freeze-in. The model has two potential DM candidates, and we focus on the +parameter space where both particles are stable. However, whether they both contribute +to the DM abundance similarly (two-component DM) or one has a negligible contribution +– 33 – + +(one-component DM) depends on their asymptotic asymmetries, where the late decays play +an important role. Such decays may significantly populate the asymmetric or symmetric +component at later times, thereby restoring annihilations, which may lead to enhanced +signals in DM indirect detection. In this case, even though the DM is symmetric at the +end, its abundance is still set by the asymmetry, and is thus independent of the annihilation +rate, contrary to the usual WIMP scenario. +We have analysed the range of model parameters that control the contribution of each +component to the DM abundance, and outline the possible scenarios, classified according +to the nature and production mechanism of each particle. We consider DM masses in the +GeV ballpark and dark asymmetries similar to the baryonic one; however, the set-up can +easily accommodate lighter (heavier) DM for a larger (smaller) dark sector asymmetry if the +branching ratio of right handed neutrinos into the dark sector is smaller (larger). We have +found that one of the main distinctive signatures is a neutrino line from S (or ψ) decays. +This would constitute a smoking gun of our model, within reach of existing or near-future +neutrino telescopes for a significant region of the parameter space of some of the scenarios. +We conclude that having an initial asymmetry in the dark sector does not necessarily +predict completely asymmetric dark matter, with its mass constrained by the dark asym- +metry. In extended models, it allows the DM component to be partially asymmetric or +symmetric, leading to more flexibility regarding the DM mass as well as the phenomeno- +logical implications. Finally, although in this work we focused and extended the cogenesis +scenario, which relates neutrino physics and dark matter, it would be interesting to consider +other frameworks in which the different possibilities outlined here may also be present. +Acknowledgments +This work is supported by the MICIN/AEI (10.13039/501100011033) grants PID2020- +113334GB-I00 and PID2020-113644GB-I00. +GL is supported by the European project +H2020-MSCA-ITN-2019/860881-HIDDeN. JHG and DV are supported by the “Generalitat +Valenciana” through the GenT Excellence Program (CIDEGENT/2020/020). +A +Decays of ϕ to SM particles +We ensure that ϕ particles produced from χ decay fast enough into SM particles, so that +the energy transfer from χ to SM radiation occurs before BBN. A rigorous bound arises +from imposing τχ + τϕ ≲ 1 sec. However, it is sufficient to check that at the time of χ +decays, corresponding to T = TD, the decays of ϕ are fast compared to the Hubble rate +and the bound in Eq. (4.17) applies (corresponding to τχ ≲ 1 sec.). The decay ϕ → SM +can occur through the Higgs portal operator λHφ|φ|2|H|2 as both the scalars take a vev. +Through this portal ϕ decays into SM fermions, mainly (if kinematically allowed) into ¯bb +or ¯cc or lighter species if mϕ < GeV. The decay rate is Γϕ = sin2 θ × Γhϕ→ ¯ff, where sin2 θ +is the mixing among ϕ and the SM Higgs boson h while Γhϕ→ ¯ff ∼ (mϕ/32π)(mf/vEW)2 is +the decay width of a SM Higgs boson with mass mϕ into SM fermions. +– 34 – + +We require that the decay is fast at T = TD (when ϕ are produced via χ decays), +i.e., Γϕ/H|T=TD > 1. +For mχ ∼ TeV, mϕ ∼ few GeV> 2mc (or eventually 2mb) and +yφ ∼ 10−12 the ratio Γϕ/H at T = TD is ≫ 1 even for small mixing angle, sin θ ≳ 10−8. +Even lighter mϕ ≳ MeV is allowed as ϕ can decay to ¯uu, ¯dd, ¯ee, with a larger mixing angle +sin θ > 5 × 10−4. Bounds from LEP constrain the mixing of a GeV-ish scalar to the SM +Higgs to be sin θ < 0.1 [66]. Therefore, the condition of Eq. (4.17) is valid. mϕ lighter than +MeV cannot decay to any SM fermion. However it could decay into 2 photons through the +effective Higgs-photon interactions. We do not study this possibility and we consider mϕ > +MeV. This also implies mS ≳ MeV. +B +Constraints on massless A′ +µ +The massless gauge bosons A′ +µ only interact with the fermion ψ0. Taking into account the +fermion mixing, the Lagrangian contains the interaction terms ¯ψψA′ (suppressed by gX), +¯ψχA′ + H.c. (suppressed by gXϵf) and ¯χχA′ (suppressed by gXϵ2 +f). These vertices give +rise to scattering processes as ¯χχ → A′A′, ¯ψψ → A′A′, ... or decays χ → A′ψ. However, +given the smallness of gX (and ϵf) these processes are extremely suppressed and no sizeable +population of A′ (which in principle could contribute to dark radiation) is produced. +We can give an upper bound on the value of the gauge coupling coming from long- +range force experiments. Indeed the mixing between A′ and ZB−L induces an interaction +geffA′ +µJµ +B−L with geff ≃ (gX/gB−L)(vφ/vB−L)2. For the reference values of this work, vφ ∼ +TeV and vB−L ∼ 1011 GeV, this corresponds to an effective coupling geff ∼ 10−16gX/gB−L. +Long-range force experiments constrain the coupling to the B−L current to be geff ≲ 10−24 +[88], which implies gX ≲ 10−8 (for gB−L ∼ O(1)). This is comparable with the condition +for ψ not to reach thermal equilibrium. +C +Implications of fermion mixing +Fermion mixing induces new interactions due to the ¯ψZDχ coupling, suppressed by ϵf, +leading to new production processes such as ¯χχ → ¯χψ. However, these scattering processes +are typically sub-dominant to decays due to suppression by yφ and ϵf as well as a phase +space suppression (taking ∆ = 1), and thus can be neglected. +In addition to the above, the mixing could also lead to an additional contribution to +ψ production. For T > vφ, the particles in the thermal bath are χ0 and ψ0 and there is no +mixing. For T < vφ, the vev of φ induces the mixing so that the states χ0 contain a small +ψ component, proportional to ϵf. Therefore, χ0 → ψ conversions contribute to the final +ψ abundance (this is analogous to the production of sterile neutrino from active neutrino +mixing). This is also freeze-in process as the population of ψ is produced non-thermally +from a small coupling (the fermion mixing). +In analogy with sterile neutrino DM, the +interaction rate of ψ is Γψ ≃ ϵ2 +fΓχ, with Γχ = nχσ¯χχ. If vφ ≲ mχ the χ particles are +non-relativistic but still in equilibrium when the mixing is generated. So, nχ = neq +χ and +these processes contribute to the symmetric component of ψ. Using the number density +at equilibrium we checked that for the typical values of our parameters (mχ ≃ 3.5 TeV, +– 35 – + +gD ≃ 0.5, vφ ∼ TeV) this contribution is at most comparable (but not larger) than the one +from decays. A detailed study of this contribution, solving the full Boltzmann Equations +(or using the density matrix formalism) is beyond the scope of this work. Therefore, in +the following we neglect this contribution, having in mind that it would change the total +(symmetric) freeze-in contribution by at most an O(1) factor. +D +Contributions to operator O6 +In Section 6, we discussed the stability of the two DM components S and ψ and the possible +observation of a neutrino line from the decay of one component into the other. As we saw, +the decay is mediated by the dimension-6 operator +O6 = ¯L ˜HSφ†ψ, +(D.1) +which is generated at low enegy by first integrating the right-handed neutrino field and +then the fermion χ, as discussed in the main text. A second contribution to O6 may arise +if we assume that the theory contains the dimension-5 operator +O5 = +¯NR(Sφ†)ψ +ΛUV +, +(D.2) +where ΛUV is a cut-off scale parametrising the UV completion of our model at scales above +vB−L. Integrating out NR gives rise to +yν +ΛUVMN1 +O6 . +(D.3) +The condition on the lifetime τ re−sc +S +> 1023 s (taking mS > mψ), is now satisfied if +ΛUV ≳ 1016GeV +� +mν +0.05eV +�1/2 �1011GeV +MN1 +�1/2 � vφ +TeV +� � +mS +50GeV +�1/2 +. +(D.4) +This scale must be at most Planckian, i.e., ΛUV ≲ MPl, which implies a bound on the mass +of the heaviest of the 2 DM particles +mS ≲ 104 TeV +�0.05eV +mν +� � +MN1 +1011GeV +� �700GeV +vφ +�2 +. +(D.5) +Finally, O6 could be directly generated by UV physics as +O6 +Λ +′2 +UV +. +(D.6) +In this case we just need +Λ′ +UV ≳ 1015 GeV +� vφ +TeV +�1/2 � +mS +50GeV +�1/4 +, +(D.7) +which gives a weaker constraint. Notice that the same bounds apply on mψ if mψ > mS. +Summarising, Eqs. (6.4) and (D.5) set an upper bound on the mass of the heavier between +ψ and S. 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B 739 (2014) 256 [1408.6845]. +– 41 – + diff --git a/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/load_file.txt b/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6c39622703dcbdc9ee2229c11bf5e97b7a6fe80 --- /dev/null +++ b/1dFQT4oBgHgl3EQfEjV1/content/tmp_files/load_file.txt @@ -0,0 +1,1811 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf,len=1810 +page_content='Prepared for submission to JHEP IFIC/23-03 Asymmetries in Extended Dark Sectors: A Cogenesis Scenario Juan Herrero-García , Giacomo Landini and Drona Vatsyayan Departamento de Física Teórica, Universidad de Valencia and IFIC, Universidad de Valencia- CSIC, C/ Catedrático José Beltrán, 2 | E-46980 Paterna, Spain E-mail: juan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='herrero@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='es, giacomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='landini@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='es, drona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='vatsyayan@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='es Abstract: The observed dark matter relic abundance may be explained by different mech- anisms, such as thermal freeze-out/freeze-in, with one or more symmetric/asymmetric com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this work we investigate the role played by asymmetries in determining the yield and nature of dark matter in scenarios with more than one dark matter particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In particular, we show that the energy density of a particle may come from an asymmetry, even if the particle is asymptotically symmetric by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' To illustrate the different effects of asymmetries we adopt a model with two dark matter components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We embed it in a co- genesis scenario that is also able to reproduce neutrino masses and the baryon asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The framework predicts a monochromatic neutrino line for some of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='13238v1 [hep-ph] 30 Jan 2023 Contents 1 Introduction 2 2 General framework 3 3 A model for neutrino masses, the baryon asymmetry, and dark matter 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 The scalar sector and spontaneous symmetry breaking 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 The gauge sector 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 The fermionic sector 10 4 Dark matter components 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Asymmetric dark matter via cogenesis 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Contribution of ψ 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Production of ψ from χ decays 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Constraints 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Contribution of S 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Production of S from freeze-out 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Production of S from χ decays 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Constraints 19 5 Dark matter relic abundance 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Scenario 1: ψ-LD-A + S-FO-A 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Scenario 2: ψ-LD-A + S-FOLD-PA 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Mixed Scenario 1-2: ψ-LD-A + S-FOLD-A 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 Scenario 3: ψ-FILD-PA + S-FO-A 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 Scenario 4: ψ-FILD-PA + S-FOLD-PA 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6 Mixed Scenario 3-4: ψ-FILD-PA + S-FOLD-A 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7 Scenario 5: ψ-FI-S + S-FO-A 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8 Scenario 6: S-FOLD-S 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 Summary of the scenarios 26 6 Phenomenological signals 28 7 A low-energy variant: The inverse seesaw 31 8 Conclusions 33 A Decays of ϕ to SM particles 34 B Constraints on massless A′ µ 35 C Implications of fermion mixing 35 – 1 – D Contributions to operator O6 36 1 Introduction The nature of Dark Matter (DM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' that makes up roughly a quarter of the energy density of our universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' along with the origin of neutrino masses and the baryon asymmetry (BAU),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' are among the most important open problems that the Standard Model (SM) fails to ex- plain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' with overwhelming experimental evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Several extensions of the SM have been proposed, where either a single particle or several stable particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', multi-component DM [1–9], make up the observed DM relic abundance, ΩDMh2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The most popular mechanisms to reproduce this relic abundance include thermal freeze-out (FO) of weakly in- teracting massive particles (WIMPs) [11–16] and freeze-in (FI) of feebly interacting massive particles (FIMPs) [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' WIMPs are initially in thermal equilibrium with the SM and undergo annihilations until they freeze-out when the annihilation rate drops below the Hubble expansion rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' therefore, the DM abundance is inversely proportional to the annihilation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the other hand, FIMPs have a negligible initial abundance and are produced mainly by tiny interactions with particles in the thermal bath so that they never thermalise, and they freeze-in once the mother particle decouples from the bath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' thus, the DM abundance is directly proportional to the production rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Depending on the interactions, a further contribution may come from late decays (LD) of the mother particle, the size of which is model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Most of these models assume that the new states are symmetric in nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', the abundance of the DM particle is the same as that of the antiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, there exist a wide variety of models that propose that the DM abundance is rather set by an initial asymmetry in the dark sector, in analogy to the visible sector (where ηB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='88 × 10−11), motivated by the closeness of baryonic and DM energy densities, ρDM ∼ 5ρB [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The asymmetry can be first generated in the visible sector and then transferred to dark sector (or vice versa) [21, 22], or an asymmetry can be generated simultaneously in both the sectors (cogenesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Such asymmetric dark matter (ADM) models (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [23, 24] for a review) aim to explain the observed baryon asymmetry and DM abundance in a common framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' An intermediate scenario between the two extremes involves the asymmetric freeze-out of a species, where the DM particle and its antiparticle freeze-out with different number densities, depending on the initial asymmetry, and the dark matter is partially asymmetric [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, a further distinction can be made regarding the nature of DM, whether it is symmetric, asymmetric or partially asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the presence of an asymmetry can have significant implications for the mechanisms discussed above in reproducing the DM abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The possibility that asymmetric single-component WIMP DM is produced in a cogen- esis scenario has been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Another model has been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [27], – 2 – with the possibility to restore the symmetric nature of DM through late decays of an extra particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [28] a model of symmetric multi-component DM, which combines freeze-out and freeze-in, accompanied by the generation of the baryon asymmetry, has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this work, we aim to generalize this picture, starting from the concrete cogenesis scenario realized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [27], by considering an extended dark sector in which we can re- alize multicomponent DM, so that one DM component, or even both, can be asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Furthermore, we combine freeze-out and freeze-in production, including the possibility of asymmetric freeze-in [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' To this end, we propose a model in which the dark sector is connected to the visible sector via a mediator particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The asymmetries in both sectors are generated via cogenesis, which yields a relation between neutrino masses, the baryon asymmetry and the DM relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We then investigate the role of the dark sec- tor asymmetry in determining the relic abundance of one/several particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For example, depending on the model parameters, the DM may either be single-component or multi- component, with either all symmetric or asymmetric components or a mixture of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Similarly, the production mechanisms may be freeze-out, freeze-in or some via freeze-out and the others via freeze-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In particular, we explore the possibility of asymmetric freeze-in, which has been over- looked in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Unlike the case of asymmetric freeze-out, it is not so straightforward and requires the presence of a richer dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Moreover, we show that in certain sce- narios the particle abundance may be set by an asymmetry even if its nature is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We find that some of them predict the existence of an observable neutrino line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Section 2, we elaborate on the framework of generating an asymmetry and the possibility of asymmetric freeze-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We discuss the com- plete model, which also generates neutrino masses and the baryon asymmetry, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The different DM candidates are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The contribution to the DM relic abundance is studied in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In section 6, we discuss the main phenomenological signatures, with special emphasis on the prediction of a monochromatic neutrino line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Section 7, we discuss a low scale variant of the model (an inverse seesaw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, we give our conclusions in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We show further details in some appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2 General framework In order to generate a dark sector asymmetry, we focus on the cogenesis of both DM and baryon asymmetries and adopt the two-sector thermal leptogenesis mechanism of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, the asymmetries are produced from out-of-equilibrium CP-violating decays of right handed neutrinos (RHNs), which are well-motivated to reproduce neutrino masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This requires the leptons and some of the dark sector particles to be charged under a lepton symmetry that is broken by Majorana masses of the RHNs, thus satisfying all the conditions for the dynamical generation of an asymmetry [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The RHNs are initially in thermal equilibrium, and once the temperature drops below the mass of the lightest RHN, MN1, the washout and other interactions leading to transfer of asymmetries between the two sectors become inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Subsequently, the asymmetries get frozen in the two sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The – 3 – leptonic asymmetry is partially converted into the baryonic one via sphalaeron processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The dark asymmetry (ηi ≡ Y + i − Y − i , where we introduce the yield as the number density upon entropy density, Yi = ni/s) is carried by a dark fermion (we denote it by χ) and once its symmetric component gets annihilated away, the asymmetric one sets the relic abundance, ΩDM ∝ ηχmχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This framework, therefore, connects neutrino mass generation with the baryon asymmetry and the DM relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' See Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [34–41] for other extensions of seesaw framework that address the three issues under the same umbrella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In our work, we go a step further and enlarge the dark sector so that the different possibilities discussed in the introduction are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For this purpose, the particle χ in our set-up is not the dark matter but rather decays to another stable fermion (say ψ), which may constitute all or part of the DM abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The schematic framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the asymmetry in χ can be transferred to ψ via its decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Indeed, the N L H χ S ψ φ yν yS yφ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Schematic framework of cogenesis and DM production in the model at T < MN1 via the indicated Yukawa interactions yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Here, L and H are the SM lepton and Higgs doublet, respectively, whereas S and φ are complex scalars belonging to the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' set-up offers a richer phenomenology as it is now possible to accommodate multi-component DM (in this case, it could be ψ and S), as well as different dynamics thanks to the presence and size of the different interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In principle, we can have four cases of equilibration between different sectors, as illustrated also in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2, we show the four cases of entropy transfer between the various sectors: the SM + NR (green), χ, S, φ (orange) and ψ (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 The first case (EE) involves equilibration among all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' It is the multi-component freeze-out scenario that has been widely considered (see for instance Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [1]), and we will not consider it here in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Cases 2 and 3, FF and FE, respectively, correspond to scenarios where χ is not in equilibrium with the SM + NR sector, which implies that yS ≪ 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the considered framework where the asymmetries are produced from the decays of heavy right-handed neutrinos, such a small Yukawa coupling would then be unable to generate a dark asymmetry comparable to the visible one, and therefore DM would have to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, we are interested in the last case (EF), where: i) the orange sector is in equilibrium with the SM + NR, 1In the scenarios considered below, φ, even if it comes from χ decays (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1), is included in the middle blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 4 – EE: χ, S φ SM +NR ψ Equilibrium Equilibrium FF: χ, S φ SM +NR ψ Freeze-in Freeze-in FE: χ, S φ SM +NR ψ Equilibrium Freeze-in EF: χ, S φ SM +NR ψ Equilibrium Freeze-in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The four scenarios for entropy transfer between the SM sector and the dark sectors formed by χ, S and ψ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Similar figure in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' ii) the dark sector asymmetry may be generated via co-genesis, of size comparable to the baryonic one, and iii) the ψ sector is not in equilibrium and is produced via freeze-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Whether the abundance produced by freeze-in is asymmetric depends on the value of Yukawa yφ, because the ψ population can be symmetric even if the mother particle χ carries an asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Let us define xi ≡ mi/T, for a species of mass mi, where T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The asymmetry freezes out at xi ∼ 20, whereas the freeze-in from early decays takes place around the mass of the mother particle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', xi ∼ 1, when both the mother particle and its antiparticle are in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' So, if the production from early decays is greater than the asymmetric yield after freeze-out, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Yψ = Y ¯ψ > Yχ ≈ ηD, then the daughter particle will be symmetric in nature, because the production from late-decays (that take place at xi ≫ 1) will be sub-dominant and negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, if the production from early decays is smaller than the asymmetry, Yψ < ηD, then the late decays that are active much later after the asymmetry has frozen-in will produce more ψ than ¯ψ, hence generating an asymmetry in ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Realising this last scenario therefore yields an example of asymmetric freeze-in, which up to our knowledge has not been studied in the literature2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As we will see, in certain scenarios, the late decays of an asymmetric particle may populate the symmetric component of a species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, the late decays play an important role in determining the final nature of a species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In order to distinguish between the scenarios, we use the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [25] to define the asymmetric ratio for a particle species as ri ≡ Y − i /Y + i with 0 < ri ≤ 1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) where +(−) denotes the particle (antiparticle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Here, the upper (lower) limit in r signi- fies that the species is completely symmetric (asymmetric) and ΩDM ∝ Y + i + Y − i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The 2In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [43], the contributions from early and late decays have been compared for a symmetric DM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 5 – asymptotic asymmetric ratio of a species i with mass mi can be written as [25] r(∞) i ≃ exp � − � πg∗ 45xf MPl ⟨σv⟩i ηD mi � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) where ⟨σv⟩i is the thermally-averaged annihilation cross section, MPl ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2×1019 GeV, mp is the mass of the proton and xf ≡ mi/T∗ ∼ 20 is the mass over freeze-out temperature ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following, we use r(∞) i ≡ ri to alleviate notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [1], different regimes appear: For ri < 10−2, the behaviour of DM is highly asymmetric (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the range, 10−2 < ri < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9, DM behaves as partially asymmetric (PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For ri > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9, DM is highly symmetric (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the rest of the paper we adopt these ranges to define the nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Further classi- fication can be made on the basis of the dominant production mechanism that determines the asymptotic nature of the dark matter, be it freeze-out (FO), early decays from freeze-in (FI), or late decays (LD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This opens up a plethora of possibilities, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' DM FO FI LD S PA A S PA A A PA S Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Possible production and final nature of dark matter components in a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Scenario Symmetric Partially Asymmetric Asymmetric (r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9) (10−2 < r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9) (r < 10−2) Freeze-out FO-S FO-PA FO-A Freeze-in FI-S FI-PA FI-A Late decays LD-S LD-PA LD-A Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Classification schemes corresponding to the behaviour and dominant production mech- anism of DM in a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We characterise the multiple combinations for the nature of one-component DM as per the name outlined in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' When multiple components are present, we prefix the name of the component to the scheme, for example, the scenario X-FI-S+Y -FO-A corresponds to the case of two DM components, X and Y , where X is symmetric and produced via freeze- in whereas Y is asymmetric and freezes out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this work, we aim to focus exclusively – 6 – on scenarios where the asymmetry is directly involved in reproducing the relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For this goal, we propose a complete model that displays the different roles played by the asymmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3 A model for neutrino masses, the baryon asymmetry, and dark matter Our initial hypothesis for the construction of a model that explains dark matter, neutrino masses and the baryon asymmetry in a common framework is that the symmetry baryon minus lepton number, B−L (under which all SM quarks have charge 1/3 and all SM leptons have charge -1), plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is one of the best-motivated symmetries beyond the SM: it is accidental and anomaly-free in the SM, and when gauged it requires the presence of three sterile neutrinos (which in turn naturally generate active neutrino masses) and is easily embedded in GUTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' When considering a complex asymmetric dark sector, however, this symmetry is not enough, and we require extra U(1)s to forbid Majorana masses and some interaction terms, as well as to annihilate the symmetric components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, we augment the SM gauge group by 3 new gauge U(1) symmetries: U(1)B−L , and the dark product U(1)D ⊗ U(1)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We add three right-handed neutrinos NR (RHNs) to cancel the gauge anomalies of the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We also extend the particle content by the addition of two Dirac dark fermions, ψ0 and χ0, and 3 scalars σ, S and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gauge bosons associated with the three new groups are Z0 B−L, Z0 D, A ′0, with gauge couplings gB−L, gD and gX, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The quantum numbers of the new fields are summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note that all the new fields are SM singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The new part of the Lagrangian of the model can thus be written as Lnew = L0 χψ + L0 kin + Lint − V (σ, S, φ, H) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) where L0 kin includes the kinetic terms of the gauge bosons and the scalars, V (σ, S, φ, H) is the most general scalar potential that one can write given the symmetries of the model, where H is the SM Higgs doublet field, and L0 χψ = ¯χ0(i /D − m0 χ)χ0 + ¯ψ0(i /D − m0 ψ)ψ0, Lint = −yαi ν ¯Lα ˜HN i R − yij σ σNic R Nj R − yi SS ¯Ni Rχ0 − yφφ ¯ψ0χ0 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) Here m0 χ,ψ are bare mass terms of the dark fermions, while the indices α = e, µ, τ and i = 1, 2, 3 run over the generations of leptons and right handed neutrinos, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We use ˜H = iσ2H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note that yν is a 3 × 3 general complex matrix, yσ is a 3 × 3 complex symmetric matrix, yS is 3-component vector and yφ is a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, several phases are unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Four phases of yσ may be removed by rephasing σ and Ni R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Two phases of yS may be removed by rephasing S and χ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' yφ can be taken real by rephasing φ or ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In accordance with the framework discussed above, we take yφ ≪ , 1 gX ≪ 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) so that ψ0 cannot thermalise with the SM bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We further assume a vanishing initial abundance nψ = 0, consistent with an inflationary epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the other hand, the particles – 7 – Field Spin U(1)B−L U(1)D U(1)X Ni R 1/2 1 0 0 σ 0 +2 0 0 χ0 1/2 1 1 0 ψ0 1/2 0 0 +1 S 0 0 1 0 φ 0 +1 1 +1 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Particle content of the model and their respective charge assignments under Lorentz and the U(1) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The first two states correspond to the sterile neutrino sector, and the last four to the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' χ0, σ, S, φ and Ni R reach thermal equilibrium with the SM thermal bath through sizeable new gauge (gB−L and gD) and scalar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note that the Lagrangian also contains the kinetic mixing between the U(1) gauge factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' An unavoidable contribution to kinetic mixing among all U(1)s arises at one-loop level with φ running in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Also, χ contributes in the case of U(1)B−L − U(1)D mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, the kinetic mixing −(κ/2)Zµν B−LZD,µν is naturally of the order of κ ≳ gDgB−L/(16π2) ∼ 10−3 gB−L gD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The kinetic mixing of U(1)X with the other U(1) factors is ≳ gXgi/(16π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Since gX ≪ 1 by assumption, this contribution can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, a kinetic mixing among ZB−L and the SM U(1)Y gauge boson is generated through a loop of SM quarks and leptons, of the order ≳ gY gB−L/(16π2) � i=q,l Yi(B − L)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that, given the conservation of U(1)em, the photon does not couple to the B − L current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In any case, the bounds on the kinetic mixing are not relevant for the range of parameters that we consider in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, let us remark that the necessary ingredients for the model to work could also be achieved with a global U(1)B−L [16, 44], explicitly violated by right-handed neutrino masses, since: i) χ is Dirac in nature because of the gauge U(1)D, ii) it has sizeable interactions with the SM (with NR, yS) to thermalise, and iii) it can undergo efficient annihilations due to the U(1)D gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, such scenario is not as theoretically appealing as the gauged B − L version that we consider, which demands the existence of 3 right-handed neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following subsections, we discuss the scalar, gauge, fermionic and dark sectors in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 The scalar sector and spontaneous symmetry breaking The details of the full scalar potential V (σ, S, φ, H) are quite involved as one can write quadratic, quartic and mixed quartic terms for each combination of the scalar fields σ, S, φ and the SM Higgs doublet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, without entering into the details of the scalar potential, we can safely assume that there is a region of the parameter space in which σ – 8 – takes a large vev, ⟨σ⟩ = vB−L ≳ 1011 GeV, which breaks the U(1)B−L symmetry by 2 units and generates Majorana masses for the sterile neutrinos as well as a mass for the U(1)B−L gauge boson, ZB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The value vB−L ≳ 1011 GeV is chosen so that the lightest RHN mass (MN1) safely obeys the equivalent Davidson-Ibarra lower bound [45] to achieve thermal Leptogenesis in the model [27], but any larger value will not change the following analysis and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, an asymmetry can be generated once the inverse decays of N1 go out of equilibrium [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note that we take MZB−L > MN1 and mσ > MN1, so that the heavy ZB−L gauge boson and the radial component of σ are therefore naturally very heavy and decay fast into quark and leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The other scalar, φ, takes a much smaller vev than that of σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ⟨φ⟩ = vφ ≪ vB−L, which breaks U(1)D ⊗ U(1)X → U(1)X+D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' After symmetry breaking, we can write φ(x) = vφ+ϕ(x)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The overall symmetry breaking pattern of the model can then be represented as U(1)B−L ⊗ U(1)D ⊗ U(1)X ⟨σ⟩ −→ U(1)D ⊗ U(1)X ⟨φ⟩ −→ U(1)X+D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) The only fields that are charged under the unbroken U(1)X+D symmetry are the fermions χ0, ψ0 with charge +1 and the scalar S with charge -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We discuss in Section 4 the conse- quences of this for the DM stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A phenomenologically relevant parameter is the mixing between the SM Higgs boson h (coming from H = (vEW + h)/ √ 2) and the scalar ϕ, which can be characterised by the mixing angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is directly related to the mixed quartic term λHφφ†φH†H in the scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the rest of the paper, we assume that this mixing angle is small (sin θ ≃ θ ≪ 1) so we can safely trade h and ϕ for the mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Finally, the scalar S does not obtain a vev, ⟨S⟩ = 0, and may therefore be a DM candidate in the model because of its charge under U(1)X+D (see discussion at the beginning of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We are interested in the regime MN3, MN2 ≫ MN1 ≫ m0 χ ≫ m0 ψ, mS > mφ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5) so that the decay channels shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1 are kinematically open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We consider values of m0 ψ and mS of similar order of magnitude, in the GeV ballpark, because we are interested in scenarios where both may contribute significantly to the DM relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 The gauge sector The symmetry breaking pattern outlined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 leads to one massless (two massive) gauge boson(s), which correspond to the unbroken (broken) generator(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The masses are given by � � � � � � � m2 A′ = 0 , m2 ZD = 2g2 Dv2 φ � 1 + O � g2 X, (vφ/vB−L)2 , κ2�� , m2 ZB−L = 8g2 B−Lv2 B−L � 1 + O � g2 X, (vφ/vB−L)2 , κ2�� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6) 3In Appendix A, we briefly discuss the mixing when studying the decays of ϕ to SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We show that even a very small mixing angle allows efficient decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 9 – where A′, ZD, ZB−L are the mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Since gX ≪ 1, κ ≪ 1 and vφ ≪ vB−L, their contribution to the masses can be neglected in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The tiny values of the parameters also suppress the mixing between the gauge bosons so that the mass eigenstates mostly coincide with the original eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The mixing among them at the leading order in the small expansion parameters (omitting Lorentz indices) can be expressed as � � � � � � � A ′0 ≃ A ′ − gX/gD ZD , Z0 D ≃ ZD + gX/gD A ′ − � (gD/4gB−L)(v2 φ/v2 B−L) + κ � ZB−L , Z0 B−L ≃ ZB−L + (gD/4gB−L)(v2 φ/v2 B−L)(1 + κgB−L/gD)Z ′ D − (gX/4gB−L)(v2 φ/v2 B−L)A′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The massless A′ µ is decoupled from all the other fields (as gX ≪ 1) and does not play any role in the following discussion (see Appendix B for a discussion on the bounds on a massless dark gauge boson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The mass/kinetic mixing among ZD and ZB−L induces an interaction of the type Zµ DJB−L µ , where JB−L µ is the B − L current,4 which may lead to decays of ZD into SM quarks and leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, the decay width is suppressed by (vφ/vB−L)4 and it is subleading with respect to other decay channels: If mZD > 2mχ, the gauge boson decays into ¯χχ or S†S at tree level, or into ϕϕ at one-loop level through a loop of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This last process depends on the coupling between ϕ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As we will show in Appendix A, the particles ϕ must have a fast decay to SM fermions, so this gives a lower bound on their mass, and therefore on the mass of ZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If 2mS < mZD < 2mχ, the gauge boson decays mainly as ZD → S†S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decays have width ∼ O(g2 DmZD/100) and are very fast in the relevant temperature regime (T < vφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 If 2mS > mZD > 2mϕ, the gauge boson can only decay into 2ϕ, see the first point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the rest of the paper we focus on the second scenario, namely 2mS < mZD < 2mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 The fermionic sector We first discuss the generation of tiny neutrino masses via the type-I see-saw mechanism [49–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The breaking of U(1)B−L gives masses to the heavy RHNs, MNi ∼ yi σ vB−L ≳ 1011 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following we use MN1 ≲ vB−L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', we take y1 σ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The masses for the active neutrinos are given by the seesaw expression, mν = −mD M−1 N mT D , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7) with mD = yνvEW/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For the dark fermions, once φ obtains a vev, a mixing is induced between χ0 and ψ0 due to the Yukawa coupling yφ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We define the fermion mixing parameter as ϵf ≡ yφvφ m0χ − m0 ψ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8) 4Note that in the absence of the mass mixing there would not be such an interaction even for κ ̸= 0 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5We ignore the possibility mZD ≳ 2mS in which the phase space of the decay closes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 10 – where yφ ≪ 1 and typically vφ < m0 χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Thus, the mixing parameter is highly suppressed, ϵf ≪ 1 (we remind that we are assuming hierarchical masses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', m0 χ ≫ m0 ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Diagonalising the fermionic sector at leading order in ϵf leads to the following masses for the the mass- eigenstates χ and ψ, � mψ = m0 ψ − ϵ2 f(m0 χ − m0 ψ) + O(ϵ3 f) , mχ = m0 χ + ϵ2 f(m0 χ − m0 ψ) + O(ϵ3 f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9) The fields in the mass basis are related to the original fields as � ψ = (1 − ϵ2 f/2) ψ0 − ϵf χ0 + O(ϵ3 f) , χ = (1 − ϵ2 f/2) χ0 + ϵf ψ0 + O(ϵ3 f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='10) As the mixing is very suppressed, ϵf ≪ 1, in the following we drop the subscripts 0, trading the original masses/fields for the physical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, let us mention that no Majorana masses for χ or ψ are generated due to the preserved gauge U(1)X+D, due to the fact that S does not take a vev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Higher dimensional operators may be written at dimension 6 and 8 for χ and ψ, respectively, Oχ = χcχσSS , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='11) Oψ = ψcψσSSφ†φ† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='12) However, as S does not take a vev, these operators do not generate Majorana masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4 Dark matter components Recall that the only fields that are charged under the remnant U(1)X+D symmetry are χ, ψ (with charge +1) and S (with charge −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As mχ ≫ mψ, mS, only ψ, S may be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The X + D charge is preserved in decays of the type ψ → S† + P or the opposite S → ¯ψ + P, where P is some uncharged state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, in principle the lightest among ψ and S is stable and would be the only DM candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, the decay ψ → S† + P (or the opposite) is suppressed by the masses of the right-handed neutrino N1 and χ in the propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' At low energies, E ≪ mχ ≪ MN1, one can integrate out both the particles and study the decay in terms of higher-dimensional operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As we analyse below in Section 6, P corresponds to active neutrinos in the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', P = ν, yielding a monochromatic neutrino line from ψ decays [55–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In that section we show that in a broad region of the parameter space, the decays are suppressed on cosmological timescales and obey current limits, and therefore both particles (ψ and S) contribute significantly to the DM relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The scenario in which the decay is fast enough and there is only 1 DM candidate has been studied extensively in the literature, so we will not discuss it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, we focus on the two-DM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The conservation at low energies of the U(1)X+D symmetry yields the constraint 0 = QX+D = ηψQψ + ηSQS =⇒ ηψ = ηS , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) where in the last step we used that the U(1)X+D charges are given by Qψ = −QS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' First, we discuss the generation of the asymmetry and then the DM production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 11 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Asymmetric dark matter via cogenesis In the following, we assume a hierarchical scenario MN2,3 ≫ MN1, which in turn implies y2,3 σ ≫ y1 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The simultaneous generation of lepton and dark sector asymmetries takes place at a high scale (T ∼ MN1) via the decays of the lightest RHN, N1, into the two channels (the asymmetry generated in the decays of N2,3 are washed out by N1 interactions), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' N1 → LH generates a lepton asymmetry, ηL, which is later reprocessed into a baryon asymmetry ηB by sphalerons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is the case of Type-I thermal leptogenesis, well studied in the literature (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [60] for a review), but with extra contributions, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' N1 → χS generates an asymmetry in the dark sector, ηD, analogous to the lepton asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Here, we focus on the regime where the asymmetry generated by N1 decays into S is not washed out, and is comparable to the asymmetry in χ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ηχ ∼ ηS ∼ ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In order to generate an asymmetry, CP needs to violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The CP asymmetry generated in the decays of N1 can be written as εL = � α ΓN1→LαH − ΓN1→¯LαH† ΓN1 , εχ = ΓN1→χS − ΓN1→¯χS† ΓN1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) where ΓN1 = ((yνy† ν)11 + |y1 S|2)MN1/(8π) is the total tree-level decay width of N1 and y1 S is the relevant Yukawa coupling for N1 → χ + S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following we refer to it as yS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, due to CPT invariance, no asymmetry can be generated at the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' CP violation arises via the interference of tree-level and one-loop level decay (vertex and self-energy corrections) amplitudes, which depends on the imaginary part of the product of the Yukawas involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, for this asymmetry to be non-zero, we require at least two distinct phases that come from the Yukawas yν and yS, so the couplings to the heavier neutrinos N2,3 are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Further, having an imaginary part in the internal loop contribution demands that the would-be decay products can be produced on-shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', the optical theorem), which is easily satisfied as we take MN1 ≫ mχ,S,L,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this cogenesis scenario, it can be seen that both εL and εχ depend on yν and yS, as the dark sector particles (χ, S) are involved in the one-loop self-energy correction for N1 → LH and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Thus, the ratio of the decay asymmetries depends on the ratio of the couplings yν and yS and may be correlated with the branching ratio of N1 decay in each sector (BrL and Brχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This relates neutrino mass generation to the baryon and DM abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [27] it has been shown that the dark asymmetry ηD can be quite different from the visible one ηB (contrary to the ADM models that predict ηB ∼ ηD) due to different branching ratios, washout effects and transfer between the sectors via inverse decays and 2 ↔ 2 scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The asymptotic asymmetries for the two sectors can be written as η∞ L = εL ξL Y eq N1(T ≪ MN1) ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6 × 10−10 , η∞ χ = εχ ξχ Y eq N1(T ≪ MN1) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) – 12 – where ξL (ξχ) is the leptonic (dark) efficiency parameter characterising the effects of washout and transfer interactions, and Y eq N1(T ≪ MN1) = 135ζ(3)/4π4g∗ is the initial equilibrium N1 yield, with g∗ ≃ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='75, the number of relativistic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The numerical value of η∞ L is selected to match the observed baryon asymmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ηB = (28/79) ηL, generated via sphaleron processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note that the presence of N − ZB−L interactions in the model may modify the usual picture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' however, the correct value of the asymmetry may be generated [46, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Typically, producing the DM relic abundance in ADM models constrains the DM mass, depending on the ratio ηD/ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, since χ is not the DM candidate in our set-up, it is not constrained to be of the order of few GeVs (for ηD ∼ ηB) and thus may be much larger, ∼ O(TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the analysis below, we consider dark asymmetries in the range ηD ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 − 1) ηB, which may be achieved if BrL ∼ Brχ and the Yukawa couplings of both sectors have similar hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Thus, we work with DM masses in the GeV ballpark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We refer the reader to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 of Ref [27], where it is shown the order of the asymmetries that can be obtained for a given value of MN1 and different branching ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A key feature of any ADM model is the presence of an interaction that efficiently depletes the symmetric component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In our set-up, in order for the DM component ψ to be asymmetric, we require that the symmetric population of χ is annihilated and only the asymmetric component survives before it decays to ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For example, this can take place via annihilations of the form ¯χχ → ZDZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the non-relativistic limit, s ≃ 4m2 χ, the thermally-averaged cross section (considering s-wave annihilations) for mχ > mZD is given by σv(¯χχ → ZDZD) = g4 D 16πm2χ � 1 − m2 ZD m2χ �3/2 � 1 − m2 ZD 2m2χ �−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) For mχ > mS, an extra contribution comes from annihilations of the form ¯χχ → ZD → S†S, which opens up the region of parameter space where mχ < mZD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' It should be noted that there is another contribution to annihilations from the channel ¯χχ → ZB−L → ¯qq (¯ll), where q (l) is a SM quark (lepton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, given the large mass of ZB−L, this turns out to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4 we show the mass ranges of ZD and χ where the latter is asymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', rχ < 10−2, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' One observes how, for larger values of the asymmetry, a larger region of the parameter space has rχ < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note also the presence of the resonance for mZD ≃ 2mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Contribution of ψ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Production of ψ from χ decays First, we study the dynamics of ψ, neglecting the contribution to the relic abundance from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We assume that the ¯χχ annihilation processes studied in the previous section are efficient enough so that rχ < 10−2 and the freeze-out abundance of χ is determined by its asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The production of ψ is driven by decays of χ: χ → ψφ for T > vφ and χ → ψϕ for T < vφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As the h − ϕ mixing is small (θ ≪ 1), it is safe to trade ϕ, h for the mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The production via decays can be divided into two types: – 13 – Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The red regions indicate the regions of mχ, mZD (left) and mS, mϕ (right) where the annihilations of χ, S respectively are strong enough to result in a fractional asymmetry rχ (S) < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Left: The star represents the benchmark point {mχ, mZD} = {3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, 500 GeV}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the gray region the annihilation channel ¯χχ → ZDZD is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Right: In the gray region, annihilations to ϕ are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' While χ is in thermal equilibrium with the SM bath (T > T∗, T∗ ≃ mχ/20 being the freeze-out temperature of χ,), the decays are symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ψ and ¯ψ are produced in equal amounts from the decays of χ and ¯χ, that are symmetric during this period, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Y + χ ≃ Y − χ = Y eq χ ≫ ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is the usual freeze-in contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We also refer to these processes as early decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We denote the abundance of ψ particles produced by freeze-in by YFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This symmetric production peaks around T ∼ mχ > vφ, so that the channel is χ → ψφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Once χ freezes out (T < T∗), the population of χ becomes asymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Y + χ ≃ ηD ≫ Y − χ ≃ rχY + χ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Such an asymmetry is then subsequently transferred to ψ via decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is the asymmetric freeze-in contribution of ψ from late decays (LD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Since the decays occur late, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e TD ≪ vφ, in this case the channel is χ → ψϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The populations are given by Y + LD = ηD 1 − rχ ≃ ηD , Y − LD = rχY + LD ≃ ηD rχ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5) where in the last step we used rχ < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the asymptotic abundance of ψ and ¯ψ can be written as Yψ ≡ Y + ψ = YFI 2 + Y + LD ≃ YFI 2 + ηD, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6) Y ¯ψ ≡ Y − ψ = YFI 2 + Y − LD ≃ YFI 2 + ηD rχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' where in the last step we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The three interesting cases for the asymmetric ratio are rψ ≃ � � � � � � � rχ if ηDrχ ≫ YFI , YFI/(2ηD) if ηDrχ ≪ YFI ≪ ηD , 1 if YFI ≫ ηD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7) – 14 – 105 9p= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 104 103 ND = NB 102 ND = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 NB 101 102 103 104 105 101 mx [GeV]101 np = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 nB, mzp = 500 GeV ms mp [GeV] 100 Λsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05 Λs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='01 Λss = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='001 10-1 101 102 100 ms [GeV]We remind that, according to our definition in Section 2, DM is asymmetric if rψ < 10−2, partially asymmetric if 10−2 < rψ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 and symmetric if rψ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, in order for ψ to be asymmetric, not only we need to require that rχ ≪ 1 but also the symmetric freeze-in contribution to the abundance, YFI/2, should be suppressed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', YFI ≪ ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the opposite regime, where the freeze-in production from early decays is dominant, the final ψ abundance is always symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decay width is given by Γχ→ψϕ ≃ y2 φmχ 32π ∆2(fψ, fϕ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8) where fψ ≡ mψ/mχ and fϕ ≡ mϕ/mχ, and ∆(fψ, fϕ) is the phase space suppression factor, ∆2(fψ, fϕ) = � 1 − f2 ψ − f2 ϕ + 2fψ �2 �� 1 − f2 ψ − f2 ϕ �2 − 4f2 ψf2 ϕ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9) with 0 ≤ ∆(fψ, fϕ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We have neglected the small corrections due to fermion mixing ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' There is an analogous expression for ϕ → φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Here onwards, we omit the arguments of the function ∆ and we consider values of the parameters such that mχ ≫ mψ, mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Within this approximation ∆ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Furthermore, in this limit the width is practically independent of mϕ or mφ, so that in the following we do not differentiate among decays into ψϕ and ψφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that the Yukawa yφ also leads to annihilation processes (such as ¯χχ → ¯χψ) which would contribute to ψ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, as long as ∆ ≃ 1, these are subleading with respect to decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The freeze-in contribution from early decays can be computed numerically by solving the Boltzmann equations for χ and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, a very good analytic estimate is given by YFI ≃ 135 8π4 � 45 πg3∗ Γχ→ψϕ MPl m2χ ≃ 6 × 10−6 y2 φ ∆2 MPl mχ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='10) where we used that the production peaks around the mass of the heaviest particle involved in the decay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', at T ≃ mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The late decays of χ would instead peak at temperature TD at which Γχ→ψϕ/H|T=TD ≃ 1, TD ≈ yφ ∆ � mχMPl 32π ≈ 10 MeV ∆ � yφ 10−12 � � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='11) In order to get asymmetric DM, we impose the condition YFI ≲ 10−2ηD (here we are assuming values of {gD, mχ, mZD} such that rχ ≲ 10−2) which implies yφ ∆ ≲ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 × 10−12 �ηD ηB �1/2 � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='12) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5, we show the parameter space in the plane yφ versus mψ where the contribution of ψ is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We fix mχ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We highlight the regions where it is symmetric, partially-asymmetric and asymmetric, subject to the constraints discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 15 – Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Parameter space for the scenario in which the DM is composed solely of ψ and we assume that S is light enough in each point of the plot so that its contribution to the DM abundance can be safely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We distinguish three different regions depending on the nature of DM (symmetric, asymmetric or partially asymmetric), labeled by the value of rψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the darker gray region DM is produced through freeze-in and is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the white region it is produced via late decays and is asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The DM relic abundance is reproduced along the blue solid (dashed) line which corresponds to ηD/ηB = 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1), whereas the horizontal dashed lines indicate the shift in the regions for ηD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Constraints The gauge coupling gX is constrained by long-range force experiments, gX ≲ 10−8, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the rest of the paper, we take gX sufficiently small so that gauge interactions can not drive thermalisation of ψ and therefore gX never plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the other side ψ could thermalise through processes involving the Yukawa coupling yφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' All these processes involve at least one χ particle (or the conjugate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The most important ones are the decays (and inverse decays) χ → ψϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The condition for non-thermalisation is Γχ→ψϕ/H|T≃mχ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Indeed, for T > mχ, the interaction rate/Hubble ratio grows when the temperature decreases, while for T < mχ decay and inverse decays become inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, we evaluate the ratio at its maximal value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', T ≃ mχ, which gives yφ ∆ ≪ 30 � mχ MPl = 5 × 10−7 � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='13) Notice that taking the limit ∆ ≪ 1 does not help in pushing towards higher values of the Yukawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Indeed, if ∆ is small, the decay channel closes and annihilations become more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Annihilations processes are independent from ∆ and give a condition analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='13) with ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, as we are considering ∆ ≃ 1, decays are more important than annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the case in which ψ is produced mainly by late decays, we require that these are peaked much after the freeze-out of χ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', TD ≪ T∗, which ensures that the population of – 16 – — nD=NB 10-8 mx= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, △ = 1 np=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 nB Symmetric DM 10-9 Freeze-in rμ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 10-10 Partially asymmetric DM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 >r ≥10-2 10-11 10-12 Late decays Asymmetric DM rμ<10-2 10-13 BBN 100 101 10-1 my [GeV]ψ is asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This gives a stronger condition, yφ ∆ ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 � mχ MPl = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 × 10−9 � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='14) If ψ is produced by early decays, peaked around Tprod ≃ mχ, it beheaves as cold DM as long as mψ ≳ keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, if the production from late decays is significant, we must take into account an additional constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Indeed, as we have a heavy particle (χ) decaying late into a lighter stable one (ψ), we must check that the DM free streaming length is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Mpc, which gives [62] mψ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 keV ⟨p/T⟩prod � 10 g∗(TD) �1/3 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7 keV mχ ∆ TD � 10 g∗(TD) �1/3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='15) where in the last line we used that the typical DM momentum at production is ⟨p⟩ ∼ mχ∆/2, while ⟨Tprod⟩ = TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='15), this translates into a lower bound on the coupling, yφ ≳ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6 × 10−14 � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV �1/2 �5 GeV mψ � � 10 g∗(TD) �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='16) This bound only applies if i) ψ reproduces the DM relic abundance, and ii) the production from late decays is dominant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Y + LD ≫ YFI, corresponding to the vertical blue lines of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We can use BBN bounds to constrain the lifetime of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If χ did not decay, its abundance today, Ωχh2, would be mχ/mψ times the would-be DM (ψ) abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In particular, for mχ ∼ O(TeV) and would-be abundance ηD ∼ 10−11, this leads to τχ ≲ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 − 1) s [63] for yφ∆ ≳ 10−13 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV mχ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='17) Here we assumed that the decay of ϕ into SM radiation (mainly hadrons) occurs instanta- neously right after χ decays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' otherwise, the bound should read τχ + τϕ ≲ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 − 1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The effects of fermion mixing are discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Contribution of S 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Production of S from freeze-out The scalar S is produced by N1 decays at high temperature and shares the same asymmetry of χ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e, ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Once produced, it thermalises with the SM bath through scalar and gauge interactions and undergoes annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following, we focus mainly on masses of S in the 1−50 GeV range, so that its contribution to the relic abundance may be of the same order as that of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this range of masses, annihilations of S into ZD may be kinematically forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, if mϕ < mS (possible by tuning the coefficients of the scalar potential, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', mϕ/vφ ≲ 10−3 and mϕ ∼ O(GeV) or lighter), then the annihilations S†S → ϕϕ are – 17 – allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Next we consider the minimal option of just the scalar portal λSφ|S|2|φ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The non-relativistic cross section for S†S → ϕϕ induced by the operator is given by 6 σv(S†S → ϕϕ) ≃ λ2 Sφ 32πm2 S � 1 − m2 ϕ m2 S �1/2 � 1 − m2 ϕ/2m2 S − 2λSφv2 φ/m2 S 1 − m2ϕ/2m2 S �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='18) Even for moderately small values of the coupling, the cross section is significant and annihi- lations are strong enough to destroy the symmetric population of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, the symmetric population of S annihilates around T (S) ∗ ∼ mS/20, leaving only the asymmetric population, with the abundance fixed by the dark asymmetry Y + S ∼ ηD, Y − S ∼ rSYS ≪ Y + S , completely analogous to the computation of χ annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4 (right panel) we show the re- gion of mϕ and mS where S is asymmetric for different values of the coupling λSφ fixing ηD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1ηB and mZD ∼ 500 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In principle, other annihilation channels for S are possible, depending on the scalar potential parameters, such as S†S → h → ¯ff (which is more suppressed for mh > mS), allowing for a larger set of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For simplicity, we focus only on annihilations into ϕϕ, which involve only one coupling λSφ and need not be very large, O(10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Production of S from χ decays Integrating out the lightest of the heavy sterile neutrinos, N1, induces the effective interac- tions yνyS ¯L ˜HSχ MN1 , y2 S χ2S2 MN1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='19) The former leads to χ → S†νL and χ → S†νLh decays, which compete with χ → ψϕ, eventually populating the S† sector, whereas, the latter induces χχ → S†S† annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Focusing on decays, we can use the estimate (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [59]) Γ(χ → S†ν) ≈ |yS|2mχ 32π � mν MN1 � � 1 − m2 S m2χ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='20) These decays are peaked around T (S) D defined as Γχ→S†ν/H|T=T (S) D = 1 (analogous to the decays into ψ computed earlier) and given by T (S) D ≃ |yS| � mνmχMPl 32πMN1 ≃ MeV � |yS| 10−3 � � mν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5TeV 1011GeV MN1 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='21) Since for the values of the parameters T (S) D < T∗, the only important decay is χ → S†νL, while the conjugate process is irrelevant as the population of ¯χ after freeze-out is negligible (recall that we are in the region of the parameter space in which rχ < 10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7 6We neglect the diagram containing the self-interaction ϕ3, which enters if the φ quartic coupling is not extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 7Also in this case we can distinguish between the late decays, peaked at T (S) D , and the early decays at T > T∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The latter produce a symmetric population of S and S† from the decays of χ and ¯χ (the analogue of the freeze-in population of ψ, peaked at T ≃ mχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, this symmetric population thermalises and undergoes annihilations leaving no imprint in the final abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 18 – We can parametrise the dominant decay channel of χ by defining the ratio of branching ratios R ≡ Br(χ → S†ν) Br(χ → ψϕ) ∼ |yS|2 y2 φ mν MN1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='22) where in the last step we used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decay of χ into ψ is the dominant channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', R ≪ 1, for |yS| ≪ � yφ 2 × 10−11 � � MN1 1011GeV �1/2 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV mν �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='23) Notice that the contribution of the 3-body decays χ → νhS† should be similar to the 2- body ones, as the decay rate gets suppressed by the phase space factor while at the same time it has an enhancement (mχ/v)2 ∼ 200 (eventually it can become dangerous for mχ ≫ TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For a generic value of R, the probability to decay into ψ is 1/(1 + R) while into S† is R/(1 + R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore the abundances of ψ, ¯ψ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6) get divided by (1 + R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Concerning S, we can distinguish two possibilities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If T (S) D < T (S) ∗ < T∗, at the time where decays into S† peak, the latter has already de- coupled from the thermal bath with an asymmetric abundance YS = ηD and therefore the S and S† population can not annihilate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Then the abundances of S, S† are YS ≡ Y + S = ηD, YS† ≡ Y − S = R 1 + R ηD , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='24) where we assume that rS < 10−2 and we have ignored it for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This corre- sponds to |yS| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 � mS GeV � � MN1 1011GeV �1/2 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV mν �1/2 �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5TeV mχ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If T (S) ∗ < T (S) D < T∗, the decays produce a population of S†, while S†S annihilations are still efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As a result there is a partial washout of ηS, which gets reduced to ηD/(1 + R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, at T < T (S) ∗ a population of S decouples with abundance Y + S = 1 1 + R ηD , Y − S ≪ Y + S , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='26) where we assume rS < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that for R ≪ 1 Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='26) are equivalent as the decays are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Constraints We can constrain the value of |yS| for the case in which R > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As for this choice χ decays mostly into S†, we must impose that the decay occurs before BBN, in analogy with Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Using the decay rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='20) we find that the BBN bound translates into the constraint |yS| ≳ 10−3 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05 eV mν MN1 1011 GeV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV mχ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='27) – 19 – For R < 1 we can choose smaller yS while BBN constrains the value of yφ, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We also find that for R > 1 the model is characterised by an interesting feature: the decays χ → S†νL, which occur before BBN and neutrino decoupling (around T ∼ O(10 MeV) for the choice of parameters we adopted and |yS| ∼ 10−2), generate also an asymmetric population of neutrinos, ∆ην = R (1 + R) ηD , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='28) which is maximal (≈ ηD) for large R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the neutrino population is more asym- metric than in the standard case, as this new contribution sums up to the usual leptonic asymmetry generated earlier on by leptogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Note, however, that since these decay processes occur below the scale of electroweak symmetry breaking, this leptonic asymmetry is not transferred to the baryonic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, if R > 1 (and T (S) D < T (S) ∗ ), the S† population that arises from late decays of χ may be warm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In such a case, if a significant fraction of the DM was made by this S† population, constraints from free streaming length (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='15 with mψ → mS and TD → T (S) D ) would give |yS| ≳ 5 MeV/mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, even in the cases in which the contribution of ψ DM is negligible, the DM is composed by a mixture of S (produced by freeze-out, always cold) and S†, where Y + S ≥ Y − S (the equality applies if R ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This corresponds to a mixture of cold/warm DM, which in general is less constrained than the full warm DM case [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following, we discuss the scenarios arising due to the different dynamics of the dark components in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' It can be checked that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 is always fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5 Dark matter relic abundance The precise values of some of the parameters do not change qualitatively the results, as- suming that we are always in the red regions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', that annihilations are efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, for definiteness, in the following we fix some of the parameters of the model, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Parameter Benchmark Value mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV mZD 500 GeV gD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 MN1 1011 GeV mν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05 eV Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Benchmark values of the model parameters used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that with this choice, rχ ≪ 10−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ¯χχ annihilations efficiently erase the symmetric population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We also assume that rS ≪ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' At the same time, depending on the values of the Yukawa couplings yφ and |yS|, and the masses of the DM particles mψ and mS, it is possible to realise the different scenarios of Table 1 while satisfying all the – 20 – constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Table 4 we outline the different scenarios, as well as provide expressions for the relic abundance of the DM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Next we analyse them one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The reader may want to skip the following discussion and go directly to the summary of scenarios in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-LD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FO-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Partially asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-LD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ + LD χ → S†νL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FOLD-PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = ηD/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηDR/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Mixed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-LD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ + LD χ → S†νL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FOLD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = ηD/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Partially asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-FILD-PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FI + LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FO-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/2 + ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Partially Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Partially Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-FILD-PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FI +LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ + LD χ → S†νL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FOLD-PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = (YFI/2 + ηD)/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/(2(1 + R)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηDR/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Partially Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Mixed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-FILD-PA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FI +LD χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ + LD χ → S†νL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FOLD-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = (YFI/2 + ηD)/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD/(1 + R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/(2(1 + R)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Symmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Asymmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ-FI-S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FI χ → ψϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FO-A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/2 + ηD ≃ YFI/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='ψ = YFI/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ≪ Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Symmetric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S-FOLD-S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Negligible production ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='FO S†S → ϕϕ + LD χ → S†νL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Y − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='S = ηD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Classification of scenarios in our model on the basis of dominant production mechanism and asymptotic nature of both dark matter components, ψ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 21 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 Scenario 1: ψ-LD-A + S-FO-A In this scenario, both components are asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For this to happen, we demand yφ ≲ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4×10−12� ηD/ηB: the freeze-in population of ψ from early decays is negligible, while the late decays are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, ψ is asymmetric with abundance Y + ψ = ηψ = ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' R ≪ 1, which means |yS| ≪ yφ/(2 × 10−11): S freezes-out once all the symmetric population has annihilated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The population of S† produced by the late decays of χ is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The abundance is determined by its asymmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e Y + S = ηS = ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the model has two asymmetric components with individual abundance ηD, where the abundance of one (ψ) is set by late decays and that of the other one (S) by freeze-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The relative abundance of the two DM species and the total abundance are given by Ωψ ΩS = mψ mS , ΩDM ΩB ≃ 5 = ηD(mψ + mS) mpηB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 (black lines), we show the region of the parameter space in which the correct DM relic abundance is reproduced in the plane mS versus mψ, for two values of the dark asymmetry ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Ωψ/ΩS increases when the curves are followed clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We can see that for ηD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 (1) ηB, dashed (solid) black line, we have mψ ≃ mS ≃ 30 (3) GeV if both species contribute similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Alternatively, we can push the mass of S down to GeV in such a way that its contribution to DM abundance is negligible and reproduce the relic abundance for mψ ≃ 50 (5) GeV for ηD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 (1) ηB, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2 Scenario 2: ψ-LD-A + S-FOLD-PA Similar to the previous scenario, here we also take yφ ≲ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 × 10−12� ηD/ηB, so that the dominant production of ψ comes from late decays, leading to its asymmetric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, here we take R ∼ O(1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', |yS| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 × 1011yφ), which makes it qualitatively distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We also assume that |yS| is small enough so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', the decays of χ to S† occur when the latter has already decoupled from the thermal bath, T (S) D < T (S) ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Due to R being order one, χ partially decays into ψ and partially into S†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The probabil- ity to decay into ψ is 1/(1+R) whereas that of into S† is R/(1+R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, ψ is highly asymmetric with abundance Y + ψ = ηψ ∼ ηD/(1 + R), whereas S becomes partially asym- metric because a population of S† is produced by χ → S†ν decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As T (S) D < T (S) ∗ , S†S annihilations are not active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore ηS reduces to ηS = ηD −R ηD/(1+R) = ηD/(1+R), while the total abundance is determined by the sum Y + S +Y − S = (1+2R)ηD/(1+R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' There- fore, the contribution to the DM abundance becomes Ωψ ΩS ≈ mψ mS 1 (1 + 2R) , ΩDM ΩB ≃ 5 = ηD(mψ + (1 + 2R)mS) (1 + R)mpηB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) – 22 – Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Values of mψ and mS for which the correct relic abundance can be reproduced for Scenario 1: ψ-LD-A + S-FO-A (black lines) and Scenario 2: ψ-LD-A + S-LD-PA (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We show ηD ≃ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1)ηB using solid (dashed) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The relative abundance of ψ with respect to S in- creases when the curves are followed clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray dashed line corresponds to mψ = mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The parameters are fixed as per Table 3, so that ¯χχ → ZDZD annihilations are efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Furthermore, we fix yφ = 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For Scenario 1 (2): |yS| = 10−3 (5 × 10−2), so that R ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='0005 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In both cases YFI ≃ 10−4ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the limit R = 0, all χ decay into ψ and we recover Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If R = 1 the relative ψ/S abundance is mψ/(3mS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The factor 3 comes from the fact that while we only have ψ and no ¯ψ, we have both S and S† with Y + ψ = Y − S = ηD/2 and Y + S = ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The limit R ≫ 1 is discussed later (Scenario 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As discussed earlier, this case leads to an enhanced background of an asymmetric neutrino population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The allowed parameter space is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 by red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, the shape of the curves is not symmetric as in Scenario 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' in the limiting case where ψ (S) dominates the abundance one needs mψ ≃ 100 GeV (mS ≃ 30 GeV) for ηD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 ηB, and masses roughly one order of magnitude smaller for ηD ≃ ηB, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3 Mixed Scenario 1-2: ψ-LD-A + S-FOLD-A We consider the same range for yφ and R but we assume that |yS| is large enough to violate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', χ decays to S† while S†S annihilations are still efficient, T (S) D > T (S) ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For ψ we find the same results of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The S† population produced by χ decays partially washes-out ηS, leaving an asymmetric population of S with abundance Y + S = ηD/(1 + R), while the S† population gets erased by S†S annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The contribution to the DM abundance is now Ωψ ΩS ≃ mψ mS , ΩDM ΩB ≃ 5 = ηD(mψ + mS) (1 + R)mpηB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) Notice that this scenario is a mixture between Scenario 1 (the nature of the DM particles is the same, both asymmetric, and the abundances are the same ones rescaled by (1 + R)) – 23 – 102 Scenario 1 Scenario 2 Np= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 NB ms [GeV] 101 ND= NB 101 100 102 my [GeV]and Scenario 2 (the ranges of yφ and R are identical and the same processes determine the final abundance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore we denote it as Mixed Scenario 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 Scenario 3: ψ-FILD-PA + S-FO-A For larger values of the Yukawa, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 × 10−12� ηD/ηB ≲ yφ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 × 10−10� ηD/ηB, the freeze-in contribution to ψ production from early decays grows and becomes comparable to the contribution from late decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' So ψ is partially asymmetric with Y + ψ ≃ YFI/2 + ηD and Y − ψ ≃ YFI/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We take R ≪ 1, so that all χ decay into ψ, partially while being in thermal equilibrium (freeze-in) and partially at a later time (late decays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The ψ abundance is Y + ψ + Y − ψ = YFI + ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the other hand, the abundance of S is determined by thermal freeze-out, once the symmetric population is annihilated away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Thus, S freezes-out with an asymmetry Y + S = ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case the relative abundance between the two DM component and the total DM abundance are given by Ωψ ΩS = mψ(ηD + YFI) mSηD , ΩDM ΩB = mψ(ηD + YFI) + ηDmS ηBmp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 Scenario 4: ψ-FILD-PA + S-FOLD-PA If R ∼ O(1) for the values of Yukawa yφ considered in the previous scenario, then χ decays half of the time into S† and the other half into ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Again, we assume that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The population of ψ, ¯ψ is produced partially by freeze-in and partially by late decays, whereas the decays into S† washout the asymmetry in S, making it partially asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, in this scenario, the late decays determine the asymptotic nature of both DM components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', it is the late decays that finally make the DM to be partially asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The corresponding expressions for the DM abundance are8 Ωψ ΩS = mψ mS ηD + YFI ηD(1 + 2R) , ΩDM ΩB = mψ(ηD + YFI) + ηD(1 + 2R)mS ηB(1 + R)mp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6 Mixed Scenario 3-4: ψ-FILD-PA + S-FOLD-A Here we consider the same range of yφ and R as in the previous scenario but a larger |yS|, which violates Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The discussion for ψ is the same as in Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In analogy with the Mixed Scenario 1-2, S†S annihilations erase the S† from the thermal bath, leaving an asymmetric population of S which survives the annihilations, so that Ωψ ΩS ≃ mψ mS YFI + ηD ηD , ΩDM ΩB ≃ 5 = mψ(YFI + ηD) + mSηD (1 + R)mpηB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6) This scenario is a mixture between Scenarios 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8Notice that the previous equations are quite general, as Scenario 1, 2, 3, 5 and 6 can be seen as particular cases of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 24 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7 Scenario 5: ψ-FI-S + S-FO-A For even larger Yukawas, in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 × 10−10� ηD/ηB ≲ yφ ≲ 5 × 10−7, the ψ sec- tor is mainly populated during freeze-in, while late decays only produce a sub-dominant component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', YFI ≫ ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the ψ population is (almost) symmetric, Y + ψ = YFI/2 + ηD ≃ YFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Concurrently, R is typically small because of the larger value of yφ and therefore S freezes with an asymmetric abundance, Y + S = ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, DM is mostly symmetric in ψ and asymmetric in S, produced by freeze-in and freeze-out, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The relative abundance of the two species and the total DM abundance are Ωψ ΩS = mψ mS YFI ηD , ΩDM ΩB ≃ 5 = mψYFI + mSηD mpηB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 7 we show the mass ranges of ψ and S for which the DM relic abundance can be reproduced for different values of ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that if S is subdominant, the mass of ψ is fixed, irrespective of the value of asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is due to the fact that the DM mass is fixed by the freeze-in contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='10), which is (almost) independent of mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the other hand, it depends on the mass of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For the values used in the figure (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1ηB < ηD < ηB, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' ), we see that DM could be mainly composed by light symmetric ψ of mass around 500 MeV, mainly by asymmetric GeV-ish S, or by a combination of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Values of mψ and mS for which the correct relic abundance can be reproduced for Scenario 5: ψ-FI-S + S-FO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray dashed line corresponds to mψ = mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We fix the parameters |yS| = 2 × 10−4 and yφ = 2 × 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' With this choice, R ≃ 5 × 10−10 and YFI ≃ 10ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As we have underlined above, for these values of yφ, R is typically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, if we lower the vB−L scale it is possible to reach R > 1 even in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We briefly discuss this possibility at the end of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8 Scenario 6: S-FOLD-S Finally, when R ≳ O(10), for whatever value of yφ compatible with it, the majority of the χ population decays into S† after freeze-out of S (but before BBN if 10−3 < |yS| < – 25 – 102 Scenario 5 Np =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 NB 10 ms [GeV] ND= NB 10- 100 10-1 my [GeV]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1(mS/GeV)) and the asymmetry is completely washed-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, the populations of S and S† survive independently, as S†S annihilations are not active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Hence, the DM relic is almost completely made up by the symmetric population of S and S†, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Y + S = Y − S , while there is a negligible abundance of ψ produced from early or late decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the scenario leads to practically only 1 DM component with abundance completely fixed by the asymmetry: Y + S + Y − S = 2ηD, leading to the prediction mS ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 GeV �ηB ηD � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8) This scenario is phenomenologically interesting because S†S annihilations get enhanced with respect to the usual freeze-out case (indeed the abundance is set by the asymmetry and not by the annihilation cross section), leading to enhanced indirect detection signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Additionally, if the S† population arises from late decays of χ there can be a mixture of cold/warm DM where S particles, coming from the thermal plasma, represent the cold component, while their anti-particles, coming from late decays, the warm one, with a pos- sible impact on structure formation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Furthermore, the additional contribution to the asymmetric background neutrino population is maximal in this case, ∆Yν ≈ ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that for larger |yS| the decays take place while S, S† are still in equilibrium so that S†S annihilations washout the asymmetry and one recovers the standard scenario of symmetric freeze-out in which the S abundance is determined by the annihilation cross section σvS†S instead of the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Depending on mS and σvS†S it may be possible to reproduce the correct relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We do not consider this possibility in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 Summary of the scenarios In the scenarios discussed above, we considered the parameter space where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Both ψ and S are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' ¯χχ → ZDZD annihilations are efficient enough so that rχ < 10−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', the late decays of χ are always asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' S†S → ϕϕ annihilations are efficient enough so that rS < 10−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', only the asym- metric population of S survives the annihilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As long as yφ lies in the range 10−13 ≲ yφ ≲ 5 × 10−7, we can summarise all the scenarios with the following expressions for the individual and total relic abundance: Ωψ ΩS = mψ(ηD + YFI) ηDmSf(R) , ΩDM ΩB = mψ(ηD + YFI) + ηDmSf(R) ηB(1 + R)mp , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9) where f(R) ≡ � 1 + 2R if T (S) D < T (S) ∗ 1 if T (S) D > T (S) ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='10) – 26 – Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Contours of DM relic abundance in the mψ - mS plane corresponding to ηD = ηB for the different scenarios discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We fix the values of the yukawa couplings (yφ, |yS|) for each scenario: Scenario 1: (10−12, 10−3), Scenario 2: (10−12, 5 × 10−2), Scenario 3: (10−10, 10−3), Scenario 4: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4 × 10−12, 10−1), Scenario 5: (2 × 10−10, 2 × 10−4), Scenario 6: (10−13, 5 × 10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray dashed line corresponds to mψ = mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8 we show contours of correct relic abundance in the mS versus mψ plane for the different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We take ηD = ηB and for each scenario we fix an appropriate value for the Yukawa couplings |yS| and yφ (Scenarios 1, 2 and 5 are the same already shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 and 7, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' DM could be mainly composed by ψ, with mψ in between hundreds of MeV and tens of GeV, mainly by the scalar S with mS ∼ GeV, or by a combination of them (both GeV-ish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For fixed R, ψ DM is heavier when is asymmetric (Scenario 1) and lighter when symmetric (Scenario 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For fixed rψ, ψ gets heavier as R gets larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the contrary, S DM gets lighter while R grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The minimal value of mS is fixed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8), corresponding to Scenario 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice the presence of a four-fold degeneracy between scenarios 1, 2, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This corresponds to the case mS = mψ, in which the mass of both the DM particles is fixed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8) (the relation is exact for Scenarios 1, 2 and 6, while for Scenario 4 it is approximately valid if YFI < ηD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Choosing another value for ηD leads to a rescaling of DM masses by a factor ηB/ηD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The main results of the paper are provided in Table 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Table 5, we show the requirements on yφ and R for each scenario and summarise the contribution to the relative and the total DM relic abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 9 we show the allowed parameter space in the plane |yS| versus yφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We fix the parameters as: mχ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, mZD = 500 GeV, gD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5, MN1 = 1011 GeV, mν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05 eV and ηD = ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We restrict our analysis to the region 10−13 ≲ yφ ≲ 5 × 10−7 (no thermalisation of ψ and BBN bound on yφ fulfilled for R < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The Yukawa |yS| is large enough to generate a sizeable dark asymmetry (and respects the BBN bound in the region in which R > 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', |yS| ≳ 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray dot- dashed (dotted) line corresponds to R = 1 for MN1 = 109 (104) GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Clearly, as N1 gets lighter, χ preferably decays into S†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that in the plot the masses of the DM particles are not fixed, but at every point there are always some values of the latter for which the correct DM relic abundance is reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 27 – 101 ND = NB Scenario 1 ms [GeV] 3 4 5 6 100 10-1 101 my [GeV]Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 10−10yφ/ � ηD/ηB R T (S) D /T (S) ∗ ΩDM/ΩB Ωψ/ΩS 1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 ≪ 1 Any ηD ηB mψ+mS mp mψ mS 2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 O(1) < 1 ηD ηB mψ+(1+2R)mS (1+R)mp mψ mS(1+2R) 1-2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 O(1) > 1 ηD ηB mψ+mS (1+R)mp mψ mS 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 ≪ 1 Any mψ(ηD+YFI)+ηDmS ηBmp mψ(ηD+YFI) mSηD 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 O(1) < 1 mψ(ηD+YFI)+ηD(1+2R)mS ηB(1+R)mp mψ(ηD+YFI) mSηD(1+2R) 3-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='064 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 O(1) > 1 mψ(ηD+YFI)+ηDmS ηB(1+R)mp mψ(ηD+YFI) mSηD 5 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 ≪ 1 Any ηD ηB mψ(YFI/ηD)+mS mp mψYFI mSηD 6 yφ ≲ 5 × 10−7 ≳ O(10) < 1 ηD ηB 2mS mp ≪ 1 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Contribution to the DM relic abundance in different scenarios depending on the values of yφ (in units of � ηD/ηB) and R for the values of parameters provided in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We take yφ < 5 × 10−7 to avoid thermalisation of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In Scenarios 2, 4 and 6 the Yukawa satisfies |yS| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1(mS/GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the Mixed Scenarios 1-2 and 3-4 we have |yS| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1(mS/GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Scenarios 1, 3 and 5 give the same result independently of this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 Phenomenological signals In principle, the models considered in this work may be difficult to test and disentangle in their current version, because of several reasons: i) Freeze-in scenarios invoke very small couplings, gX ≪ 1, yφ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' ii) The considered scale of B−L breaking is very large, vB−L ≫ vEW, so collider searches are not an option;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' iii) Asymmetric DM yields suppressed indirect detection signals in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Moreover, in our scenarios, the symmetric component is typically erased into the dark sector, via ¯χχ → ZDZD and S†S → ϕϕ, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' iv) The mixing of ϕ with the Higgs, generated by λHφH†Hφ†φ, was taken to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, there are a few distinctive signals of S, through the usual Higgs portal, λHSH†HS†S: Direct detection in the case in which the DM is mainly composed of S, which currently sets the limits λHS ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='01 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Higgs invisible decays, for mS < mh/2, which currently sets the limits λHS ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='01 [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, it could be that S was produced via Higgs decays, but it did not constitute a dominant part of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Similarly, indirect detection signals from annihilations are present if the final abun- dance is composed of S and partially-asymmetric (Scenarios 2 and 4) or symmetric (Scenario 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Scenario 6 is interesting, since it requires larger than usual annihilation – 28 – Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Parameter space of the scenarios discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We fix mχ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, mZD = 500 GeV, gD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5, MN1 = 1011 GeV, mν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05 eV and ηD = ηB and 10−13 ≲ yφ ≲ 5 × 10−7 (no thermalisation and BBN bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The Yukawa |yS| is large enough to generate a sizeable dark asymmetry (and respects the BBN bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that in every point the masses of the DM particles, mψ and mS, are not fixed, and it is not possible to get the correct relic abundance for arbitrary values of mψ and mS, but only for particular values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', along the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The black diamond (red circle) is the benchmark point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 6 for scenario 1 (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' the purple star corresponds to the benchmark of Scenario 5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Scenario 4 is not visible in the plot but would appear in the intersection between 2 (red region) and 3 (blue region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Depending on the value of mS a portion of region 2 (3) could convert into the Mixed Scenario 1-2 (3-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, for mS ≳ GeV, this requires |yS| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray dot-dashed (dotted) line corresponds to R = 1 for MN1 = 109 (104) GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Clearly, as N1 gets lighter, χ preferably decays into S†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Above the red (blue) [purple] line the heavier between S and ψ, for fixed mass max(mS, mψ) = 3 GeV, has a lifetime, re-scaled by its relative number density, of 1023 (1024) [1025] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the region above the red line is excluded, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' rates, and is very predictive, with a mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 GeV, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, on-shell S-annihilations into muons, pions and electrons are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Typically, such light thermal DM is severely constrained by its energy injection in the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' There- fore, it may be interesting to further investigate for which values of λHS is Scenario 6 allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Last but not least, there is a very interesting phenomenological signal of our model: the presence of a monochromatic flux of neutrinos coming from the late decay of the heaviest of the two DM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Let us assume mS > mψ (the opposite case mψ > mS is completely analogous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' At low energies, E ≪ mχ ≪ MN1, the decay S → ¯ψ + νL is generated by the dimension-6 operator O6 = ¯L ˜HSφ†ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) – 29 – 10-1 6 ND = NB Scenario Scenario 2 10-2 R Ts(n1 DM/ns) [s] 1023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 S Scenario 3 1024 10-3 — - 10 1025 Scenario 1 Scenario5 rμ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 rμ<10-2 10-4 10-12 10-10 10-9 10-13 10-11 ysThis operator arises by first integrating the right-handed neutrino field NR, which generates the interaction given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='19), and then at the lower scale the fermion χ, giving rise to the interaction 9 ySyφyν MN1mχ O6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) Once H and φ acquire vevs, the decay S → ¯ψ + νL is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decay width reads Γ(S → ¯ψ + νL) ≈ |yS|2y2 φmS 32π � vφ mχ �2 � mν MN1 � � 1 − m2 ψ m2 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) To guarantee that both S and ψ are cosmologically stable and contribute to the DM abundance, the lifetime of the heavier particle needs to be larger than the age of the Universe, τS > tU ∼ 4 × 1017 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, there are stronger constraints if the decay products include an active neutrino, τS ≳ 1023 s (for a GeV-ish single-component DM) [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If S decays at late times, it leads to a very distinctive signature: a neutrino line peaked at mS/2 ∼ O(GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, depending on the relative abundance of ψ and S, some region of the parameter space could be excluded, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' To quan- tify this, we compare the experimental bound with the re-scaled lifetime of the particle, τ re−sc S = τS (n1DM/nS) > 1023 s, where n1DM (nS) is the single-component DM (S) number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This constraint gives a condition on the parameters, yφ|yS| ≲ 3 × 10−12 �3 GeV mS �1/2 �700GeV vφ � � mχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5TeV � �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV mν �1/2 � MN1 1011GeV �1/2 � ΩDM ΩS(yφ, |yS|, mS, ηD) �1/2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) valid if mS > mψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In general the condition is not linear in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If ΩS/ΩDM ≪ 1, the constraint is extremely weak: DM is made only by ψ and there are no decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' On the contrary, if ΩS/ΩDM ∼ O(1), the condition becomes linear in the Yukawas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the opposite regime, mψ > mS, there is an equivalent condition to that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4), with mS → mψ and ΩS → Ωψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If the two states are almost degenerate mS ≃ mψ there is a strong phase space suppression so that the constraint gets weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that, even taking into account that the lifetime is re-scaled by the relative number density, this constraint is much stronger than the one coming from τS > tU, so that it automatically guarantees the stability of both S and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The neutrino line emerging from S → ¯ψνL (ψ → S†νL) decay is the main smoking gun of our scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As an illustrative example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 9 we show three different lines in red, blue and purple corresponding respectively to τ re−sc S = 1023, 1024, 1025 s, for fixed max(mS, mψ) = 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For this choice of mass, as we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8, ΩS/ΩDM ∼ Ωψ/ΩDM ∼ O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) gives a simple linear constraint on the Yukawas and the DM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The corresponding region above the red line is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the region 9Operator O6 may also be generated by a UV completion of our model at scales above vB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This produces additional constraints, weaker than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We discuss them in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 30 – between the red and the purple lines the signal is close to the experimental sensitivity and could lead to the observation of a neutrino line in existing or near-future neutrino telescopes (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [67–72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' From the figure we deduce that Scenarios 3 and 5 are therefore the most testable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that, for different choices of max(mS, mψ), the picture can be more complicated because Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) becomes non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' By performing a numerical scan over the relevant Yukawa and DM mass ranges, we have checked that, as expected, there is always an excluded region in the upper right corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8, which corresponds to yφ ≳ 10−10, |yS| ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 7 A low-energy variant: The inverse seesaw So far, we have assumed that active neutrinos get masses through the Type-I seesaw at a very high scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this framework, the lepton and dark asymmetries are generated in a similar way as in high-scale leptogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the following, we discuss the possibility to embed our models into a low-scale leptogenesis scenario, trying to lower the scale of B − L of the scenarios considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is an alternative path, which may yield interesting phenomenological signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, there can be DM interactions with the SM, mediated by ZB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Apart from direct and indirect detection signals, χ may be produced at colliders via ¯qq → ZB−L → ¯χχ and then decay, χ → ψϕ or χ → S†ν, yielding missing energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='10 For R ≲ 1, if the mixing of ϕ with the Higgs is in the correct range, displaced vertices may be produced from ϕ → SM SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Searches for long-lived particles are very active, with lots of experiments running or designed for the following years [73–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' It has been shown that leptogenesis at a low scale is possible, for example via resonant leptogenesis, with O(10 TeV) right-handed neutrino masses [76–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Another possibility is to adopt an inverse see-saw (ISS) mechanism to give mass to neutrinos [79, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Next we consider this option within a B − L set-up, see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' [81–85] and the review [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We can modify our model by replacing the scalar σ with two new fields: a scalar σ′ and three copies of a fermion SL, with the quantum numbers outlined in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Field Spin U(1)B−L U(1)D U(1)X SL 1/2 0 0 0 σ′ 0 +1 0 0 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Quantum numbers of the new states in the inverse seesaw variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The Lagrangian is the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) (excluding the terms involving σ), with the addition of LISS = SLi/∂SL − σ′SLyσ′NR − 1 2SLµSc L + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) where µ is a 3 × 3 complex symmetric matrix which can be taken to be real and diagonal without loss of generality, and yσ′ is a 3 × 3 complex matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If the new scalar takes a vev, ⟨σ′⟩ = vB−L, B − L is spontaneously broken and SL and NR form a pseudo-Dirac pair, with mass MD = yσ′ vB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, neutrinos get a mass through the inverse see-saw 10Notice that there is no ZB−LS†S vertex in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 31 – mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We focus on the range MD > mD ≫ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The mass of the active neutrinos is given by mν ≃ mD M−1 D µ (M−1 D )T mT D , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) so that light neutrino masses may be reproduced with small values of MD ∼ vB−L ∼ O (TeV) for small enough values of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Both low-scale variants result into a massive gauge boson, ZB−L, with a mass in the 5 − 10 TeV range, allowed by experimental constraints [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, the larger B − L gauge interactions allow to erase the symmetric χ population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the U(1)D gauge interactions are not needed, and in the limit gD → 0, U(1)D acts as a global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='11 Notice that the presence of a global U(1)D is still crucial to stabilise the DM particles and forbid dangerous operators, such as ¯ψφNR, which would generate ψ−NR mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, U(1)D could also be replaced by a Z2 symmetry, under which all fields but χ, S and φ are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In such a case, the symmetry stabilising the DM particles is a Z′ 2, coming from the combination of the broken U(1)X and the original Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Parameter space for the low-scale variant with the gauge coupling fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 and ηD = ηB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In the red region the fractional asymmetry rχ < 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The gray region is excluded because ZB−L is too light whereas in the purple region ZB−L is lighter than χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Since we expect mZB−L ≳ mN1, in the purple region N1 → χS decays are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The star indicates the benchmark point {mχ, mZB−L} = {3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, 10 TeV}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Thanks to the low scale, the annihilations ¯χχ → ZB−L → ¯qq (¯ll) are strong enough to erase the symmetric population of χ and make the model more testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The cross section 11In the limit gD → 0, the unbroken U(1)X+D symmetry is now global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The would-be Goldstone boson of the broken U(1)X−D is now eaten by A′, which becomes massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A′ is light because its mass is proportional to the gauge coupling gX, mA′ ∼ gXvφ ≪ vφ (recall that gX is tiny by assumption to avoid ψ thermalisation), and it does not thermalise because it has only gauge interactions driven by gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A′ couples to the B − L current proportionally to (v2 φ/v2 B−L)gX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The interactions of a massive gauge boson are less constrained than those of a massless one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, as now vφ ≲ vB−L, one obtains a constraint of the order of gX ≲ 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 32 – ND = nB, 9p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 105 104 103 X > 7-8zu 102 101 102 103 104 101 mx [GeV]for the ¯χχ → ZB−L → ¯ff process (assuming we are far enough from the resonance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', 4m2 χ − m2 ZB−L ≫ ΓZB−LmZB−L), reads σv(¯χχ → ¯ff) = NC(f)g4 B−Lq2 B−L(f) 2π � 1 − m2 f m2χ 2m2 χ + m2 f (4m2χ − m2 ZB−L)2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) where NC(f) = 3 (1) and qB−L(f) = 1/3 (−1) for quarks (leptons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The different scenarios for DM are equivalent to the ones studied in the previous sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, the decay χ → S†νL is now enhanced (with respect to the previous case) by the larger value of (mν/MN1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, we need smaller values of |yS| to realise a scenario with R ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, too small |yS| may be problematic for the generation of the dark asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A precise lower bound is not well established and could be computed in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Assuming |yS| ≳ 10−4, the most natural scenario is that the decays of χ into S† occur while the latter are still in equilibrium, at T (S) D > T (S) ∗ , (partially) washing-out the asymmetry ηS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, if yφ is small enough such that R > 10, χ mostly decay into S† and ηS gets completely erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Subsequently S undergoes a standard symmetric freeze-out and its abundance is determined by the annihilation cross section σvS†S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If yφ is larger, other scenarios can be realised, such as Scenarios 3, Mixed 3-4 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Eventually, if both yφ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='9 × 10−10� ηD/ηB and R > 1, a possibility not realised when vB−L ≫ TeV, a new scenario appears: ψ is symmetric and is produced mainly by freeze-in with abundance Y + ψ ≃ Y − ψ = YFI/(2(1 + R)) while the decays into S† partially washout ηS, leaving an asymmetric abundance of S, Y + S = ηD/(1 + R) ≫ Y − S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is basically a generalisation of Scenario 5 with all the abundances rescaled by a 1+R factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that, in any case, in order to have multicomponent DM, S must be light enough to satisfy the neutrino constraints, corresponding to the decay S → ψνL discussed in Section 6, which get stronger the smaller the vB−L scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Interestingly, in this case regions of smaller yφ and |yS| could be probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' 8 Conclusions Most probably, the dark sector is very rich, with a plethora of theoretical possibilities re- garding the number of stable particles, their nature, and their production mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this work we classify and analyse the different options by adopting a cogenesis model that simultaneously explains neutrino masses, the baryon asymmetry and the DM relic abun- dance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' While neutrino masses and the baryon asymmetry are produced via the standard Type-I seesaw mechanism and leptogenesis (with some extra contributions), respectively, we find that in such a framework there is a variety of viable scenarios for explaining the nature and abundance of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Once the decays of right handed neutrinos into the visible and dark sectors generate the asymmetries, some dark sector particles undergo asymmetric freeze-out and others are produced via freeze-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The model has two potential DM candidates, and we focus on the parameter space where both particles are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, whether they both contribute to the DM abundance similarly (two-component DM) or one has a negligible contribution – 33 – (one-component DM) depends on their asymptotic asymmetries, where the late decays play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Such decays may significantly populate the asymmetric or symmetric component at later times, thereby restoring annihilations, which may lead to enhanced signals in DM indirect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In this case, even though the DM is symmetric at the end, its abundance is still set by the asymmetry, and is thus independent of the annihilation rate, contrary to the usual WIMP scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We have analysed the range of model parameters that control the contribution of each component to the DM abundance, and outline the possible scenarios, classified according to the nature and production mechanism of each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We consider DM masses in the GeV ballpark and dark asymmetries similar to the baryonic one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' however, the set-up can easily accommodate lighter (heavier) DM for a larger (smaller) dark sector asymmetry if the branching ratio of right handed neutrinos into the dark sector is smaller (larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We have found that one of the main distinctive signatures is a neutrino line from S (or ψ) decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This would constitute a smoking gun of our model, within reach of existing or near-future neutrino telescopes for a significant region of the parameter space of some of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We conclude that having an initial asymmetry in the dark sector does not necessarily predict completely asymmetric dark matter, with its mass constrained by the dark asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In extended models, it allows the DM component to be partially asymmetric or symmetric, leading to more flexibility regarding the DM mass as well as the phenomeno- logical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Finally, although in this work we focused and extended the cogenesis scenario, which relates neutrino physics and dark matter, it would be interesting to consider other frameworks in which the different possibilities outlined here may also be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Acknowledgments This work is supported by the MICIN/AEI (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='13039/501100011033) grants PID2020- 113334GB-I00 and PID2020-113644GB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' GL is supported by the European project H2020-MSCA-ITN-2019/860881-HIDDeN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' JHG and DV are supported by the “Generalitat Valenciana” through the GenT Excellence Program (CIDEGENT/2020/020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A Decays of ϕ to SM particles We ensure that ϕ particles produced from χ decay fast enough into SM particles, so that the energy transfer from χ to SM radiation occurs before BBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A rigorous bound arises from imposing τχ + τϕ ≲ 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, it is sufficient to check that at the time of χ decays, corresponding to T = TD, the decays of ϕ are fast compared to the Hubble rate and the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='17) applies (corresponding to τχ ≲ 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decay ϕ → SM can occur through the Higgs portal operator λHφ|φ|2|H|2 as both the scalars take a vev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Through this portal ϕ decays into SM fermions, mainly (if kinematically allowed) into ¯bb or ¯cc or lighter species if mϕ < GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' The decay rate is Γϕ = sin2 θ × Γhϕ→ ¯ff, where sin2 θ is the mixing among ϕ and the SM Higgs boson h while Γhϕ→ ¯ff ∼ (mϕ/32π)(mf/vEW)2 is the decay width of a SM Higgs boson with mass mϕ into SM fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 34 – We require that the decay is fast at T = TD (when ϕ are produced via χ decays), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', Γϕ/H|T=TD > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For mχ ∼ TeV, mϕ ∼ few GeV> 2mc (or eventually 2mb) and yφ ∼ 10−12 the ratio Γϕ/H at T = TD is ≫ 1 even for small mixing angle, sin θ ≳ 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Even lighter mϕ ≳ MeV is allowed as ϕ can decay to ¯uu, ¯dd, ¯ee, with a larger mixing angle sin θ > 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Bounds from LEP constrain the mixing of a GeV-ish scalar to the SM Higgs to be sin θ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1 [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, the condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='17) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' mϕ lighter than MeV cannot decay to any SM fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However it could decay into 2 photons through the effective Higgs-photon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We do not study this possibility and we consider mϕ > MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This also implies mS ≳ MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' B Constraints on massless A′ µ The massless gauge bosons A′ µ only interact with the fermion ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Taking into account the fermion mixing, the Lagrangian contains the interaction terms ¯ψψA′ (suppressed by gX), ¯ψχA′ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (suppressed by gXϵf) and ¯χχA′ (suppressed by gXϵ2 f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' These vertices give rise to scattering processes as ¯χχ → A′A′, ¯ψψ → A′A′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' or decays χ → A′ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, given the smallness of gX (and ϵf) these processes are extremely suppressed and no sizeable population of A′ (which in principle could contribute to dark radiation) is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' We can give an upper bound on the value of the gauge coupling coming from long- range force experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Indeed the mixing between A′ and ZB−L induces an interaction geffA′ µJµ B−L with geff ≃ (gX/gB−L)(vφ/vB−L)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For the reference values of this work, vφ ∼ TeV and vB−L ∼ 1011 GeV, this corresponds to an effective coupling geff ∼ 10−16gX/gB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Long-range force experiments constrain the coupling to the B−L current to be geff ≲ 10−24 [88], which implies gX ≲ 10−8 (for gB−L ∼ O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is comparable with the condition for ψ not to reach thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' C Implications of fermion mixing Fermion mixing induces new interactions due to the ¯ψZDχ coupling, suppressed by ϵf, leading to new production processes such as ¯χχ → ¯χψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' However, these scattering processes are typically sub-dominant to decays due to suppression by yφ and ϵf as well as a phase space suppression (taking ∆ = 1), and thus can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In addition to the above, the mixing could also lead to an additional contribution to ψ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For T > vφ, the particles in the thermal bath are χ0 and ψ0 and there is no mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' For T < vφ, the vev of φ induces the mixing so that the states χ0 contain a small ψ component, proportional to ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, χ0 → ψ conversions contribute to the final ψ abundance (this is analogous to the production of sterile neutrino from active neutrino mixing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' This is also freeze-in process as the population of ψ is produced non-thermally from a small coupling (the fermion mixing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' In analogy with sterile neutrino DM, the interaction rate of ψ is Γψ ≃ ϵ2 fΓχ, with Γχ = nχσ¯χχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' If vφ ≲ mχ the χ particles are non-relativistic but still in equilibrium when the mixing is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' So, nχ = neq χ and these processes contribute to the symmetric component of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Using the number density at equilibrium we checked that for the typical values of our parameters (mχ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5 TeV, – 35 – gD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5, vφ ∼ TeV) this contribution is at most comparable (but not larger) than the one from decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A detailed study of this contribution, solving the full Boltzmann Equations (or using the density matrix formalism) is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, in the following we neglect this contribution, having in mind that it would change the total (symmetric) freeze-in contribution by at most an O(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' D Contributions to operator O6 In Section 6, we discussed the stability of the two DM components S and ψ and the possible observation of a neutrino line from the decay of one component into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' As we saw, the decay is mediated by the dimension-6 operator O6 = ¯L ˜HSφ†ψ, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='1) which is generated at low enegy by first integrating the right-handed neutrino field and then the fermion χ, as discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' A second contribution to O6 may arise if we assume that the theory contains the dimension-5 operator O5 = ¯NR(Sφ†)ψ ΛUV , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='2) where ΛUV is a cut-off scale parametrising the UV completion of our model at scales above vB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Integrating out NR gives rise to yν ΛUVMN1 O6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='3) The condition on the lifetime τ re−sc S > 1023 s (taking mS > mψ), is now satisfied if ΛUV ≳ 1016GeV � mν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV �1/2 �1011GeV MN1 �1/2 � vφ TeV � � mS 50GeV �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) This scale must be at most Planckian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=', ΛUV ≲ MPl, which implies a bound on the mass of the heaviest of the 2 DM particles mS ≲ 104 TeV �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='05eV mν � � MN1 1011GeV � �700GeV vφ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5) Finally, O6 could be directly generated by UV physics as O6 Λ ′2 UV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='6) In this case we just need Λ′ UV ≳ 1015 GeV � vφ TeV �1/2 � mS 50GeV �1/4 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='7) which gives a weaker constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Notice that the same bounds apply on mψ if mψ > mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Summarising, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='4) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content='5) set an upper bound on the mass of the heavier between ψ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' Therefore, this shows that it is quite natural that both ψ and S are stable on cosmological scales, and that they both contribute to the DM relic abundance and respect current limits from neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFQT4oBgHgl3EQfEjV1/content/2301.13238v1.pdf'} +page_content=' – 36 – References [1] A.' metadata={'source': 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@@ -0,0 +1,4358 @@ +arXiv:2301.01687v1 [hep-th] 4 Jan 2023 +Tropical Mirror Symmetry: Correlation functions +Andrey Losev +Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of Sciences, USTC, No.96, +JinZhai Road Baohe District, Hefei, Anhui, 230026, P.R.China +National Research University Higher School of Economics, Laboratory of Mirror Symmetry, +NRU HSE, 6 Usacheva str., Moscow, Russia, 119048 +Vyacheslav Lysov +Okinawa Institute of Science and Technology, +1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan +Abstract +We formulate the mirror symmetry for correlation functions of tropical observables. +We prove the tropical mirror correspondence for correlation functions of evaluation ob- +servables on toric space. The key point of the proof is the localization of correlation +functions for mirror states in type-B higher topological quantum mechanics on trees. +The correlation functions localize to the correlation functions of holomorphic func- +tions, defined recursively in Landau-Ginzburg-Saito theory with exponential mirror +superpotential and tropical good section. + +Contents +1 +Introduction +3 +2 +Mirror Correspondence +5 +2.1 +Toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Gromov-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.3 +Tropical GW invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.4 +Mirror relation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3 +Mirror for correlation functions +11 +3.1 +Landau-Ginzburg-Saito theory . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.2 +Correlation functions in Landau-Ginzburg-Saito theory . . . . . . . . . . . . +14 +3.3 +Cohomology and pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.4 +Contact terms from good section +. . . . . . . . . . . . . . . . . . . . . . . . +17 +4 +HTQM on trees +19 +4.1 +Higher topological quantum mechanics . . . . . . . . . . . . . . . . . . . . . +19 +4.2 +HTQM on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4.3 +Amplitudes on trees +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +4.4 +Amplitudes in homotopy notation . . . . . . . . . . . . . . . . . . . . . . . . +22 +4.5 +Deformation of HTQM by a state . . . . . . . . . . . . . . . . . . . . . . . . +24 +4.6 +Diagrammatic representation of deformed theory . . . . . . . . . . . . . . . . +31 +5 +Correlation functions in HTQM +32 +5.1 +3-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +5.2 +4-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +5.3 +Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +5.4 +Invariance theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5.5 +Recursion relation for correlation functions . . . . . . . . . . . . . . . . . . . +38 +6 +Mirror for HTQM +41 +6.1 +A-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +6.2 +Correlation functions in A-model +. . . . . . . . . . . . . . . . . . . . . . . . +42 +6.3 +Dual variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +6.4 +B-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +1 + +7 +Localization of mirror states +48 +7.1 +Mirror states vs divisor states . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +7.2 +Spectral sequence for QW-cohomology +. . . . . . . . . . . . . . . . . . . . . +51 +7.3 +Pairing and localization of states +. . . . . . . . . . . . . . . . . . . . . . . . +51 +7.4 +Higher pairings in B-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +7.5 +B-model deformation by a holomorphic function . . . . . . . . . . . . . . . . +57 +7.6 +Higher pairing for mirror states . . . . . . . . . . . . . . . . . . . . . . . . . +58 +8 +Correlation functions for mirror states +60 +8.1 +4-point function invariance and holomorphic representatives +. . . . . . . . . +61 +8.2 +n-point function invariance and holomorphic representative . . . . . . . . . . +63 +8.3 +3-point functions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +65 +8.4 +4-point functions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +66 +8.5 +Contact terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +8.6 +5-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +8.7 +Parallel transport of a good section . . . . . . . . . . . . . . . . . . . . . . . +73 +8.8 +Localization of correlation functions . . . . . . . . . . . . . . . . . . . . . . . +73 +2 + +1 +Introduction +The real tropical numbers is a set of real numbers, extended by {−∞} with arithmetic +operations: tropical addition x +T y = max(x, y) and tropical multiplication x ∗T y = x + y. +The set of tropical numbers is a semigroup with respect to tropical addition. The tropical +numbers appeared at different times at several branches of mathematics: Maslov introduced +notion of tropical integration, while computer scientist Imre Simon introduced and adjective +tropical. +Mikhalkin [1], [2], [3] used tropical numbers to study geometric problems. The collection +of problems and methods become the Tropical Geometry research area. In many cases the +enumerative problems in algebraic geometry over complex numbers have tropical counter- +parts. In particular, we can define the tropical Gromov-Witten invariants by counting graphs +passing through some cycles, see Mikhalkin [1,2]. +Mikhalkin [2] observed that the number of tropical curves in P2 of degree 3, passing +through 8 points in general position is 12. This number, matches with counting of degree-3 +algebraic curves passing through 8 points. Gathmann and Markwig [4] showed that equality +generalizes for all Gromov-Witten invariants of P2. The matching of tropical and complex +invariants was formalized into: +Theorem (tropical correspondence): Gromov–Witten invariant coincides with its trop- +ical counterpart. +We propose to analyze the tropical correspondence theorem in context of mirror symme- +try. +Theorem (mirror symmetry): The Gromov–Witten invariants for X equal to the corre- +lation functions in Landau-Ginzburg-Saito theory on mirror space X∨. +The mirror symmetry for toric space X relates [5] the GW invariants to Landau-Ginzburg- +Saito theory with exponential superpotentials, see [6] for review. +Theorem (tropical mirror symmetry): The tropical Gromov–Witten invariants for toric +space X of complex dimension N equal to the correlation functions in Landau-Ginzburg- +Saito theory on mirror space X∨ = C∗N with certain exponential superpotential, canonical +holomorphic top form and mirror K. Saito’s good section. +3 + +Our main result is the proof of the tropical mirror symmetry theorem. There are two possible +application of our result: We can assume the tropical correspondence theorem and use our +proof as a new proof of mirror symmetry for the toric spaces. Alternatively, we can assume +the mirror symmetry theorem and use our proof as a proof for the tropical correspondence +for toric spaces. +In our proof we use several key ideas: tropical numbers as a scaling of the exponential +map, quantum mechanics representation of the tropical GW invariants, mirror relation for +quantum mechanics as summation over divisor states and localization for HTQM correlation +functions. +Real tropical numbers can be constructed from the field of real numbers with usual +addition and multiplication by performing the exponential map X = e +x +ǫ followed by the +limit ǫ → 0. The exponential map turns multiplication into addition for any value of ǫ, while +the tropical addition requires the limit +lim +ǫ→0 ǫ ln(e +x +ǫ + e +y +ǫ ) = max(x, y) = x +T y. +The scaling of exponential map generalizes to the complex geometry. The scaling construc- +tion also provides an heuristic proof of the Tropical Correspondence theorem. +The Gromov-Witten invariants over complex numbers can be described using A-type +twisted topological string theory [7]. In our work [8] we proved that the tropical GW in- +variants can be described using the higher topological quantum mechanics (HTQM) with +circle action on graphs, introduced in [9]. Certainly, the HTQM can be constructed as the +ǫ → 0 limit of maximally degenerate complex structure (very long strings) in topological +string theory, but the fact that it captures all tree-level tropical GW invariants is a novel +result. Moreover in [10] (see also [9]) it was shown that the quantum mechanics, similar to +the one we described in previous paper, provides a solution to the WDVV equations. +Authors of [11] used the 2d CFT to relate the sum over holomortex insertions in A-type +topological string with superpotential deformation of the 2d CFT. In our paper [8] we showed +that the sum over divisor states in A-type HTQM amplitudes equals to the amplitudes in B- +type HTQM. The B-type HTQM is the deformation of the A-type HTQM by the boundary +divisor states. In particular we showed that for the case of toric spaces the B-type HTQM +has exponential superpotential. +The topological nature of the B-type topological string allows for the drastic simplification +for correlation functions of certain observables. In particular, the correlation functions of the +4 + +mirrored evaluation observables can be written using the recursive construction, discussed +in section 8 of present work. The base of the recursion, the 3-points functions are evaluated +in terms of the residue formula, constructed from superpotential and holomorphic top form. +Such simplification is a reflection of a localization-like phenomenon. Typically, localization +in field theory and quantum mechanics requires path integral formulation, while we showed +that the localization construction can be realized using the operator formalism in quantum +mechanics. +The recursive construction for implies that the superpotential alone is not enough to +evaluate the correlation functions for more than 3 observables! The additional data is given in +the form of good section, introduced and studied by K. Saito [12]. The correlation functions in +Landau-Ginzburg-Saito theory have non-trivial dependence on the choice of good section. As +a part of the proof for the main theorem we derive the good section for mirror superpotentials +from the tropical mirror correspondence. +The structure of our paper is as follows: In section 2 we briefly review the mirror symme- +try and formulate it in the form of the equality for correlation functions. In section 3.1 we +review the recursive construction for correlation functions in Landau-Ginzburg-Saito theory. +In section 4 we define the Higher Topological Quantum Mechanics on trees, describe the +amplitudes and describe a deformation of HTQM by special state. In section 5 we introduce +the notion of correlation functions for HTQM and discuss their properties. +In section 6 +we briefly review the HTQM representation for tropical GW invariants and mirror relation +between A- and B- types HTQMs. In section 7 we will introduce a notion of localization +for mirror states, while in the last section we will use the localized states to evaluate the +correlation functions in mirror HTQM. +2 +Mirror Correspondence +The mirror symmetry describes a relation between the A- and B-models. +The A-model +of our interest is the theory of the Gromov-Witten invariants for a Kahler manifold X of +dimension N. The B-model is the theory of complex structure deformations on the dual +complex manifold X∨ of same dimension N. In well-known examples both X and X∨ are +compact Calabi-Yau 3-folds. +We can generalize the correspondence away from three-dimensional Calabi-Yau spaces +by relaxing the compactness condition on X∨. The complex structure deformations in that +case are also parametrized by a holomorphic a function W and the corresponding B-model in +commonly referred to as the Landau-Ginzburg (LG) theory with holomorphic superpotential +5 + +W. +In our paper we will discuss the A-model on a toric space X of dimension N. Furthermore, +we will perform a certain scaling procedure to the geometry, which transforms the usual GW +invariants into the tropical ones. The corresponding mirror B-model becomes the LG theory +on X∨ = C∗N with exponential superpotential. +In the rest of this section we will briefly review the definition of toric manifolds, tropical +GW invariants and give a detailed formulation of the mirror relation. +2.1 +Toric varieties +Toric manifold X is a compactification of C∗N. We can represent C∗N = RN × TN with +the radial part RN, equipped with standard coordinates ri, i = 1, .., N and angular part, +N-dimensional torus TN = (S1)N, with standard angular coordinates φi. Equivalently, we +can say that the C∗N is a trivial N-dimensional toric fibration over RN. We describe the +compactification of C∗N using the fibraton data. +• The radial part is compactified by the hyperplanes at infinity. Each hyperplane is given +in terms of the inside-pointing N-dimensional normal vector ⃗ba with components bi +a. +Each normal vector has integer components i.e. bi +a ∈ Z. The two normal vectors with +proportional components describe the same hypersurface. We can fix this ambiguity +by choosing the primitive normal vector, such that gcd(b1 +a, b2 +a, ..., bN +a ) = 1. For toric +space X we will denote the set of such normal vectors by BX = {⃗ba}. +• In order to get a compactification of a complex manifold, we require that one of the +circles S1 ⊂ TN inside the toric fibration shrinks to zero when we approach each of +the compactifying hypersurfaces. The choice of a circle is given by a class in π1(TN) = +H1(TN, Z). For the hyperplane with normal vector ⃗ba the corresponding class is the +class of � bi +adφi. +We will refer to the compacting hyperplanes as compactifying divisors, and to BX as the +set of all compactifying divisors for toric space X. +2.2 +Gromov-Witten theory +The Gromov-Witten invariant NX +β (C1, ..., Cn) counts the number of algebraic curves of +degree-β, genus-0 in complex space X, passing through the cycles C1, ..., Cn. The integral +6 + +representation +NX +β (C1, ..., Cn) = +� +M0,n(X,β) +n� +α=1 +ev∗ +αγα, +(2.1) +uses M0,n(X, β), the moduli space of degree β curves in X with n marked points, compact- +ified by quasi-maps, equipped with the evaluation map +evα : M0,n(X, β) → X : (φ : P1 → X, z1, ..., zn) �→ φ(zα). +(2.2) +The γα are special representatives (smoothed out delta functions on cycles) of the Poincare +dual to the cycles Cα. In mathphysics literature the commonly used notation for the same +GW invariant is ⟨γ1, .., γn⟩X +β,0 and we will use it in our paper. +For a given cycles γk ∈ H∗(X) we can organize the genus-0 GW invariants of different +degrees into a single expression +⟨γ1, .., γn⟩X = +� +β∈H1,1(X) +qβ⟨γ1, .., γn⟩X +β,0 +(2.3) +where q, describes the Kahler moduli of X. For generic X the GW invariants are formal +series in q, but for toric X they simplify to the polynomials in q. +Remark: In present paper we will restrict our attention to the GW invariants for 3 or +more observables i.e n ≥ 3. There are interesting geometric invariants of X, with natural +description in the form of GW invariants for two observables, but most of them can be +reformulated as invariants with 3 or more observables. +2.3 +Tropical GW invariants +On toric manifold X we can perform a tropical limit: a coordinate transformation in the +form of scaling (rk, φk) → (rk/ǫ, φk) followed by the ǫ → 0 limit. For more details see our +previous paper [8] . The limit of a smooth algebraic curve of genus zero in toric space X is a +circle bundle over a tree. The tree is embedded by a piece-wise linear map into radial part of +X. The limiting curve is known as a tropical curve and was extensively studied by Mikhalkin +and collaborators [2] in context of Tropical Geometry. The moduli of a tropical curve are +position of a root vertex, lengths and twists of internal edges of a tree. We discussed the +moduli space in our work [8], while for more detailed review see Mikhalkin’s book [3]. +We can take the tropical limit for the differential forms γα to define the tropical A-model +7 + +observables. Note that the tropical limit turns smooth forms on toric space X into singular +forms on radial part of X. +We can define the tropical GW invariants as the integral over tropical moduli space +of tropical observables. In our work [8] we showed that the tropical GW invariants can be +written as a sum of amplitudes in Higher Topological Quantum Mechanics (HTQM) on trees. +We will provide the detailed definition of HTQM and describe the amplitudes in section 4. +In many examples it was observed that the tropical GW invariants match with the con- +ventional GW invariants. This observation was formalized into: +Theorem (tropical correspondence): For the toric space X and smooth differential +forms γk ∈ H∗ +dR(X) the Gromov-Witten invariant ⟨γ1, .., γn⟩X +β matches with tropical Gromov- +Witten invariant ⟨γtrop +1 +, .., γtrop +n +⟩X +β for the tropical limit γtrop +k +of the forms γk. +Evidence: There are several different types of evidence for the theorem: +• Mikhalkin [2] counted the number degree-3 tropical curves in P2 passing through the +8 points in general position, by presenting the corresponding graphs, counted with +proper multiplicities. The total number he obtained was 12, what matched with the +well known N3 = 12 answer for the same problem for algebraic curves. +• Gathmann and Markwig [13] derived the recursion formula for number of tropical +curves of degree-d, passing through the 3d − 1 points on P2. Their result matches +with the Kontsevich-Manin recursion formula [14] for algebraic curves. Such matching +essentially gives a prof of tropical correspondence for P2. +• There are several results on match for the descendant GW invariants: The case of P1 +was discussed by B¨ohm, Goldner and Markwig [15]. Markiw and Rau [16] proved the +equality for descendant invariants for P2. +• Our construction of tropical numbers and geometric objects as a scaling limit ǫ → 0 in +cylindrical coordinates serves as an heuristic proof of the theorem. The GW invariants +do not depend on the choice of coordinates, hence remain the same for any non-zero +value of ǫ. +• In [8] we derived the tropical mirror superpotential for toric X and it matches with +exponential mirror superpotential derived in [5], [17] for the same toric space X. Given +8 + +that the mirror symmetry relation holds for toric spaces we can use our result as an +evidence in favor of the tropical correspondence theorem. +2.4 +Mirror relation +The mirror of the N-dimensional toric manifold X is a non-compact N-dimensional Calabi- +Yau X∨ = C∗N with holomorphic superpotential. +We will use the toric representation +C∗N = RN × TN with radial coordinates rj and angular (holomorphic) coordinates Yj. The +holomorphic top form in these coordinates is Ω = dY 1 ∧ .. ∧ dY N. Let us formulate several +relations, implied by the mirror correspondence theorem. +Definition: The Jacobi ring for superpotential W is +JW = RC∗N/IW, +(2.4) +where RC∗N is the ring of holomorphic functions on C∗N. In our coordinates RC∗N is the ring +of periodic functions of Y . The IW is the ideal generated by the partial derivatives of W +IW = +�∂W +∂Yj +� +. +(2.5) +Remark: If W has isolated critical points then JW is finite-dimensional. +Proposition (mirror for observables): The de Rahm cohomology of toric space X iso- +morphic (as a vector spaces !) to the Jacobi ring +J : H∗ +dR(X) → JWX : γ �→ Jγ +(2.6) +for mirror superpotential +WX = +� +⃗b∈BX +q⃗b ei⟨⃗b,⃗Y ⟩, +(2.7) +where the sum is taken over the compactifying divisors BX of X. +We can refine the mirror relation for observables to the isomorphism of rings. +Proposition (mirror for Frobenius rings): The quantum cohomology of toric space +X isomorphic (as graded-commutative Frobenius rings) to the Jacobi ring for mirror super- +9 + +potential. +The Frobenius ring structure (CA, gA) on quantum cohomology ring of X is determined +by the 3-point GW invariants of X +CA +αβδ = ⟨γα, γβ, γδ⟩X, +gA +αβ = ⟨γα, γβ, 1⟩X = +� +X +γα ∧ γβ, +(2.8) +while the Frobenius ring structure (CB, gB) for Jacobi ring can be formulated using the +residue formula. Hence the mirror relation for rings can be formulated as equality +CB +αβδ = +� +dWX=0 +Φγα Φγβ Φγδ +det ∂k∂lWX +, gB +αβ = +� +dWX=0 +Φγα Φγβ +det ∂k∂lWX +, +(2.9) +where Φγ is a representative of a class Jγ. One can show that (2.9) is independent on the +choice of representative. There is a further generalization of the mirror relation for rings, +which includes all GW invariants of X. +Let γ1, .., γn be a basis in H∗ +dR(X) and T k − linear coordinates on this space in this basis. +We can organize all genus-0 GW invariants for X into the generating function +FA(T, q) = +∞ +� +j1,..,jn=0 +⟨γ1, .., γ1 +� �� � +j1 +, ..., γn, .., γn +� �� � +jn +⟩X +n +� +k=1 +(T k)jk +jk! +. +(2.10) +Parameters q represent the Kahler moduli dependence for X in A-model. +The B-model generating function was defined in works of K.Saito [12], Blok-Varchenko [18] +and Dijkgraaf-Verlinde-Verlinde [19]. Namely +∂3FB(T, q) +∂T α ∂T β ∂T δ = +� +dW =0 +Φα(T) · Φβ(T) · Φδ(T) +det ∂k∂lW(T) +, +(2.11) +where W(T) is the deformation of mirror superpotential in special coordinates, introduced +by K. Saito, the holomorphic functions Φα(T) are partial derivatives of superpotential, i.e. +Φα(T) = ∂W +∂T α. +(2.12) +The partial derivatives ∂k∂lW of W are taken in coordinates Y , where the holomorphic top +form is dY1 ∧ .. ∧ dYN. +10 + +Theorem (mirror correspondence): For toric space X the generating function of the +GW invariants equals to the B-model generating function (2.11) for deformations of mirror +superpotential WX i.e. +FA(T, q) = FB(T, q). +(2.13) +Remark: The equality above is an equality for the formal series in q and T. For the toric +GW invariant each coefficient in T-expansion is a polynomial in q, rather than the formal +series. Therefore for the case of toric A-model the corresponding mirror B-model expression +should also be a polynomial. Hence we will focus on proving the equality of two polynomials +for the coefficients of formal series in T. In the next section we will give a description for +the B-model coefficients in the generating function expansion. +3 +Mirror for correlation functions +For arbitrary three holomorphic functions we define the 3-point correlation function +⟨Φα, Φβ, Φγ⟩W = +� +dW =0 +Φα · Φβ · Φγ +det ∂k∂lW . +(3.1) +Then the generating function of B-model (2.11) can be written in the form +∂3FB(T, q) +∂T α ∂T β ∂T γ = ⟨Φα(T), Φβ(T), Φγ(T)⟩W (T) . +(3.2) +In particular, the relation above implies that the cubical term in T-expansion is given by +FB(T, q) = 1 +3!T αT βT γ ⟨Φα, Φβ, Φγ⟩W + O(T 4), +(3.3) +where Φα = Φα(0) are representatives for the classes in Jacobi ring JW for W = W(0). The +quartic term in the expansion for FB can be expressed in the form +∂4FB +∂T α ∂T β ∂T γ ∂T δ +��� +T=0 = +∂ +∂T δ +��� +T δ=0 ⟨Φα, Φβ, Φγ⟩W +T δΦδ + +� ∂ +∂T δ +��� +T δ=0Φα(T), Φβ, Φγ +� +W ++ +� +Φα, +∂ +∂T δ +��� +T δ=0Φβ(T), Φγ +� +W ++ +� +Φα, Φβ, +∂ +∂T δ +��� +T δ=0Φγ(T) +� +W +. +(3.4) +11 + +The expression above has the following interpretation: The first term is the change in 3- +point correlation function under the change of W in the direction of Φδ. The last three +terms describe the change of the functions Φα(T) under the transport in direction of Φδ. We +can introduce connection CW, so the change of a function Φα(T) in direction Φδ is given by +∂ +∂T δ +��� +T δ=0Φα(T) = CW(Φδ, Φα). +(3.5) +The combination of four terms in the formula (3.4) is known as the recursion formula for +the 4-point function in 2-dimensional topological theory. The commonly used form for the +recursion relation is (see also [20]) +⟨Φα, Φβ, Φγ, Φδ⟩W = d +dǫ +��� +ǫ=0⟨Φα, Φβ, Φγ⟩W +ǫΦδ + ⟨CW(Φδ, Φα), Φβ, Φγ⟩W ++ ⟨Φα, CW(Φδ, Φβ), Φγ⟩W + ⟨Φα, Φβ, CW(Φδ, Φγ)⟩W. +(3.6) +Note that the 4-point function depends on a choice of holomorphic function representatives +for the classes in Jacobi ring. +The connection coefficients CW(Φδ, Φα) are referred to in [20] as the contact terms. The +fourth derivative of the generating function can be written in terms of the 4-point function, +defined by (3.6) +∂4FB +∂T α ∂T β ∂T γ ∂T δ +��� +T=0 = ⟨Φα, Φβ, Φγ, Φδ⟩W. +(3.7) +The recursion relation (3.6) can be further generalized to the case of n-point functions. In +physics literature the recursion relations of that type are commonly discussed in context of +the Landau-Ginzburg theory [19], so we will adopt the same name for our review of it. We +will give a detailed version of the recursion formula and contact terms later in this section. +The expansion coefficients of the generating function become the n-point functions, i.e. +∂nFB(T, q) +∂T α1...∂T αn +��� +T=0 = ⟨Φα1, ..., Φαn⟩W. +(3.8) +The mirror relation for generating functions implies the mirror relation for the correlation +functions: +Proposition (mirror for correlation functions): For toric space X the GW invari- +ants fo cycles γα equal to the correlation function of special representatives SJγα of the +12 + +corresponding the Jacobi ring classes Jγα for the mirror superpotential. +⟨γ1, ..., γn⟩X = ⟨SJγ1, ..., SJγn⟩W X. +(3.9) +3.1 +Landau-Ginzburg-Saito theory +The study of Landau-Ginzburg theory was motivated by the theory of critical phenomena, +which later grown into 2D CFT and eventially become part of the topological string theory. +For review see [17], namely the (2, 2)-supersymmetric sigma models on non-compact spaces +in B-type twisting. After the B-type twisting, the theory is not superconformal and further +requires setting the anti-holomorphic superpotential to zero, while keeping the holomorphic +superpotential W fixed. B-twisting is anomalous that results in appearence of the holomor- +phic top form. +Definition: The Landau-Ginzburg-Saito theory on complex space X with superpotential +W is a collection of the following data: +1. non-compact complex manifold X of dimension N; +2. non-degenerate holomorphic top form +Ω ∈ ΩN,0(X); +(3.10) +Note that the pair (X, Ω) can be constructed from non-compact Calabi-Yau manifold. +3. holomorphic function W : X → C, called superpotential; +4. Good section S : JW → RX. +The readers, familiar with mathphysics literature on the Landau-Ginzburg models, might +not be familiar with the notion of good section and its significance for the LGS theory. +Indeed all of the LGS correlation functions can be obtained from the T-expansion of the +generating function FLG(T) defined via the 3-point function +∂3FLG(T) +∂T α ∂T β ∂T γ = +� ∂W +∂T α, ∂W +∂T β , ∂W +∂T γ +� +W (T) +. +(3.11) +However, in order to use the formula (3.11) we need to define the good times T, which typ- +ically have complicated functional relation to the deformations of W, linear on Jacobi ring. +13 + +The good section data is equivalent to the choice of good times T, but is has more straight- +forward meaning for the recursive definition of correlation functions. In some simple cases +(polynomial superpotentials) the good section can be constructed from the superpotential +if we impose an extra requirement of homogeneity. For more information about the good +times and good section relation see [12] and [20]. +The most studied LGS theories have complex manifold CN, with coordinates xj, canonical +holomorphic top form Ω = dx1 ∧..∧dxN and polynomial superpotential W(x). The simplest +LGS model of this class has single variable (N = 1) and polynomial superpotential of degree +k. The image of good section in this case is well known and consists of monomials of degree +up to k − 2 +Im S = C⟨1, x, .., xk−2⟩. +(3.12) +Note that in case of polynomial superpotential with two variables, the good section is known +only for limited classes of superpotentials. +Our main interest is the mirror of the A-model for toric manifold X of complex dimension +N, given in terms of compactifying divisors BX. The corresponding LGS theory has complex +manifold C∗N equipped with cylindrical coordinates: the radial coordinates rj ∈ R and +holomorphic angular coordinates Yj ∈ S1. The holomorphic top form in this coordinates is +Ω = dY1 ∧ ... ∧ dYN, +(3.13) +while the superpotential is the exponential function, written using the primitive normal +vectors +WX = +� +⃗b∈BX +q⃗b ei bkYk. +(3.14) +The good sections has not been constructed for all LGS theories with exponential superpo- +tentials. The simplest exponential superpotential is the mirror superpotential for X = P1 +WP1 = eiY + qe−iY . +(3.15) +The image of good section is +Im SP1 = C⟨1, eiY ⟩. +(3.16) +3.2 +Correlation functions in Landau-Ginzburg-Saito theory +Definition: For holomorphic functions Φα, α = 1, .., n > 2 in LGS theory on C∗N with +14 + +superpotential W the n-point correlation function ⟨Φ1, ..., Φn⟩W is defined recursively: +• the 3-point function is given by the residue formula +⟨Φ1,Φ2, Φ3⟩W = +� +dW =0 +Φ1Φ2Φ3 +det ∂j∂kW +(3.17) +• The (n+1)-point function is defined recursively via n-point functions and their deriva- +tives according to formula below +⟨Φ1, Φ2, .., Φn, Φn+1⟩W = d +dǫ +��� +ǫ=0⟨Φ1, Φ2, .., Φn⟩W +ǫΦn+1 + ⟨CW(Φ1, Φn+1), Φ2, .., Φn⟩W ++ ⟨Φ1, CW(Φ2, Φn+1), .., Φn⟩W + .. + ⟨Φ1, Φ2, .., CW(Φn, Φn+1)⟩W +(3.18) +Earlier we saw that the n-point correlation functions represent the coefficient in the generat- +ing function (3.8) hence they are symmetric under permutation of all arguments Φ1, Φ2, ..., Φn. +The 3-point function in our definition is manifestly symmetric. The symmetry of higher point +functions is rather obscure from the recursive definition and require certain properties of the +contact terms CW. Following the literature [12] and [20], we will formulate this properties +in terms of the K. Saito’s connection on Brieskorn cohomology. +Proposition: The correlation functions in (3.18) are symmetric if +• connection is symmetric; +• connection is flat; +• connection preserves the metric. +In [20] was proposed a construction of CW in terms of the K. Saito’s good section S. Such +connection is manifestly symmetric, while the flatness and metric preservation are derived +from the properties of a good section. +3.3 +Cohomology and pairing +In order to give a definition of good section and contact terms we will introduce a cohomology +theory, motivated by topological string theory of type B. +15 + +Let us consider a graded vector space +VLGS = RC∗N ⊗ C[ψi +Φ] +(3.19) +for parity-odd variables ψi +Φ. On VLGS there is a pair of graded-commuting differentials +QW = ∂W +∂Yj +∂ +∂ψj +Φ +, +G− = +∂ +∂Yj +∂ +∂ψj +Φ +. +(3.20) +Remark: VLGS is isomorphic to the space of polyvector fields on C∗N, hence it is equipped +with parity-odd symplectic structure and holomorphic top form. The G− is a Batalin–Vilkovisky +(BV) operator on VLGS, which generalizes the divergence on vector fields to polyvector fields. +Remark: The local holomorphic observables of dimension-0 in topological string theory +of type B can be identified with polyvector fields. The G− is the action of the superpartner +to certain U(1)-rotation, which preserves insertion positions for these observables. +Definition: On VLGS there is C[[z]]-valued Saito’s higher residue pairing +K(v1, v2) = +� +S1N dNY +� +RN dNr +� +dNψΦdNψR v1 ∧ e−iΛ{QW +zG−+dR,L}v2, +(3.21) +where Λ is a real parameter, +L = +N +� +k=1 +rkψk +Φ +(3.22) +is the localization function and +dR = ψj +R +∂ +∂rj +(3.23) +is the radial de Rham operator. +Remark: The integral form of the K. Saito’s pairing (3.21) was proposed in [20]. +Definition: The C-valued higher pairings K(k) are defined as expansion coefficients in z- +expansion of K i.e. +K(v1, v2) = +∞ +� +k=0 +zk K(k)(v1, v2). +(3.24) +In section 7.4 we will discuss a similar pairing in details, so for now let us list some properties +16 + +of the pairing without the proof: +• The operators QW − zG− and QW + zG− are conjugated with respect to the pairing +(3.21) i.e. +K((QW − zG−)v1, v2) = −(−1)|v1|K(v1, (QW + zG−)v2). +(3.25) +Hence we can descend the pairing to the pairing on cohomology H∗(QW − zG−) ⊗ +H∗(QW + zG−). One can show that all cohomology for QW ± zG− are holomorphic +functions. +• The pairing on H∗(QW − zG−) ⊗ H∗(QW + zG−) is independent of Λ. Hence we +can choose Λ → ∞, what localizes the pairing on a sum over critical points of W. In +particular, the first two pairings on holomorphic functions +K(0)(Φ1, Φ2) = (2πi)N � +dW =0 +Φ1Φ2 +det ∂i∂jW +(3.26) +and +K(1)(Φ1, Φ2) = (2πi)N 1 +2 +� +dW =0 +(∂k∂lW)−1(Φ1 ∂k∂lΦ2 − Φ2 ∂k∂lΦ1) +det ∂m∂nW +. +(3.27) +• We can use the pairing K(0) to establish an isomorphism between the Jacobi ring JW +and H∗(QW). +Remark: The cohomology of QW + zG− were introduced by K. Saito under the name of +Brieskorn lattice. +3.4 +Contact terms from good section +The construction of Jacobi ring comes with canonical projection πW : RC∗N → JW. Given +a pair of homolorphic functions Φ1 and Φ2 we can project their product Φ1Φ2 to the class +πW(Φ1Φ2) in Jacobi ring JW. The section (which inverts πW) SW : JW → RC∗N turns this +class into holomorphic function SW πW(Φ1Φ2). The difference +Φ1Φ2 − SW πW(Φ1Φ2) +(3.28) +17 + +is trivial in Jacobi ring. An isomorphism between the JW and H∗(QW) means that there +exists a map ΣW : RC∗N → VLGS such that +Φ1Φ2 − SWπW(Φ1Φ2) = QWΣW(Φ1Φ2), +(3.29) +and +ΣWSW = 0. +(3.30) +The choice of such ΣW is known as the choice of homotopy for QW. +Definition: We define a contact term fo Φ1 and Φ2 in LGS theory with section SW +CS +W(Φ1, Φ2) = ±G−ΣW(Φ1Φ2). +(3.31) +In other terms the product of two functions Φ1Φ2 can be decomposed into the sum of the +image of SW and a linear combination of ∂1W, .., ∂NW, i.e. +Φ1Φ2 = SWπW(Φ1Φ2) + σk∂kW +(3.32) +The ΣW(Φ1Φ2) has the form σk(Y )ψk +Φ, so G−-action on it is +G−ΣW(Φ1Φ2) = ∂σk(Y ) +∂Yk +, +(3.33) +i.e. just a divergence of the vector field σk(Y )∂Yk. Note that for a given SW the decomposition +in (3.32) does not uniquely fixes the σk(Y ). The freedom of choice σ is fixed by the choice +of homotopy ΣW. +Note that the dependence of contact term CW on the choice of homotopy ΣW is (QW + +zG−)-exact. It was shown that the correlation functions are well-defined in H∗(QW +zG−), +so the choice of homotopy does not affect the recursion formula. +We can define a natural projection π : VLGS ⊗ C[[z]] → VLGS, given by an evaluation at +z = 0. The projection π is a chain map, hence it induces projection on cohomology +π : H∗(QW + zG−) → H∗(QW). +(3.34) +The section SW induces a section SSaito : JW = H∗(QW) → H∗(QW + zG−). Indeed, every +holomorphic function is both QW- and G−-closed, hence it describes a class in H∗(QW + +18 + +zG−), which we take as an image of the SSaito-map. +Definition: The good section SSaito : H∗(QW) → H∗(QW + zG−) is +• a section for π i.e. +π ◦ SSaito = IdH∗(QW ); +(3.35) +• the higher pairings (3.24) vanish for all pairs Φ1, Φ2 ∈ Im SSaito i.e +K(k)(Φ1, Φ2) = 0, ∀ k > 0; +(3.36) +• For a given section SSaito we can construct the corresponding contact term and con- +nection. The good section SSaito is preserved under the parallel transport respect to +this connection. +4 +HTQM on trees +In previous work [8] we showed that the tropical Gromow-Witten invariants can be described +using the higher topological quantum mechanics (HTQM) on tree graphs. In this section +we will briefly review the definition on the HTQM on trees, describe the amplitudes and +construct a family of HTQMs as a deformation of HTQM by a certain type of states. +4.1 +Higher topological quantum mechanics +Definition: The higher topological quantum mechanics, HTQM (with the circle action) +(V, Q, G±) is a collection of the following data: +1. Z2-bi-complex (V, Q, G−), namely: +• Z2-graded vector space V , can be infinite-dimensional. There is a decomposition +of V = V0 ⊕ V1 into even V0 and odd V1 under the grading. We will use the +notation |v| ∈ Z2 to describe the grading of a vector v ∈ V ; +• pair of differentials Q, G− : V → V , such that +– grading-odd operators: |Qv| = |G−v| = |v| + 1, +– square to zero: Q2 = G2 +− = 0; +• two differentials graded-commute, i.e. {Q, G−} = 0. +19 + +2. Unnormalized homotopy G+ : V → V , such that +• grading-odd operator: |G+v| = |v| + 1, +• squares to zero G2 ++ = 0, +• {G+, G−} = 0. +In case V is infinite-dimensional we impose certain consistency conditions on HTQM data +(V, Q, G±). We define the Hamiltonian operator H = {Q, G+} : V → V . The consistency +conditions are formulated in terms of Hamiltonian: +• The hamiltonian H is such that the evolution operator e−tH is well defined for t ≥ 0 +in the following sense: +– it is a solution to the ODE +(∂t + H)e−tH = 0, +e−0·H = 1, +t ∈ R+ ∪ {0}; +(4.1) +– forms a 1-parameter semi-group with multiplication +e−t1He−t2H = e−(t1+t2)H, ∀ t1, t2 ∈ R+ ∪ {0}. +(4.2) +• We require that the t → ∞ limit of the evolution operator exists and is the projector +on ker H, i.e. +lim +t→+∞ e−tH = Π0. +(4.3) +• The projector Π0 obeys +Π0G± = G±Π0 = 0. +(4.4) +4.2 +HTQM on trees +Definition: The HTQM (V, Q, G±, µ2, g) on a connected tree Γ with distinct root vertex is +the collection of the following data +1. 1-valent vertices are assigned the HTQM states i.e. va ∈ V, a = 1, .., n1 = |V1(Γ)|. +2. 2-valent vertices are assigned observables Oα ∈ V ⊗ V ∗ α = 1, .., n2 = |V2(Γ)|. +3. 3-valent vertices are assigned the multiplication µ2 : V ⊗ V → V such that the triple +(Q, G−, µ2) obeys +20 + +• µ2 is graded commutative +µ2(v, w) = (−1)|v||w|µ2(w, v); +(4.5) +• µ2 is associative +µ2(µ2(v, w), u) = µ2(v, µ2(w, u)); +(4.6) +• Leibniz rule for (µ2, Q) +Qµ2(v, w) = µ2(Qv, w) + (−1)|v|µ2(v, Qw); +(4.7) +• the pair (G−, µ2) obeys the 7-term relation for all v, u, w ∈ V +G−µ2(µ2(v, w), u) = µ2(G−µ2(v, w), u) + (−1)|w|(|v|−1)µ2(w, G−µ2(v, u)) ++ (−1)|v|µ2(v, G−µ2(w, u)) − µ2(G−v, µ2(w, u)) − (−1)|v|µ2(v, µ2(G−w, u)) +− (−1)|u|+|v|µ2(v, µ2(w, G−u)). +(4.8) +4. the root 3-valent vertex assigned the multiplication +µ0 +3 = g ◦ µ2 : V ⊗3 → R +(4.9) +constructed from Frobenius structure (g, µ2, Q), where the scalar product, commonly +referred to as the pairing, g : V ⊗ V → R obeys the following properties: +• non-degeneracy on V ; +• the graded-symmetry +g(v, w) = (−1)|v||w|g(w, v); +(4.10) +• Q-invariance +g(Qv, w) + (−1)|v|g(v, Qw) = 0; +(4.11) +• G±-invariance +g(G±v, w) = (−1)|v|g(v, G±w); +(4.12) +• evolution invariance +g(e−tHv, w) = g(v, e−tHw). +(4.13) +21 + +4.3 +Amplitudes on trees +Definition: The evolution operator in HTQM (V, Q, G±) is +U(t, dt, dϕ) = e−tH+G+dt+G−dϕ ∈ Ω∗(R+ × S1) ⊗ End(V ). +(4.14) +Definition: For each tree Γ we define pre-amplitude +PAΓ : V ⊗n1(Γ) ⊗ (V ⊗ V ∗)⊗n2(Γ) → Ω∗(M(Γ)), +(4.15) +where M(Γ) is the moduli space of trees Γ, defined in (4.19). Each connected tree Γ defines +a contraction in tensor algebra +⟨ ⟩Γ : (V ⊗ V ∗)⊗E ⊗ V ⊗n1 ⊗ (V ∗ ⊗ V )⊗n2 ⊗ (V ∗⊗2 ⊗ V )⊗(n3−1) ⊗ V ∗⊗3 → R, +(4.16) +where n3 is the number of 3-valent vertices in Γ. The pre-amplitude on a tree Γ with states +va and operators Oα is +PAΓ(va; Oα) = +� +U⊗I(Γ) ⊗ 1⊗n1 +n1 +� +a=1 +va +n2 +� +α=1 +Oα ⊗ µ⊗(n3−1) +2 +⊗ µ0 +3 +� +Γ +. +(4.17) +Note that the pre-amplitude on tree Γ has no evolution operators on external edges (edges +attached to leaves of a tree Γ). +Definition: The amplitude on connected tree Γ is an integral +AΓ = +� +M0(Γ) +PAΓ, +(4.18) +over moduli space of connected tree +M0(Γ) = +� +R+ × S1�I(Γ) . +(4.19) +4.4 +Amplitudes in homotopy notation +Definition: The propagator K : V → V for HTQM (V, Q, G±) is +K = lim +T→∞ +� T +0 +dt e−tH G+ = +� ∞ +0 +dt e−tH G+. +(4.20) +22 + +Note, that the integral has potential divergence, when the exponent vanishes for states from +ker H. The G+ in the expression (4.20) and the HTQM property (4.4) evaluates G+v = 0 +on all v ∈ ker H, hence Kv = 0 for such states. +The propagator K is a homotopy i.e. +{Q, K} = +� ∞ +0 +dt e−tH {Q, G+} = − +� ∞ +0 +d +� +e−tH� += e−tH��� +0 − e−tH��� +∞ = 1 − Π0. +(4.21) +We can perform the moduli space integral in the amplitude definition (4.18) and express the +amplitude using propagators +AΓ(va; Oα; K) = +� +(2πKG−)⊗I(Γ) +n1(Γ) +� +a=1 +va +n2(Γ) +� +α=1 +Oα ⊗ µ⊗(n3(Γ)−1) +2 +⊗ µ0 +3 +� +Γ +. +(4.22) +Note that the factors 2πG− originate from the angular parts of the moduli space integral. +It is very convenient to introduce a graphical representation for amplitudes: we use solid +lines for edges, equipped with propagator 2πKG− and dashed lines for edges without the +propagator. We label 1-valent vertices by the corresponding states va, 2-valent vertices by +the corresponding operators Oα. For each tree there is a single special vertex, responsible +for the pairing in HTQM, which can be either the 3-valent special vertex equipped with +µ0 +3-multiplication, or the 2-valent vertex, equipped with the pairing g. The graphical rep- +resentation is not unique. Below we present three graphical representations for the same +amplitude. +µ2 +µ0 +3 +v1 +v2 +v3 +v4 +O1 +O2 +µ2 +µ2 +v1 +v2 +v3 +v4 +O1 +O2 +g +µ0 +3 +µ2 +v1 +v2 +v3 +v4 +O1 +O2 +The amplitude, evaluated from the left representation is +AΓ = (2π)3 µ0 +3(v4, KG−O1v3, KG−µ2(v1, KG−O2v2)). +(4.23) +We can use the Frobenius structure for (g, µ2) to evaluate the same amplitude, while moving +the 3-valent special vertex with µ0 +3 to the edge with v4-state and turning it into 2-valent +vertex with pairing g, i.e. +AΓ = (2π)3 g(v4, µ2(KG−O1v3, KG−µ2(v1, KG−O2v2))). +(4.24) +23 + +We can revaluate the same amplitude by moving the µ0 +3 to the different 3-vertex +AΓ = (2π)3 µ0 +3(v1, KG−O2v2, KG−µ2(v4, KG−O1v3)). +(4.25) +The two representation (4.23) and (4.25) are related by the KG−-flip +g(KG−v, w) = g(v, KG−w). +(4.26) +We can derive the flip formula using the definition (4.20) of propagator +g(KG−v, w) = lim +T→∞ +� T +0 +dt g +� +e−tHG+G−v, w +� += lim +T→∞ +� T +0 +dt g +� +G+G−v, e−tHw +� += lim +T→∞ +� T +0 +dt g +� +v, G+G−e−tHw +� += g(v, KG−w). +(4.27) +In the equality we used the integral representation (4.20) for the propagator, the G±- +invariance of the pairing (4.12) and the evolution invariance of the pairing (4.13). +4.5 +Deformation of HTQM by a state +In our work [8] on tropical mirror we argued that the HTQM on trees admits a “state-operator +correspondence”-type relation. We can turn a HTQM state Ψ ∈ V into an operator +OΨ = µ2(Ψ, ·) : V → V +(4.28) +acting as the µ2-multiplication by Ψ. Such relation allowed us to indirectly study the HTQM +deformation by a state, by the means of turning state into an operator first. For the discus- +sion in later parts of the paper we introduce the notion of the HTQM, deformed by a state +below. +Proposition (HTQM deformation by a state): Given HTQM (V, Q, G±, µ2, g) on trees +and an even state ǫΨ such that +QΨ = G−Ψ = 0 +(4.29) +there is an one-parameter family (V, Qǫ, G±, µ2, g) of HTQMs on trees with differential +Qǫ = Q − [G−, µ2(ǫΨ, ·)] + O(ǫ2), +(4.30) +24 + +i.e. the action on states is given by +Qǫv = Qv − G−µ2(ǫΨ, v) + µ2(ǫΨ, G−v) + O(ǫ2). +(4.31) +Proof: For a proof we need to check that the family (V, Qǫ, G±, µ2, g) satisfies the definitions +from sections 4.1 and 4.2. The gradings of G− and Q are odd, while the grading of ǫΨ is +even hence the grading of Qǫ is odd i.e. +|Qǫ| = |[G−, µ2(ǫΨ, ·)]| = 1 + |ǫΨ| = 1. +(4.32) +The Qǫ is a differential to the leading order in ǫ, what follows from the square-evaluation +QǫQǫv = Q2v − QG−µ2(ǫΨ, v) + Qµ2(ǫΨ, G−v) − G−µ2(ǫΨ, Qv) + µ2(ǫΨ, G−Qv) ++ G−µ2(ǫΨ, G−µ2(ǫΨ, v)) − G−µ2(ǫΨ, µ2(ǫΨ, G−v)) − µ2(ǫΨ, G−µ2(ǫΨ, G−v)) += 1 +2µ2(G−µ2(ǫΨ, ǫΨ), G−v) − 1 +2G−µ2(G−µ2(ǫΨ, ǫΨ), v) = O(ǫ2). +(4.33) +In the equality we used Leibniz rule (4.7), associativity of the multiplication µ2 and the +7-term relation (4.8). +Remark: The QǫQǫ = 0 holds for all orders in ǫ for the states ǫΨ, such that +G−µ2(ǫΨ, ǫΨ) = 0. +(4.34) +The condition above does not follow from the G−Ψ = 0, since the pair (G−, µ2) does not +obey the Leibniz rule. +The Qǫ and G− form a pair of graded-commuting differentials. +Indeed, we can simplify +the graded commutator +{Qǫ, G−}v = QG−v − G−µ2(ǫΨ, G−v) + µ2(ǫΨ, G−G−v) + G−Qv +− G−G−µ2(ǫΨ, v) + G−µ2(ǫΨ, G−v) = 0. +(4.35) +The pair (Qǫ, µ2) obey DGA: By construction µ2 is an associative, graded commutative +multiplication and Qǫ is the differential, so we just need to check the Leibniz rule for (µ2, Qǫ). +25 + +Indeed, we can evaluate +µ2(Qǫv, w) + (−1)|v|µ2(v, Qǫw) − Qǫµ2(v, w) += −µ2(G−µ2(ǫΨ, v), w) + µ2(µ2(ǫΨ, G−v), w) − (−1)|v|µ2(v, G−µ2(ǫΨ, w)) ++ (−1)|v|µ2(v, µ2(ǫΨ, G−w)) − µ2(ǫΨ, G−µ2(v, w)) + µ2(G−µ2(v, w), ǫΨ) ++ (−1)|w|(|v|−1)µ2(w, G−µ2(v, ǫΨ)) + (−1)|v|µ2(v, G−µ2(w, ǫΨ)) +− µ2(G−v, µ2(w, ǫΨ)) − (−1)|v|µ2(v, µ2(G−w, ǫΨ)) = 0 +(4.36) +In the equality we used the Leibniz rule (4.7) and 7-term relation (4.8). +The Qǫ-invariance of the pairing follows from +g(Qǫv, w) + (−1)|v|g(v, Qǫw) = −g(G−µ2(ǫΨ, v), w) + g(µ2(ǫΨ, G−v), w) ++ (−1)|v|g(v, µ2(ǫΨ, G−w)) − (−1)|v|g(v, G−µ2(ǫΨ, w)) = 0. +(4.37) +We used the Q-invariance of the pairing (4.11) and G± invariance of the pairing (4.12) and +Frobenius structure for (g, µ2). +The Hamiltonian in deformed HTQM is given by +Hǫ = {Qǫ, G+} = H − {G+, [G−, µ2(ǫΨ, ·)]} = H − ǫ VΨ. +(4.38) +The last equality introduces VΨ, the linear in ǫ correction to the Hamiltonian in deformed +theory. The Cauchy problem for the evolution operator +(∂t + H − ǫVΨ)e−tHǫ = 0, +e−0·Hǫ = 1, +t ∈ R+ +(4.39) +can be solved in power series in ǫ. The evolution operator for deformed theory is +e−tHǫ = e−tH + +� t +0 +ds e(s−t)H ǫVΨ e−sH + O(ǫ2) += e−tH + +� t +0 +ds e(s−t)H{G+, [G−, µ2(ǫΨ, ·)]}e−sH + O(ǫ2). +(4.40) +We can rewrite the integral in more symmetric form +� t +0 +ds esH−tH ǫVΨ e−sH = +� +t1>0, t2>0, t1+t2=t +dt1dt2 e−t2H ǫVΨ e−t1H. +(4.41) +The symmetric formula is a common formula for perturbation theory in quantum mechanics +26 + +and has the following interpretation: The t1 and t2 describe the decomposition of the interval +[0, t] into two sub-intervals: [0, t1] of length t1 and [t1, t] of length t2. Each interval is equipped +with the evolution operator e−t1H and e−t2H, while the splitting point carries the insertion +of the deformation ǫVΨ of the Hamiltonian. +The evolution operators in deformed theory form a semi-group, what follows from the +analysis of the composition of two perturbative solutions (4.40) with times t and s +e−sHǫe−tHǫ = e−(t+s)H + +� t +0 +dt1 e−(t+s−t1)H ǫVΨ e−t1H ++ +� s+t +t +dt1 e−(s+t−t1)H ǫVΨ e−t1H + O(ǫ2) += e−(t+s)Hǫ + O(ǫ2). +The evolution with respect to Hǫ preserves the pairing. Indeed, we can evaluate +g(e−tHǫv, w) = g(v, e−tHw) + +� +t1+t2=t +dt1dt2 g +� +e−t2H ǫVΨ e−t1Hv, w +� += g(v, e−tHw) + +� +t1+t2=t +dt1dt2 g +� +v, e−t1H ǫVΨ e−t2Hw +� += g(v, e−tHǫw). +(4.42) +We used the evolution invariance of the pairing (4.13) and the invariance of the pairing with +respect to the ǫVΨ-action, what follows from the Qǫ-invariance of the pairing (4.37) and +G±-invariance of the pairing (4.12) and the following relation +g(ǫVΨ v, w) = g({G+, Qǫ − Q}v, w) = g(G+(Qǫ − Q)v, w) + g((Qǫ − Q)G+v, w) += (−1)2|v|+2g(v, (Qǫ − Q)G+w) + (−1)2|v|+2g(v, G+(Qǫ − Q)w) += g(v, ǫVΨ w). +(4.43) +We use the semi-continuity of the kernel: The kernel of H can only decrease under the +small deformation, hence dim ker Hǫ ≤ dim ker H. The possible decrease of the kernel is due +to the obstruction for deformation of v0 ∈ ker H into v0ǫ ∈ ker Hǫ. Given a state v0 ∈ ker H +we can deform it by a O(ǫ)-term to construct a state +v0ǫ = v0 + ǫv1 + O(ǫ2) ∈ ker Hǫ, +(4.44) +27 + +where the deformation ǫv1 is a solution to +Hǫv0ǫ = −ǫVΨv0 + ǫHv1 + O(ǫ2) = O(ǫ2). +(4.45) +The solution for ǫv1 can be written in the form +ǫv1 = +� ∞ +0 +dt e−tHG+G−µ2(ǫΨ, v0) = KG−µ2(ǫΨ, v0). +(4.46) +Indeed, we can check +Hǫv1 = +� ∞ +0 +dt H e−tHG+G−µ2(ǫΨ, v0) = − +� ∞ +0 +d +� +e−tHG+G−µ2(ǫΨ, v0) +� += −e−tHG+G−µ2(ǫΨ, v0) +��� +t=∞ +t=0 = G+G−µ2(ǫΨ, v0) − Π0G+G−µ2(ǫΨ, v0) += G+G−µ2(ǫΨ, v0) = ǫ VΨv0, +(4.47) +where we replaced the t → ∞-limit by a projector Π0 and used (4.4) to eliminate the term. +The deformation (4.46) exists for all v0 ∈ ker H, hence dim ker Hǫ = dim ker H. +Remark: The integral in (4.46) acquires most of its value near t = 0, since for large t, +the exponential operator e−tH is close to the projector Π0. Hence we can write an approxi- +mation for the integral in the form of the finite region integral +ǫv1 = +� ∞ +0 +dt e−tHG+G−µ2(ǫΨ, v0) ≈ +� T +0 +dt e−tHG+G−µ2(ǫΨ, v0). +(4.48) +Let us choose a basis v0 +k in ker H. Since ker H and ker Hǫ are of the same dimension then +the corresponding deformed states v0ǫ +k form a basis in ker Hǫ. We use the non-degeneracy of +the pairing g to identify the vector space V and its dual V ∗ to express the projector +Π0 = +� +v0 +kv0 +k +∗. +(4.49) +The projector Πǫ +0 on ker Hǫ is written using the deformed states v0ǫ +k +Πǫ +0 = +� +v0ǫ +k v0ǫ +k +∗ = Π0 + +� +ǫv1 +k v0 +k +∗ + ǫ +� +v0 +k ǫv1 +k +∗ + O(ǫ2). +(4.50) +28 + +The t → ∞ limit of the deformed evolution operator (4.40) is +lim +t→∞e−tHǫ = Π0 + lim +t→∞ +� t +0 +ds esH−tH ǫVΨ e−sH + O(ǫ2). +(4.51) +We decompose the integration interval [0, t] into three regions and evaluate the limit for the +integration over each region: +• Right side of the interval: The t2 is small, i.e t2 ∈ [0, T] for some finite T, while +t1 ≈ t is very large and we can replace the corresponding exponential factor by the +projector Π0. The integral (4.41) in this region evaluates into +lim +t→∞ +� T +0 +dt2 e−t2H ǫVΨ e−tHet2H = +� T +0 +dt2 e−t2H ǫVΨ · Π0 += +� T +0 +dt2 e−t2HǫVΨ +� +v0 +kv0 +k +∗ = +� � T +0 +dt2 e−t2HG+G−µ2(ǫΨ, v0 +k)v0 +k +∗ +≈ +� � ∞ +0 +dt2 e−t2HG+G−µ2(ǫΨ, v0 +k)v0 +k +∗ = +� +KG−µ2(ǫΨ, v0 +k)v0 +k +∗ += +� +ǫv1 +k v0 +k +∗. +(4.52) +• Left side of the interval: The t1 is such that t1 < T, while t2 ≈ t is very large and +we can replace the corresponding exponential factor by the projector. The integral +(4.41) in this region evaluates into +lim +t→∞ +� T +0 +dt1 e−tHet1H ǫVΨ e−t1H = +� T +0 +dt1 Π0 ǫVΨ e−t1H += +� T +0 +dt1 +� +v0 +kv0 +k +∗ ǫVΨe−t1H = +� +v0 +k +�� T +0 +dt1 e−t1HG+G−µ2(ǫΨ, v0 +k) +�∗ +≈ +� +v0 +k +�� ∞ +0 +dt1 e−t1HG+G−µ2(ǫΨ, v0 +k) +�∗ += +� +v0 +k +� +KG−µ2(ǫΨ, v0 +k) +�∗ += +� +v0 +k ǫv1 +k +∗. +(4.53) +• Middle of the interval: For the middle region t1 ∈ [t/2 − T, t/2 + T], i.e. both t1 +and t2 are large. The integral (4.41) in this region evaluates into +lim +t→∞ +� t/2+T +t/2−T +dt2 e−t2H ǫVΨ e−t1H = +� T +−T +dt1 Π0 ǫVΨ Π0 += +� T +−T +dt1 0 = O(T) · 0 = 0. +(4.54) +29 + +The sum over three contributions +lim +t→∞ e−tHǫ = Π0 + +� +ǫv1 +k v0 +k +∗ + +� +v0 +k ǫv1 +k +∗ = Πǫ +0 +(4.55) +matches with our expression (4.50) for projector on ker Hǫ. +We can use the (4.50)-representation to check the properties of projector in deformed +theory +Πǫ +0G±v = Π0G±v + +� � +ǫv1 +k v0 +k +∗ + v0 +k ǫv1 +k +∗� +G±v += +� +ǫv1 +k g(v0 +k, G±v) + +� +v0 +k g(ǫv1 +k, G±v) = 0 +(4.56) +and +G±Πǫ +0v = G±Π0v + G± +� � +ǫv1 +k v0 +k +∗ + v0 +k ǫv1 +k +∗� +v += +� +(G± ǫv1 +k) v0 +k +∗ + +� +(G±v0 +k) ǫv1 +k +∗ = 0. +(4.57) +In equalities we used the G±-invariance of the pairing, G±v0 +k = 0 and expression (4.46) for +v1 +k to evaluate +G± ǫv1 +k = G± +� ∞ +0 +dt e−tHG+G−µ2(ǫΨ, v0 +k) = 0. +(4.58) +■ +Definition: For any state v in HTQM (V, Q, G±) its leading order deformation by Q-, +G−-closed state Ψ is +vǫ = v + KG−µ2(ǫΨ, v) + O(ǫ2). +(4.59) +Proposition (preservation of closeness): If v is Q- and G−-closed, then the deformed +state vǫ is Qǫ- and G−- closed i.e. +Qǫvǫ = G−vǫ = O(ǫ2). +(4.60) +Proof: The G−-closeness if fairly straightforward +G−vǫ = G−v + G−KG−µ2(ǫΨ, v) = −G2 +−Kµ2(ǫΨ, v) = 0. +(4.61) +30 + +The Qǫ-action on the deformed state +Qǫvǫ = Qǫv + QKG−µ2(ǫΨ, v) + O(ǫ2) += Qv − G−µ2(ǫΨ, v) + QKG−µ2(ǫΨ, v) + O(ǫ2) += −G−µ2(ǫΨ, v) + (1 − Π0)G−µ2(ǫΨ, v) + O(ǫ2) = O(ǫ2). +(4.62) +In the equality we used the homotopy formula (4.21), Leibniz rule (4.7) for even state ǫΨ +and the projector Π0 property (4.4) from the HTQM definition. +■ +Remark: The formula (4.59) for deformation of a state v, describes a connection on Q+zG−- +cohomology, fibered over the space of HTQM deformations. +4.6 +Diagrammatic representation of deformed theory +We can express the propagator for deformed theory as expansion in ǫ in terms of propagators +in original theory +Kǫv = +� ∞ +0 +dt e−tHǫG+v = +� ∞ +0 +dt e−tHG+v ++ +� ∞ +0 +dt +� +t1+t2=t, t1, t2>0 +dt1dt2 e−t1H{G+, [G−, µ2(ǫΨ, ·)]}e−t2HG+v + O(ǫ2) += Kv + +� +(R+)2 dt1dt2 e−t1HG+[G−, µ2(ǫΨ, ·)]e−t2HG+v + O(ǫ2) += Kv + K[G−, µ2(ǫΨ, ·)]Kv + O(ǫ2). +The propagator for deformed theory further simplifies if we use it in KG−-combination i.e. +KǫG−v = KG−v + KG−µ2(ǫΨ, KG−v) + O(ǫ2). +(4.63) +We can give a graphical representation for a propagator in deformed theory (denoted as thick +solid line) in terms of the diagrams in the original theory += ++ +ǫΨ +In case G−µ2(ǫΨ, ǫΨ) = 0 the KG− in deformed theory can be written to all orders in +31 + +ǫ in the form +KǫG−v = KG− + KG−µ2(ǫΨ, KG−v) + KG−µ2(ǫΨ, KG−µ2(ǫΨ, KG−v)) + . . . +(4.64) +with graphical representation += ++ +ǫΨ ++ +ǫΨ +ǫΨ ++ . . . +The diagrammatic expression for the deformed state (4.59), denoted as the thick dashed +line, is the sum of two terms +vǫ += +v ++ +ǫΨ +v +In case G−µ2(ǫΨ, ǫΨ) = 0 the higher order terms of the state deformation take the form +vǫ += +v ++ +ǫΨ +v ++ +ǫΨ +ǫΨ +v + . . . +The diagrammatic expression above describes the sum +vǫ = v + KG−µ2(ǫΨ, v) + KG−µ2(ǫΨ, KG−µ2(ǫΨ, v)) + ... +(4.65) +5 +Correlation functions in HTQM +In this section we introduce the correlation functions for the states in HTQM. The correla- +tion functions obey certain nice properties such as symmetry in all arguments, Q-invariance +and recursion relation. +Definition: The tree-level connected n-point correlation function for states Ψ1, ..., Ψn in +HTQM (V, Q, G±, g, µ2) is +⟨Ψ1, ..., Ψn⟩Q = +� +Γ,σ∈Sn +A0 +Γ(Ψσ(1), ..., Ψσ(n); K) +|Aut(Γ)| +, +(5.1) +32 + +where |Aut(Γ)| is the symmetry factor for the tree Γ and K is a homotopy (4.20). The sum- +mation is taken over all distinct 3-valent connected trees Γ with n leaves and over possible +assignment of states Ψ1, ..., Ψn on leaves of Γ. +Remark: The sum over permutations in (5.1) makes the correlation function ⟨Ψ1, ..., Ψn⟩Q +manifestly symmetric in all arguments. +Remark: The sum over trees, weighted with the symmetry factors, in correlation func- +tions (5.1) is a signature of their relation to amplitudes in certain Quantum Field Theory. +We conjecture that the QFT is a BCOV-like theory [21], see also [10]. We are working on +further investigation of this conjecture. +Remark: The HTQM states Ψa, relevant for the mirror symmetry, are even, i.e. |Ψa| = 0. +Hence, for simplicity, we will assume that |Ψa| = 0 for the rest of this section. Such assump- +tion will drastically reduce the complexity of the sign factors in various expressions. +5.1 +3-point function +There is a single tree with 3-valent vertices and three leaves. Hence, the 3-point correlation +function is just a sum over 3! possible permutations of states Ψ1, Ψ2, Ψ3 on the leaves of a +tree below +Ψ1 +Ψ2 +Ψ3 +The amplitudes for each permutation are identical, hence we have a sum of 3! = 6 identi- +cal terms. The symmetry factor of the graph above is |Aut Γ3| = 3! = 6, so the 3-point +correlation function simplifies to +⟨Ψ1, Ψ2, Ψ3⟩Q = 1 +3! · 3! µ0 +3(Ψ1, Ψ2, Ψ3) = µ0 +3(Ψ1, Ψ2, Ψ3). +(5.2) +33 + +The 3-point function (5.2) is invariant under the shift Ψ3 → Ψ3 + Qχ, given that QΨ1 = +QΨ2 = 0. Indeed, we can evaluate the difference +⟨Ψ1, Ψ2, Qχ⟩Q = g(Qχ, µ2(Ψ1, Ψ2)) = g(χ, Qµ2(Ψ1, Ψ2)) += g(χ, µ2(QΨ1, Ψ2)) + g(χ, µ2(Ψ1, QΨ2)) = 0. +(5.3) +We used the Q-invariance of the pairing (4.11) and Leibniz rule (4.7) to simplify the expres- +sion. +5.2 +4-point function +There is a single tree with 3-valent vertices and four leaves. Hence the 4-point correlation +function is just a sum over 4! possible distributions of states Ψ1, Ψ2, Ψ3, Ψ4 on the leaves of +a tree. The 24 terms in a sum of three groups of 8, with equal amplitudes in each group. +The 3 (independent) amplitudes are depicted below and are commonly referred to as the +s−, t−, u−diagrams +Ψ4 +Ψ1 +Ψ3 +Ψ2 +Ψ4 +Ψ1 +Ψ2 +Ψ3 +Ψ3 +Ψ1 +Ψ2 +Ψ4 +The 4-point correlation function is the sum of three contributions +⟨Ψ1, Ψ2, Ψ3, Ψ4⟩Q = AΓ4(Ψ1, Ψ2, Ψ3, Ψ4) + AΓ4(Ψ1, Ψ3, Ψ2, Ψ4) + AΓ4(Ψ1, Ψ4, Ψ2, Ψ3). (5.4) +The number 8 equals to the symmetry factor |Aut Γ4| = 8 of a tree and is constructed as +2·2·2. The amplitudes on graphs, related by symmetry, are the same. Indeed, the amplitude +for the first tree +AΓ4(Ψ1, Ψ2, Ψ3, Ψ4) = g(µ2(Ψ3, Ψ4), 2πKG−µ2(Ψ1, Ψ2)) += g(µ2(Ψ3, Ψ4), 2πKG−µ2(Ψ2, Ψ1)) = g(µ2(Ψ4, Ψ3), 2πKG−µ2(Ψ1, Ψ2)). +(5.5) +is invariant under the exchange of two pairs of states Ψ1 ↔ Ψ2 and Ψ3 ↔ Ψ4. Indeed, the Ψa +are even states and µ2 is graded-symmetric. The last factor of 2 is related to the reflection +34 + +of the tree, i.e. exchange of two pairs Ψ1, Ψ2 and Ψ3, Ψ4 i.e. +g(µ2(Ψ4, Ψ3), KG−µ2(Ψ1, Ψ2)) = g(KG−µ2(Ψ4, Ψ3), µ2(Ψ1, Ψ2)). +(5.6) +The invariance of the amplitude follows from the KG−-flip relation (4.26). +The 4-point amplitude is invariant under the shift Ψ4 → Ψ4 + Qχ, given that +QΨa = G−Ψa = G−χ = 0, +a = 1, 2, 3. +(5.7) +Indeed, we can evaluate +2 +2π⟨Ψ1, Ψ2, Ψ3, Qχ⟩Q = +� +σ∈S3 +g(Qχ, µ2(Ψσ(3), KG−µ2(Ψσ(1), Ψσ(2)))) += +� +σ∈S3 +g(χ, Qµ2(Ψσ(3), KG−µ2(Ψσ(1), Ψσ(2)))) += +� +σ∈S3 +g(χ, µ2(Ψσ(3), {Q, K}G−µ2(Ψσ(1), Ψσ(2)))) = +� +σ∈S3 +g(χ, µ2(Ψσ(3), G−µ2(Ψσ(1), Ψσ(2)))) += 2g(χ, G−µ2(µ2(Ψ1, Ψ2), Ψ3)) = −2g(G−χ, µ2(µ2(Ψ1, Ψ2), Ψ3)) = 0. +(5.8) +In rewriting the equality we used the Q-invariance of the pairing (4.11), Leibniz rule (4.7), +the homotopy formula (4.21), the projector property (4.4), the 7-term relation (4.8) for even +G−-closed states in the form +G−µ2(µ2(Ψ1, Ψ2), Ψ3) = µ2(G−µ2(Ψ1, Ψ2), Ψ3) + µ2(Ψ2, G−µ2(Ψ1, Ψ3)) ++ µ2(Ψ1, G−µ2(Ψ2, Ψ3)). +(5.9) +The last equality in (5.8) uses the G±-invariance of pairing (4.12) and completes the proof +of invariance. +5.3 +Generating function +We introduce formal parameters tk, such that t2 +k = 0 and combine Ψk into even state +Ψ = Ψ(t) = +� +tkΨk. +(5.10) +35 + +The n-point correlation function of Ψ has t-expansion with coefficients being the n-point +correlation functions i.e. +⟨Ψ, ..., Ψ +� �� � +n +⟩Q = n! · ⟨t1Ψ1, t2Ψ2, .., tnΨn⟩Q. +(5.11) +Definition: The n-point correlation function of Ψ can be organized into generating function +F0(Ψ, K) = +∞ +� +k=3 +1 +k!⟨Ψ, ..., Ψ +� �� � +k +⟩Q. +(5.12) +The generating function equals to the the sum over connected 3-valent trees +F0(Ψ, K) = +� +Γ +AΓ(Ψ, ..., Ψ; K) +|Aut(Γ)| +. +(5.13) +The diagrammatic expression for generating function is +Ψ +Ψ +Ψ ++ 1 +8 +1 +6 +Ψ +Ψ +Ψ +Ψ ++ 1 +8 +Ψ +Ψ +Ψ +Ψ +Ψ ++ 1 +8 +Ψ +Ψ +Ψ +Ψ +Ψ +Ψ ++ 1 +48 +Ψ +Ψ +Ψ +Ψ +Ψ +Ψ ++ . . . +5.4 +Invariance theorem +In our analysis for the 3- and 4-point functions we observed the invariance under the shift +by a Q-exact term, given that the states were Q- and G-closed. The invariance generalizes +to the n-point function. +Theorem (invariance): Given the states Ψ1, Ψ2, ..., Ψn, χ of HTQM on trees (V, Q, G±, µ2, g) +such that +QΨα = G−Ψα = G−χ = 0, α = 1..n +(5.14) +36 + +the (n + 1)-point correlation function vanishes i.e. +⟨Ψ1, ..., Ψn, Qχ⟩Q = 0. +(5.15) +Proof: We use the generating function (5.12) to prove the theorem. The change in generating +function can be expressed +δF0(Ψ) = F0(Ψ + δΨ) − F0(Ψ) = g(δΨ, ˜γ) + O(δΨ)2 +(5.16) +via the state ˜γ, defined as a sum over 3-valent rooted trees, weighted with symmetry factors. +The first several trees of the sum ˜γ are presented below +1 +2 +Ψ +Ψ ++ 1 +2 +Ψ +Ψ +Ψ ++ 1 +2 +Ψ +Ψ +Ψ +Ψ ++ 1 +8 +Ψ +Ψ +Ψ +Ψ ++ . . . +The diagrammatic sum for ˜γ takes the form +˜γ = 1 +2µ2(Ψ, Ψ) + 1 +2µ2(Ψ, 2πKG−µ2(Ψ, Ψ)) + 1 +2µ2(Ψ, 2πKG−µ2(Ψ, 2πKG−µ2(Ψ, Ψ))) ++ 1 +8µ2(2πKG−µ2(Ψ, Ψ), 2πKG−µ2(Ψ, Ψ)) + . . . +(5.17) +Note that ˜γ is an even state since Ψ is even and KG− is even operator. We also introduce +a related even state +γ = Ψ + 2πKG−˜γ, +(5.18) +used in [10], which is G−-closed +G−γ = G−Ψ − 2πKG2 +−˜γ = 0. +(5.19) +The ˜γ and γ obey the “root-cutting” relation +˜γ = 1 +2µ2(γ, γ). +(5.20) +37 + +The representation (5.20) for ˜γ is very convenient for deriving the recursive formula +Q˜γ = 1 +2Qµ2(γ, γ) = µ2(γ, Qγ) = µ2(γ, QΨ + 2πQKG−˜γ) += µ2(γ, 2πG−˜γ) + µ2(γ, 2πKG−Q˜γ) = πµ2(γ, G−µ2(γ, γ)) + µ2(γ, 2πKG−Q˜γ) += π +3 G−µ2(γ, µ2(γ, γ)) + µ2(γ, 2πKG−Q˜γ). +(5.21) +In our derivation we used Leibniz rule (4.7), homotopy formula (4.21), the projector property +(4.4) and the 7-term relation (4.8) for even, G−-closed state γ in the form +G−µ2(γ, µ2(γ, γ)) = 3µ2(γ, G−µ2(γ, γ)). +(5.22) +We can replace the Q˜γ in the last expression and use G2 +− = 0 to get +Q˜γ = π +3 G−µ2(γ, µ2(γ, γ)) + µ2(γ, 2πKG−µ2(γ, 2πKG−Q˜γ)). +(5.23) +We can iterate the process for of the Q˜γ-replacement to arrive into +Q˜γ = π +3G−µ2(γ, µ2(γ, γ)). +(5.24) +The invariance of the correlation functions follows from +F0(Ψ + Qχ) − F0(Ψ) = g(Qχ, ˜γ) = g(χ, Q˜γ) = π +3 g(χ, G−µ2(γ, µ2(γ, γ))) += −π +3 g(G−χ, µ2(γ, µ2(γ, γ))) = 0. +(5.25) +We used the Q-invariance of the pairing (4.11), the G±-invariance of the pairing (4.12) and +the assumption of the theorem that G−χ = 0. +■ +5.5 +Recursion relation for correlation functions +Let us consider the HTQM (V, Q, G±, µ2, g) and four Q-, G−-closed states Ψa, a = 1, .., 4. +The 4-point correlation function (5.4) of such states can be written in the following form +⟨Ψ1, Ψ2,Ψ3, Ψ4⟩Q = µ0 +3 (2πKG−µ2(Ψ4, Ψ1), Ψ2, Ψ3) + µ0 +3 (Ψ1, 2πKG−µ2(Ψ4, Ψ2), Ψ3) ++ µ0 +3 (Ψ1, Ψ2, 2πKG−µ2(Ψ4, Ψ3)) += d +dǫ +��� +ǫ=0µ0 +3(Ψǫ +1, Ψǫ +2, Ψǫ +3) = d +dǫ +��� +ǫ=0⟨Ψǫ +1, Ψǫ +2, Ψǫ +3⟩Qǫ, +(5.26) +38 + +where we used the leading order deformation (4.59) of states Ψ1, Ψ2, Ψ3 by the state Ψ4 i.e. +Ψǫ +a = Ψa + 2πKG−µ2(ǫΨ4, Ψa) + O(ǫ2), a = 1, 2, 3. +(5.27) +We can describe the relation (5.26) using diagrammatic representation +Ψǫ +1 +Ψǫ +2 +Ψǫ +3 +d +dǫ|ǫ=0 += +Ψ1 +Ψ2 +Ψ3 +Ψ4 ++ +Ψ1 +Ψ2 +Ψ3 +Ψ4 ++ +Ψ1 +Ψ2 +Ψ3 +Ψ4 +Theorem (recursion relation): The (n + 1)-point correlation function for the Q-, G−- +closed states Ψ0, Ψ1, .., Ψn in HTQM on trees (V, Q, G±, µ2, g) can be expressed as a derivative +of n-point correlation function in HTQM, deformed by the state Ψ0, i.e. +⟨Ψ1, ..., Ψn, Ψ0⟩Q = d +dǫ +��� +ǫ=0⟨Ψǫ +1, ..., Ψǫ +n⟩Qǫ. +(5.28) +The deformed HTQM (V, Qǫ, G±, µ2, g) has differential +Qǫ = Q − 2π[G−, µ2(ǫΨ0, ·)] + O(ǫ2), +(5.29) +while deformed states are +Ψǫ +a = Ψa + 2πKG−µ2(ǫΨ0, Ψa) + O(ǫ2). +(5.30) +Proof: The key idea in our proof is to use the generating function for the amplitudes (5.12). +The generating function in deformed theory is +F0(Ψǫ, Kǫ) = +∞ +� +k=3 +1 +k!⟨Ψǫ, ..., Ψǫ +� +�� +� +k +⟩Qǫ. +(5.31) +Let us recall the formula for the change of generating function under variation of the external +state Ψ +F0(Ψ + ǫδΨ, K) = F0(Ψ, K) + g(˜γ, ǫδΨ) + O(ǫ2), +(5.32) +39 + +where the state ˜γ is a sum over rooted trees (5.17). Similarly we can derive the change of +generating function under the change of propagator K +F0(Ψ, K + ǫδK) = F0(Ψ, K) + πg(˜γ, ǫδK˜γ) + O(ǫ2). +(5.33) +The full change of the generating function is presented on the picture below +Ψ +Ψ +Ψ +... +δ += +δΨ +g +Ψ +Ψ +... ++ +g +δK +Ψ +Ψ +... +Ψ +Ψ +... +We use the state deformation formula (4.59) for δΨ and the propagator in deformed theory +(4.63) for δK in terms of the state Ψ0, i.e. +ǫδKG− = KǫG− − KG− = 2πKG−µ2(ǫΨ0, KG−·), +ǫδΨ = Ψǫ − Ψ = 2πKG−µ2(ǫΨ0, Ψ). +(5.34) +Using the KG−-flip (4.26) and relations for the sum over rooted trees (5.18) we can rewrite +the generating function in deformed theory +F0(Ψǫ, Kǫ) − F0(Ψ, K) = g(˜γ, ǫδΨ) + πg(˜γ, ǫδK˜γ) + O(ǫ2) += g(ǫΨ0, µ2(2πKG−˜γ, Ψ)) + πg(ǫΨ0, µ2(2πKG−˜γ, KG−˜γ)) + O(ǫ2) += g(ǫΨ0, ˜γ) − 1 +2g(ǫΨ0, µ2(Ψ, Ψ)) + O(ǫ2) += F0(Ψ + ǫΨ0, K) − F0(Ψ, K) − 1 +2g(ǫΨ0, µ2(Ψ, Ψ)) + O(ǫ2). +(5.35) +The derivative of the relation becomes +∞ +� +k=3 +d +dǫ +��� +ǫ=0 +1 +k!⟨Ψǫ, ..., Ψǫ +� +�� +� +k +⟩Qǫ = d +dǫ +��� +ǫ=0F0(Ψ + ǫΨ0, K) − 1 +2 g(Ψ0, µ2(Ψ, Ψ)) += 3 +3!⟨Ψ, Ψ, Ψ0⟩Q + +∞ +� +k=4 +1 +k!k⟨Ψ, ..., Ψ +� �� � +k−1 +, Ψ0⟩Q − 1 +2⟨Ψ, Ψ, Ψ0⟩Q += +∞ +� +k=3 +1 +k!⟨Ψ, ..., Ψ +� �� � +k +, Ψ0⟩Q. +(5.36) +40 + +The equality holds at each order in Ψ so +d +dǫ +��� +ǫ=0⟨Ψǫ, ..., Ψǫ +� +�� +� +k +⟩Qǫ = ⟨Ψ, ..., Ψ +� �� � +k +, Ψ0⟩Q +(5.37) +what completes the proof of the theorem. +■ +6 +Mirror for HTQM +6.1 +A-model +Tropical Gromov-Witten theory on toric manifold X of complex dimension N defines the +A-type HTQM on trees, also denoted as the A-model in this section. In this section we will +describe the HTQM data (V, Q, G±, g, µ2) for A-model. +Let us consider pair |ω, ⃗m⟩, of tropical form ω on C∗N, and N-dimensional integer-valued +vector ⃗m. The pairing on |ω, ⃗m⟩ is the integration of corresponding form forms +g(|ω1, ⃗m1⟩, |ω2, ⃗m2⟩) = δ⃗m1+⃗m2,⃗0 +� +C∗N ω1 ∧ ω2. +(6.1) +The vector space VA is the space of tropical differential forms on C∗N, equipped with the +integer vector, i.e. +VA = Ωtrop(C∗N) ⊗ R⟨⃗m | ⃗m ∈ ZN⟩. +(6.2) +The Z2-grading of the state |ω, ⃗m⟩ is the grading of the form ω, a degree of the differential +form mod 2. The differential Q is the de Rham operator on C∗N +Q|ω, ⃗m⟩ = |dω, ⃗m⟩. +(6.3) +The G± on states is a contraction with the constant radial (angular) vector field ⃗m +G+|ω, ⃗m⟩ = |ιR +⃗mω, ⃗m⟩ = |ιmi∂riω, ⃗m⟩, +G−|ω, ⃗m⟩ = |ιΦ +⃗mω, ⃗m⟩. +(6.4) +The multiplication µ2 : V ⊗V → V is the wedge product on differential forms supplemented +with addition of corresponding vectors +µ2(|ω1, ⃗m1⟩, |ω2, ⃗m2⟩) = |ω1 ∧ ω2, ⃗m1 + ⃗m2⟩. +(6.5) +41 + +The pair (Q, µ2) is essentially an external derivative and the wedge product, hence obeys +the DGA properties. The pair (G−, µ2) obeys the 7-term relation. +The Hamiltonian +H = {Q, G+} +(6.6) +is the Lie derivative along the constant radial vector field ⃗m +H|ω, ⃗m⟩ = {Q, G+}|ω, ⃗m⟩ = |{d, ιR +⃗m}ω, ⃗m⟩ = |LR +⃗mω, ⃗m⟩. +(6.7) +The evolution operator e−tH, defined as solution to +(∂t + H)e−tH = (∂t + LR +⃗m)e−tH = 0 +(6.8) +is a 1-parameter family of diffeomorphisms Φt +⃗m : ri �→ ri − mit +e−tH|ω, ⃗m⟩ = |(Φt +⃗m)∗ω, ⃗m⟩. +(6.9) +The composition property (4.2) naturally holds for diffeomorphisms. Since the vector fields +are constant vector fields the corresponding flows do not develop any singularities, hence +composition is valid for all values of t. +6.2 +Correlation functions in A-model +In our paper [8] we showed that the tropical GW invariant of genus-0 and degree-β on toric +space X is the sum of the A-type HTQM amplitudes +⟨γ1, ..., γn⟩X +β = +� +Γ +AΓ(Ψγ1, .., Ψγn, Ψ⃗b1, .., Ψ⃗bB; K) +(6.10) +with two types of states: +• Evaluation states Ψγk = |γk,⃗0⟩, constructed from the tropical forms γk on X. The +space Ωtrop(X) of tropical forms on X is a subspace of tropical forms on C∗N with +good behaviour at compactifying divisors. +• Divisor states Ψ⃗ba = |1,⃗ba⟩, where ⃗ba are primitive normal vectors for compactifying +divisors of X. +Using the definition (5.1) we can replace the sum over the amplitudes by the correlation +function and formulate the HTQM representation for the tropical GW invariants in the form: +42 + +Theorem (HTQM representation of tropical GW): For toric space X, given in terms +of boundary divisors BX, and tropical cycles γk, the tropical GW invariant matches with the +HTQM correlation function i.e. +⟨γ1, ..., γn⟩X +β = +1 +d1! · .. · dB!⟨Ψγ1, .., Ψγn, Ψ⃗b1, .., Ψ⃗b1 +� +�� +� +d1 +, .., Ψ⃗bB, .., Ψ⃗bB +� +�� +� +dB +⟩Q. +(6.11) +Here Ψγa are the evaluation states for tropical cycles γa, Ψ⃗b are divisor states for boundary +divisors from BX of dimension B = dim BX. The degree of the map β ∈ H1,1(X) determines +the number da of divisor states Ψ⃗ba of a given type via the corresponding tropical intersection +number of β and boundary divisor with normal vector ⃗ba. +The GW theory is defined on classes of cycles Cα. The change of a cycle within the same +class leads to the shift of an evaluation observable (Poincare-dual form) γ by an exact form +γ → γ + dλ. +(6.12) +The shift of evaluation observable γ changes the corresponding A-model state Ψγ by a Q- +exact term +Ψγ → Ψγ+dλ = Ψγ + QΨλ. +(6.13) +Furthermore, since the states Ψγ and Ψλ carry trivial integer vector, they are G−-closed, i.e. +G−Ψλ = G−|λ,⃗0⟩ = 0, G−Ψγ = G−|γ,⃗0⟩ = 0 +(6.14) +The forms γa are closed forms so is the corresponding HTQM states Ψγ1. Hence we can use +the invariance theorem (5.4) to verify that the tropical GW invariants in HTQM represen- +tation (6.11) are defined on cohomology classes of γa. +6.3 +Dual variables +It is convenient to introduce angular variables Yj ∈ S1, dual to the integer vector components +mi ∈ Z. We introduce a Fourier transform of a state +� +⃗m∈ZN +c⃗m|ω⃗m, ⃗m⟩ �→ Ψ = +� +⃗m∈ZN +ei⟨⃗m,⃗Y ⟩c⃗m|ω⃗m, ⃗m⟩ ∈ VB = Ωtrop(C∗N) ⊗ C∞(TN). +(6.15) +43 + +For convenience we describe the differential forms on X using Grassmann variables: +drj = ψj +R, +dφj = ψj +Φ. +(6.16) +The differentials on mirror states in new notations become first and second order differential +operators +QΨ = +� +⃗m∈ZN +ei⟨⃗m,⃗Y ⟩c⃗m|dω⃗m, ⃗m⟩ = dRΨ = ψk +R +∂ +∂rk Ψ, +G−Ψ = +� +⃗m∈ZN +ei⟨⃗m,⃗Y ⟩c⃗m|ιΦ +⃗mω, ⃗m⟩ = −i ∂ +∂Yk +∂ +∂ψk +Φ +Ψ, +G+Ψ = +� +⃗m∈ZN +ei⟨⃗m,⃗Y ⟩c⃗m|ιR +⃗mω, ⃗m⟩ = −i ∂ +∂Yk +∂ +∂ψk +R +Ψ. +(6.17) +The multiplication µ2, on forms becomes multiplication of functions on superspace with +coordinates r, Y, ψR, ψΦ +µ2(Ψ1, Ψ2) = Ψ1 · Ψ2, +(6.18) +while the pairing is the integration over superspace +g(Ψ1, Ψ2) = +� +dµ Ψ1Ψ2, +(6.19) +where Berezin integration measure for dimC X = N is +dµ = dNr dNY dNψΦdNψR. +(6.20) +The integrating region (for Grassmann-even variables) is the N-dimensional torus (S1)N for +Y -variables and Euclidean space RN for r-variables. +Remark: The differential operator representation (6.17) of the HTQM data (V, Q, G±, µ2, g) +allows for an easy check of HTQM definitions from sections 4.1 and 4.2. In particular the +7-term relation (4.8) for G− is a property of the second order differential operator in repre- +sentation (6.17). +The divisor states |1,⃗b⟩ become exponential functions of Y +Ψ⃗b(Y ) = ei⟨⃗b,⃗Y ⟩ = eibkYk. +(6.21) +44 + +For the tropical form +γ = γi1..ikj1..jl(r) dφi1 ∧ .. ∧ dφik ∧ drj1 ∧ .. ∧ drjl +(6.22) +the corresponding evaluation state is +Ψγ = γi1..ikj1..jl(r) ψi1 +Φ · .. · ψik +Φ · ψj1 +R · .. · ψjl +R. +(6.23) +Example: The evaluation state for U(1)-invariant Poincare dual of the point on P1 is +γ = 1 +2πδ(r − r0)dφdr �→ Ψγ = |γ, 0⟩ = 1 +2πδ(r − r0)ψΦψR. +(6.24) +6.4 +B-model +In our work [8] we showed that the deformation of A-model, the HTQM (VA, Q, G±, µ2, g), +by divisor states Ψ⃗b for toric X is also an HTQM (VB, QX, G±, µ2, g), which we will denote +as the B-type HTQM or B-model for short. +Remark: The two-dimensional version of the deformation by compactifying divisors of +toric space was discussed in A-I-B mirror paper [11]. However, the deformation of observ- +ables was not discussed there. +Divisor states obey +G−µ2(Ψ⃗b1, Ψ⃗b2) = 0, +(6.25) +hence the deformation of the differential Q, which we discussed in section 4.5 holds beyond +the linearized level. The deformation of the A-model differential by all divisors BX of X is +QX = Q − 2π +� +⃗b∈BX +[G−, µ2(q⃗bΨ⃗b, ·)] = ψj +R +∂ +∂rj + 2πi +� +⃗b∈BX +q⃗b +∂Ψ⃗b +∂Yj +∂ +∂ψj +Φ +. +(6.26) +The same differential can be written as +QX = Q + 2πi∂WX(Y ) +∂Yj +∂ +∂ψj +Φ += QWX, +(6.27) +45 + +using the mirror superpotential +WX(Y ) = +� +⃗b∈BX +q⃗b Ψ⃗b = +� +⃗b∈BX +q⃗b ei⟨⃗b,⃗Y ⟩. +(6.28) +We can absorb some q⃗b by the redefinition of Yj to obtain more familiar (at least for the PN +case) form of the superpotential with fewer parameters q⃗b. +Remark: The exponential mirror superpotentials for toric spaces were derived by Given- +tal [5] and by Hori and Vafa [17] using different methods. +Definition: For toric space X the deformation (4.59) of an A-model evaluation state Ψγ by +the divisor states Ψ⃗b is a mirror state ΨX +γ . We can write the mirror state using the A-model +notations in the form +ΨX +γ = Ψγ + 2πKG−µ2(WX, Ψγ) + (2π)2KG−µ2(WX, KG−µ2(WX, Ψγ)) + ... +(6.29) +For an A-model state Ψγ corresponding to the tropical form γ ∈ Ωk,l(X) the sum (6.29) +terminates after min(k, l) + 1 terms. Indeed, the action of KG− lowers the degree of the +form by (1, 1), hence it can be applied to (k, l)-form Ψγ at most min(k, l) times. +By preservation of closeness proposition (4.60), the mirror state ΨX +γ is QX- and G−-closed, +i.e. +QXΨX +γ = G−ΨX +γ = 0. +(6.30) +Example: For X = P1 the space of boundary normal vectors is BP1 = {+1, −1}, hence the +mirror superpotential equals +WP1(Y ) = +� +b∈BP1 +qb eibY = q+eiY + q−e−iY . +(6.31) +We can absorb the q+ by constant shift of Y to arrive into more familiar form of the super- +potential +WP1 = eiY + qe−iY +(6.32) +with q = q+q−. The B-model differential is +QP1 = ψR +∂ +∂r + 2πi∂WP1 +∂Y +∂ +∂ψΦ += ψR +∂ +∂r − 2π(q+eiY − q−e−iY ) ∂ +∂ψΦ +. +(6.33) +46 + +The mirror state ΨP1 +γ +for the A-model evaluation observable γ ∈ H∗ +dR(P1) contains three +terms +ΨP1 +γ = Ψγ + 2πKG−µ2(WP1, Ψγ) += Ψγ + 2πq+ +� ∞ +0 +dt e−tH G+G−(eiY Ψγ) + 2πq− +� ∞ +0 +dt e−tH G+G−(e−iY Ψγ). +(6.34) +Indeed, the top form on P1 has degree (1, 1), hence we can apply the deformation only once +for each of two boundary divisors. The mirror state for the constant form γ = 1 ∈ H0 +dR(P1) +has a trivial deformation +ΨP1 +1 = Ψ1 = 1. +(6.35) +Consider the Poincare dual of the point φ = φ0 and r = r0 in tropical coordinates on P1 +P0 = δ(φ − φ0)δ(r − r0) dφdr. +(6.36) +The U(1)-averaging over φ0 of the form P0 is +P = 1 +2πδ(r − r0) dφdr ∈ H1,1 +dR(P) +(6.37) +The corresponding A-model state equals +ΨP = 1 +2πδ(r − r0)ψΦψR. +(6.38) +The corresponding mirror state is +ΨP1 +P = 1 +2πδ(r − r0)ψΦψR + q+eiY +∞ +� +0 +dt δ(r − r0 − t) + q−e−iY +∞ +� +0 +dt δ(r − r0 + t) += 1 +2πδ(r − r0)ψΦψR + q+eiY Θ(r − r0) + q−e−iY Θ(r0 − r). +(6.39) +In our work [8] we proved the mirror symmetry for the A-type HTQM (VA, Q, G±, µ2, g) +and the B-type HTQM (VB, QX, G±, µ2, g). The proof was essentially a summation over +the divisor states Ψ⃗b for all divisors of the toric space X. The HTQM mirror implies the +following statement for the correlation functions. +47 + +Theorem (tropical mirror for HTQMs): On toric space X, with boundary divisors +⃗b1, ...,⃗bB the sum over divisor states in the A-model’s correlation function of evaluation states +Ψγa equals to the correlation function of the corresponding mirror states ΨX +γa in B-model with +mirror superpotential WX i.e. +∞ +� +k=0 +1 +k! ⟨Ψγ1, ..., Ψγn, WX, .., WX +� +�� +� +k +⟩Q = ⟨ΨX +γ1, ..., ΨX +γn⟩QWX . +(6.40) +Note that the q-dependence in the A-model is due to the divisor states in WX, while in +B-model both evaluation states and differential QX have nontrivial q-dependence. +7 +Localization of mirror states +In the A-model formulation evaluation states and divisor states look very different and does +not have any simple relation. In case X = P1 the evaluation state for the point observable +at r = r0 and divisor states are +ΨP = 1 +2πδ(r − r0)ψΦψR, +Ψ+ = eiY , Ψ− = e−iY . +(7.1) +In section 6.4 we constructed the P1-mirror state for the evaluation state of a point observable +ΨP1 +P = 1 +2πδ(r − r0)ψΦψR + q+eiY Θ(r − r0) + q−e−iY Θ(r0 − r). +(7.2) +The expression (7.2) for mirror state turns into a pure divisor state Ψ± in the limit r0 → ±∞, +i.e. +lim +r0→−∞ΨP1 +P = q+eiY = q+Ψ+ +lim +r0→+∞ΨP1 +P = q−e−iY = q−Ψ−. +(7.3) +There is a natural geometric interpretation of this relation: The point at finite position +r = r0 becomes a compactifying divisor (a point at infinity in case of P1) when we move r0 +to infinity. +The relation between mirror states and divisor states plays a key role for the localization +of the B-model correlation functions. Note, that the limit of the state, often referred to as +the point-wise limit, in general, may not commute with the amplitude/correlation function +evaluation for the same state. We will discuss this potential problem carefully in the next +48 + +section. +7.1 +Mirror states vs divisor states +The pair of A-model states, corresponding to the Poincare duals of points r0 and r1 are +related by a Q-exact term +Ψ1 − Ψ0 = 1 +2πδ(r − r1)ψΦψR − 1 +2πδ(r − r0)ψΦψR += Q +� +− 1 +2πΘ(r − r1)ψΦ + 1 +2πΘ(r − r0)ψΦ +� += Qχ01. +(7.4) +The tropical form +χ01 = 1 +2π [Θ(r − r0) − Θ(r − r1)] ψΦ. +(7.5) +is a tropical form on P1 since the difference of two Θ-functions has finite support. We can +take a limit r1 → −∞ for χ01 to define +χ+ = +lim +r1→−∞ χ01 = 1 +2π(Θ(r − r0) − 1)ψΦ = − 1 +2πΘ(r0 − r)ψΦ. +(7.6) +The tropical form χ+ can be used to turn the mirror state into the boundary divisor state, +i.e. +q+Ψ+ = ΨP1 +P + QP1χ+. +(7.7) +The relation (7.7) is equivalent to the statement that the mirror state ΨP1 +P and the holomor- +phic function Ψ+ = eiY represent the same cohomology as a forms in VB, i.e. +ΨP1 +P = q+Ψ+ ∈ H∗(QP1, VB). +(7.8) +Note that the tropical form χ+ is G−-closed, i.e. +G−χ+ = G− +� +− 1 +2πΘ(r0 − r)ψΦ +� += 0, +(7.9) +hence we can formulate a stronger equality in cohomology +ΨP1 +P = q+Ψ+ ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]). +(7.10) +49 + +We can take r1 → +∞ limit for (7.5) to define a different tropical form +χ− = 1 +2πΘ(r − r0)ψΦ +(7.11) +such that +q−Ψ− = ΨP1 +P + QP1χ− +(7.12) +Similarly to the previous case +G−χ− = G− +� 1 +2πΘ(r − r0)ψΦ +� += 0, +(7.13) +hence +ΨP1 +P = q−Ψ− ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]). +(7.14) +We can choose more general QP1-exact term, so that the mirror state will become a holo- +morphic function, which is not one of the compactifying divisor states for P1 +ˆΨP = ΨP1 +P + QP1 ˆχ = ΨP1 +P + QP1 +� 1 +2πΘ(r − r0)ψΦ + 1 +2πqe−2iY ψΦ +� += q2e−3iY . +(7.15) +The key difference from the χ± is that the tropical form ˆχ is not G−-closed, i.e. +G− ˆχ = +� 1 +2πΘ(r − r0)ψΦ + q +2π e−2iY ψΦ +� += − q +π e−2iY ̸= 0. +(7.16) +hence ˆΨP and ΨP belong to different classes in (QP1 + zG−)-cohomology. Indeed, a simple +computation shows that +q2e−3iY = eiY + qz +π e−2iY ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]). +(7.17) +In the remaining part of this section we will show that all mirror states admit holo- +morphic function representatives in H∗(QW + zG−) and discuss some properties of these +representatives. +50 + +7.2 +Spectral sequence for QW-cohomology +A mirror state describes a class in H∗(QW, VB). The QW-differential +QW = ψj +R +∂ +∂rj + 2πi∂W +∂Yj +∂ +∂ψj +Φ += dR + 2πi QW +(7.18) +is a sum of two (graded-) commuting differentials, the radial de Rham differential +dR = ψj +R +∂ +∂rj , +d2 +R = 0 +(7.19) +and the LGS differential (3.20) +QW = ∂W +∂Yk +∂ +∂ψk +Φ +, +Q2 +W = 0. +(7.20) +Hence we can use a spectral sequence to evaluate the cohomology of the QW. The spectral +sequence converges at the second step +H∗(QW, VB) = H∗(QW, H∗(dR, VB)), +(7.21) +since the only cohomology of the radial de Rham operator on RN are constant (r-independent) +forms of degree 0 in ψR and is isomorphic to the LSG vector space (3.19). The QW is dif- +ferential in LGS theory, hence we can express the cohomology via the Jacobi ring (2.4) for +superpotential W +H∗(QW, VB) = JW. +(7.22) +An isomorphism (7.22) means that for every QW-closed B-model state we can find a holo- +morphic function from the same QW-cohomology class on VB. +7.3 +Pairing and localization of states +Definition: The holomorphic germ for a B-model state Ψ is an evaluation of the state Ψ at +r = ψ = 0, i.e. +Φ(Y ) = Ψ(r, Y, ψΦ, ψR) +��� +ψ=r=0. +(7.23) +51 + +Definition: For states Ψ1, Ψ2 in B-model with superpotential W we introduce a pairing +gΛ +W(Ψ1, Ψ2) = g +� +Ψ1, eΛQW (L)Ψ2 +� +, +(7.24) +where g is the B-model pairing, Λ is a real parameter and L is a localization function +L = +N +� +k=1 +rkψk +Φ. +(7.25) +Note: The exponent in the pairing evaluates into +QW(L) = +N +� +k=1 +ψk +Φψk +R + 2πi +N +� +k=1 +rk ∂W +∂Yk +. +(7.26) +Hence the QW(L) in (7.26) is an oscillating function in parity-even variables r and Y what +makes the radial integral converging. +The pairing gΛ +W matches with the B-model pairing (6.19) for Λ = 0. The pairing gΛ +W is +QW-invariant, indeed +gΛ +W(QWΨ1, Ψ2) = g +� +QWΨ1, eΛQW (L)Ψ2 +� += −(−1)|Ψ1|g +� +Ψ1, QWeΛQW (L)Ψ2 +� += −(−1)|Ψ1|gΛ +W(Ψ1, QWΨ2). +(7.27) +We used the QW-invariance of the B-model pairing and an identity +QWeQW (L) = QWeQW L+LQW = eQW (L)QW. +(7.28) +The QW-invariance of the pairing implies that it is well-defined on H∗(QW, VB). +Lemma: On QW-closed states the pairing gΛ +W is independent of Λ. +Proof: The derivative of the pairing evaluates into +d +dΛgΛ +W(Ψ1, Ψ2) = d +dΛg +� +Ψ1, eΛQW (L)Ψ2 +� += g +� +Ψ1, (QWL + LQW)eΛQW (L)Ψ2 +� += g +� +Ψ1, LeΛQW (L)QWΨ2 +� +− (−1)|Ψ1|g +� +QWΨ1, eΛQW (L)Ψ2 +� += 0 +(7.29) +52 + +We used the QW invariance of the B-model pairing and QW-closeness of both Ψ1 and Ψ2 +and an identity (7.28). +■ +The Λ-independence means that the pairing gΛ +W matches with the B-model pairing g on +H∗(QW, VB). The B-model pairing g is non-degenerate on VB and QW-invariant, hence g is +non-degenerate on QW-cohomology, so is the gΛ +W. +Proposition: The QW-closed state Ψ and its holomorphic germ Φ are in the same QW- +cohomology class i.e. +Ψ = Φ ∈ H∗(QW, VB). +(7.30) +Proof: Since QW(L) in (7.26) is an oscillating function in parity-even variables r and Y , then +the integral will localize near critical points of QW(L). Moreover, the localization exponent +is scaled by Λ, so we can choose large Λ to eliminate the subleading corrections to the saddle +point formula. The critical points of QW(L) are determined from +∂ +∂rk QW(L) = 2πi∂W +∂Yk += 0, +∂ +∂Ym +QW(L) = 2πi +N +� +k=1 +rk +∂2W +∂Yk∂Ym += 0, +∂ +∂ψk +Φ +QW(L) = ψk +R = 0, +∂ +∂ψk +R +QW(L) = −ψk +Φ = 0. +(7.31) +Under the assumption that the critical points of W are isolated, the rank of +∂2W +∂Yl∂Ym is maximal, +hence the integral in the limit Λ → ∞ localizes to the critical points of W and origin in all +other variables +rk = ψk +R = ψk +Φ = 0, Y = Y0, dW(Y0) = 0. +(7.32) +The ratio of determinants for the integration around the critical point is +det +�∂2QW(L) +∂ψk +Φ∂ψl +R +� +· +(2π)N +det +� +∂2QW (L) +∂Yk∂rl +� = +(2π)NΛN +(2πΛ)N det +� +∂2W +∂Yk∂Yl +�. +(7.33) +The leading order saddle point formula for the pairing integral localizes the gΛ +W-pairing to +gΛ +W(Ψ1, Ψ2) = +� +dW =0 +1 +det +� +∂2W +∂Yk∂Yl +� Ψ1 · Ψ2 +��� +r=ψ=0. +(7.34) +Note that the Λ-dependence drops out from the leading saddle point formula. We observe +that the pairing gΛ +W for B-model states localizes to the corresponding holomorphic germs. In +53 + +particular, we have an equality +0 = gΛ +W(Ψ, Ψ′) − gΛ +W(Φ, Ψ′) = gΛ +W(Ψ − Φ, Ψ′). +(7.35) +The equality holds for all QW-closed states Ψ′ and pairing gΛ +W is non-degenerate on H∗(QW, VB), +hence Ψ and Φ represent the same class in QW-cohomology, i.e. +Ψ = Φ ∈ H∗(QW, VB), +(7.36) +what completes the proof of the proposition. +■ +7.4 +Higher pairings in B-model +Definition: For a B-model with superpotential W we can introduce a C[[z]]-valued pairing +on B-model states +KW(Ψ1, Ψ2) = +� +dµ Ψ1 · eΛ{QW +zG−,L}Ψ2 = g +� +Ψ1, eΛ{QW +zG−,L}Ψ2 +� +, +(7.37) +with the integration measure (7.24), real parameter Λ and localization function +L = +N +� +k=1 +rkψk +Φ. +(7.38) +The pairing KW matches with the B-model pairing (6.19) for Λ = 0. +The pairing KW +is a pairing between H∗(QW − zG−) and H∗(QW + zG−). Indeed, the conjugation formula +KW((QW − zG−)Ψ1, Ψ2) = g +� +(QW − zG−)Ψ1, eΛ{QW +zG−,L}Ψ2 +� += −(−1)|Ψ1|g +� +Ψ1, (QW + zG−)eΛ{QW +zG−,L}Ψ2 +� += −(−1)|Ψ1|g +� +Ψ1, eΛ{QW +zG−,L}(QW + zG−)Ψ2 +� += −(−1)|Ψ1|KW(Ψ1, (QW + zG−)Ψ2). +(7.39) +Lemma: On H∗(QW − zG−) ⊗ H∗(QW + zG−) the pairing (7.37) is independent on Λ. +54 + +Proof: The derivative of the pairing evaluates into +d +dΛKW(Ψ1, Ψ2) = d +dΛg(Ψ1, eΛ{QW +zG−,L}Ψ2) = g(Ψ1, {QW + zG−, L}eΛ{QW +zG−,L}Ψ2) += g(Ψ1, (QW + zG−)LeΛ{QW +zG−,L}Ψ2) + g(Ψ1, L(QW + zG−)eΛ{QW +zG−,L}Ψ2) += −(−1)|Ψ1|g((QW − zG−)Ψ1, LeΛ{QW +zG−,L}Ψ2) + g(Ψ1, LeΛ{QW +zG−,L}(QW + zG−)Ψ2) += 0. +In the relation we used QW-invariance of the B-model pairing (4.11), G−-invariance of the +B-model pairing (4.12) an equality +(QW + zG−)eΛ{QW +zG−,L} = eΛ{QW +zG−,L}(QW + zG−). +(7.40) +to complete the proof. +■ +The Λ-independence means that the pairing KW matches with the B-model pairing g on +H∗(QW, ker G−). The B-model pairing g is non-degenerate on VB and QW-, G−-invariant, +hence g is non-degenerate on H∗(QW, ker G−), so is the KW. +Remark: We can introduce an expansion for the pairing in z to define higher pairings +KW(Ψ1, Ψ2) = +∞ +� +k=0 +zk K(k) +W (Ψ1, Ψ2), +(7.41) +such that the K(0) +W is identical to the gΛ +W from the section 7.3. +The argument of exponent in (7.37) is a sum of two terms: a function +{QW, L} = QW(L) = +N +� +k=1 +ψk +Φψk +R + 2πi +N +� +k=1 +rk ∂W +∂Yk +(7.42) +and first-order differential operator +{G−, L} = −i +N +� +k=1 +rk ∂ +∂Yk +. +(7.43) +55 + +We can use the Zassenhaus formula +eA+B = eA eB e− 1 +2 [A,B] e +1 +6(2[B,[A,B]]+[A,[A,B]]) . . . +(7.44) +to express the localization exponent as product of the QW(L)-localizaton exponent and +differential operator with coefficients in z, i.e. +eΛ{QW +zG−,L} = eΛQW (L) · D(z, Λ, r, Y, ∂Y ). +(7.45) +The representation (7.45) allows us to conclude that the higher pairings also localize to +ψ = r = 0, i.e. the value of the pairing KW is the same for QW-, G−-closed states Ψ, Ψ′ and +their holomorphic germs Φ, Φ′, i.e. +KW(Ψ, Ψ′) = KW(Φ, Φ′). +(7.46) +The localization exponent to first order in z +eΛ{QW +zG−,L} = eΛQW (L) +� +1 + πzΛ2 +N +� +k,l=1 +rkrl ∂2W +∂Yk∂Yl +− izΛ +N +� +k=1 +rk ∂ +∂Yk ++ O(z2) +� +(7.47) +allows us to evaluate the K(1) +W -pairing +K(1) +W (Ψ1, Ψ2) = +� +dµ Ψ1 · eΛQW (L) +� +πΛ2 rkrl ∂2W +∂Yk∂Yl +− iΛrk ∂ +∂Yk +� +Ψ2 += −π +� +dW =0 +(∂Yk∂YlW)−1(Ψ1∂Yk∂YlΨ2 − Ψ2∂Yk∂YlΨ1) +det ∂Yk∂YlW +��� +r=ψ=0. +(7.48) +Remark: In case of single Y -variable the pairing simplifies into +K(1) +W (Ψ1, Ψ2) = −2π +� +W ′=0 +1 +2 +Ψ1Ψ′′ +2 − Ψ′′ +1Ψ2 +(W ′′)2 +��� +r=ψ=0. +(7.49) +Remark: The pairings K(0) +W and K(1) +W on holomorphic functions match with the correspond- +ing higher pairing components (3.26) and (3.27) for the LGS theory. +In proposition (7.30) we showed that QW-closed B-model state Ψ and its holomorphic germ +represent the same class in H∗(QW, VB). The pairing KW allows us to make a stronger +statement: +56 + +Proposition: The QW-, G−-closed state Ψ and its holomorphic germ Φ are in the same +class of H∗(QW + zG−, VB) and there exists a tropical form (z-independent) χ such that +Ψ = Φ + QWχ, +G−χ = 0. +(7.50) +Proof: The pairing of both representatives with arbitrary Ψ′ ∈ H∗(QW −zG−) are identical +i.e. +KW(Ψ′, Ψ) = KW(Ψ′, Φ) +=⇒ KW(Ψ′, Ψ − Φ) = 0. +(7.51) +The non-degeneracy of the pairing leads to +Ψ − Φ = 0 ∈ H∗(QW + zG−), +(7.52) +what implies the the existence of the tropical form χ such that +Ψ − Φ = (QW + zG−)χ. +(7.53) +By construction Ψ and Φ are independent of z and we can choose χ, which is z-independent, +hence G−χ = 0, what concludes the proof. +■ +Example: The holomorphic germ for the P1-mirror state (6.39) is +Φγ = ΨP1 +γ +��� +r=ψ=0 = q+eiY Θ(−r0) + q−e−iY Θ(r0). +(7.54) +For r0 < 0 it coincides with the Ψ+-divisor, while for r0 > 0 it coincides with Ψ−-divisor. +For both cases we showed that the corresponding tropical forms χ± are G−-closed. +7.5 +B-model deformation by a holomorphic function +In this section we will describe a one-parameter family of deformations for the B-type HTQM +(VB, QW, G±, µ2, g) by a holomorphic function Φ. The holomorphic function belongs to the +B-model space of states, hence we can use construction from section 4.5 for the leading order +57 + +deformation +Qǫ +W = QW − 2π[G−, µ2(ǫΦ, ·)] = QW + ǫ 2πi ∂Φ +∂Yk +∂ +∂ψk +Φ += Q + 2πi∂W +∂Yk +∂ +∂ψk +Φ ++ 2πiǫ ∂Φ +∂Yk +∂ +∂ψk +Φ += QW ǫ. +(7.55) +The last equality allows us to describe the deformed differential Qǫ +W in the form of differential +in B-model with deformed superpotential +W ǫ = W + ǫ Φ. +(7.56) +The deformation of a B-model state is defined as (a finite) expansion in W and A-model +propagators K +Ψǫ = Ψ + 2πKWG−µ2(ǫΦ, Ψ) + O(ǫ2) += Ψ + 2πKG−µ2(ǫΦ, Ψ) + 2πKG−µ2(W, 2πKG−µ2(ǫΦ, Ψ))... + O(ǫ2). +(7.57) +The deformation of a mirror state Ψ = ΨW +γ simplifies into +[ΨW +γ ]ǫ = ΨW +γ + 2πKWG−µ2(ǫΦ, ΨW +γ ) + O(ǫ2) += Ψγ + 2πKG−µ2(W, Ψγ) + 2πKG−µ2(ǫΦ, Ψγ) ++ 2πKG−µ2(W, 2πKG−µ2(W, Ψγ)) + 2πKG−µ2(ǫΦ, 2πKG−µ2(W, Ψγ)) ++ 2πKG−µ2(W, 2πKG−µ2(ǫΦ, Ψγ)) + ... + O(ǫ2) += Ψγ + 2πKG−µ2(W + ǫΦ, ΨW +γ ) ++ 2πKG−(W + ǫΦ, 2πKG−µ2(W + ǫΦ, ΨW +γ )) + ... + O(ǫ2) += ΨWǫ +γ +Hence we conclude that the deformation [ΨW +γ ]ǫ of the mirror state ΨW +γ +by a holomorphic +function in B-model with superpotential W is a mirror state ΨWǫ +γ +for the same γ but in +B-model with deformed superpotential W ǫ. +7.6 +Higher pairing for mirror states +The holomorphic germs for mirror states in B-model can be used to construct a good section +in the corresponding LGS theory. +58 + +Definition: The tropical good section in mirror LGS theory for toric space X with su- +perpotential W is a linear span of holomorphic germs for mirror states +Im Strop +W += C⟨ΦW +γ | γ ∈ H∗ +dR(X)⟩. +(7.58) +Remark: A similar construction of the good section from holomorphic germs of harmonic +states was described in section 7.2 and 7.3 of [20]. +The definition (3.36) of a good section in LGS requires vanishing of higher pairings. +Proposition (higher pairing for tropical good section): Higher LGS pairings (3.21) +for tropical good section vanish. Moreover +KW(ΦW +γ1, ΦW +γ2) = +� +X +γ1 ∧ γ2, +∀γ1, γ2 ∈ H∗ +dR(X). +(7.59) +Proof: By construction in (6.29) the mirror states ΨW +γ +are G−- and QW-closed, hence +ΨW +γ ∈ H∗(QW ± zG−) and the higher B-model pairing (7.37) for such states is independent +of Λ so we can relate it at different values of Λ. In particular, we use +• Λ → ∞ limit gives us the LG pairing for holomorphic functions i.e. +KW(ΨW +γ1, ΨW +γ2) = KW(ΦW +γ1, ΦW +γ2) = KW(ΦW +γ1, ΦW +γ2); +(7.60) +• Λ = 0 gives us the B-model pairing g i.e. +KW(ΨW +γ1, ΨW +γ2) = g(ΨW +γ1, ΨW +γ2). +(7.61) +The mirror states ΨW +γk can be written in the following schematic form +ΨW +γ = Ψγ + G−χW +γ +(7.62) +for the A-model state +χW +γ = −2πKµ2(W, Ψγ) − 2πKµ2(W, 2πKG−µ2(W, Ψγ)) + . . . +(7.63) +59 + +The B-model pairing g on representation (7.62) simplifies into +g(ΨW +γ1, ΨW +γ2) = g(Ψγ1, Ψγ2) + g(Ψγ1, G−χW +γ2) + g(G−χW +γ1, Ψγ2) + g(G−χW +γ1, G−χW +γ2) += g(Ψγ1, Ψγ2) + g(G−Ψγ1, χW +γ2) − g(χW +γ1, G−Ψγ2) − g(χW +γ1, G2 +−χW +γ2) += g(Ψγ1, Ψγ2). +(7.64) +We used the G±-invariance of the pairing (4.12) to simplify the expression +The A-model pairing on evaluation states Ψγ is the intersection of the cohomology classes +on X i.e. +g(Ψγ1, Ψγ2) = +� +X +γ1 ∧ γ2. +(7.65) +The proof of the proposition is the following: We use (7.60) to turn LGS higher pairing into +the B-model higher pairing on mirror states. We use (7.61) to turn the B-model higher pairing +into ordinary pairing which according to (7.64) simplifies to the intersection of cohomology +classes. The higher parings are z-expansion of KW hence +K(k) +W (ΦW +γ1, ΦW +γ2) = 0, ∀ k > 0. +(7.66) +what completes the proof of the proposition. +■ +Example: The toric description of P2 consists of three compactifying divisors with primitive +normal vectors: (1, 0), (0, 1) and (−1, −1) so the mirror superpotential +WP2 = eiY1 + eiY2 + qe−iY1−iY2 +(7.67) +The de Rham cohomology of P2 are one-dimensional in degrees 0, 2, and 4. The image of +tropical good section is +Im Strop +P2 += C⟨1, qe−iY1−iY2, eiY1+iY2⟩. +(7.68) +8 +Correlation functions for mirror states +The correlation function invariance theorem from section 5.4 tells us that +⟨Ψ1, ..., Ψn, QXχ⟩QX = 0 +(8.1) +60 + +for QX- G−-closed states Ψa and a G−-closed state χ. In previous section we showed that +there exists a tropical form χ such that it can turn a mirror state into a holomorphic germ. +Moreover, in section 7.5 we saw that the deformation of B-model by a holomorphic func- +tion leaves it in the same class but with different superpotential. Hence, our strategy for +evaluating correlation functions of mirror states in B-model will be the following: +1. Replace one mirror state by the corresponding holomorphic germ. +2. Deform the superpotential and mirror states by the holomorphic germ. +3. Apply the recursion formula for correlation functions to reduce the number of argu- +ments by one. +4. Repeat steps 1-3 till the number of arguments reaches 3. +The simplification of the theory deformation while using the holomorphic germ rather +than the mirror state itself suggest an alternative strategy for the correlation function evalu- +ation: We replace all mirror states by the corresponding holomorphic germs and evaluate the +correlation function. The application of this strategy to the 4pt functions immediately gives +zero answer. The reason for that is our definition of the B-model amplitudes via an expan- +sion in A-model amplitudes. In A-model the propagator acts trivially on the holomorphic +functions what is the source of zero answers. +To resolve the puzzle we recall that the A-model amplitudes for tropical GW invariants +have a particular number of divisor states, determined by the degree. The mirror states +can provide additional divisor states, but if we already have too many (from holomorphic +germs) the amplitude vanishes due to the degree selection for the tropical GW invariants. +Since, all non-trivial GW invariants have degree bigger than zero, the corresponding A-model +amplitudes have at least one divisor state, hence we can always replace a single mirror state +by the holomorphic germ. Later we will see that a single replacement is just enough to prove +our main theorem. +8.1 +4-point function invariance and holomorphic representatives +Using the symmetries of the amplitudes we can rewrite the 4pt function in the form +2⟨Ψ1, Ψ2, Ψ3, Ψ4⟩QW = g(Ψ4, +� +σ∈S3 +µ2(Ψσ(1), KWG−µ2(Ψσ(2), Ψσ(3)))) += g(Ψ4, Ψ123) = g(Φ4 + QWχ+, Ψ123) = g(Φ4, Ψ123) = 2⟨Ψ1, Ψ2, Ψ3, Φ4⟩QW +(8.2) +61 + +The equality on the second line requires +g(QWχ+, Ψ123) = 0. +(8.3) +In (7.7) we showed that χ for P1-model does not belong to tropical forms on P1, hence the +difference +g(QWχ+, Ψ123) − g(χ+, QWΨ123) +(8.4) +might not be zero. The difference is controlled by the boundary term of the r-integration +g(QWχ+, Ψ123) − g(χ+, QWΨ123) = +� +dµ Q(χ+Ψ123) = +� +dµ dR(χ+Ψ123) += lim +r→−∞ +� +S1 dY Θ(r4 − r) Ψ123 +��� +ψ=0 = lim +r→−∞ +� +S1 dY Ψ123 +��� +ψ=0. +(8.5) +We can repeat the analysis for the case different holomorphic representative Φ′ +4 = qe−iY and +χ− +g(QWχ−, Ψ123) − g(χ−, QWΨ123) = lim +r→+∞ +� +S1 dY Ψ123 +��� +ψ=0. +(8.6) +Both boundary terms vanish if Y -independent part of Ψ123 +��� +ψ=0 has finite support. We can +evaluate the Ψ123 for P1-mirror states +� +S1dY Ψ123 +��� +ψ=0 = 1 +2 +� +σ∈S3 +� +S1 dY µ2(Ψσ(3), 2πKWG−µ2(Ψσ(1), Ψσ(2))) +��� +ψ=0 += πq +� +σ∈S3 +� +Θ(min(rσ(1), rσ(2)) − r))Θ(r − rσ(3)) + Θ(r − max(rσ(1), rσ(2)))Θ(rσ(3) − r) +� +and observe that the products of Θ-functions indeed have the finite support. +We can perform the same analysis for ⟨Ψ1, Ψ2, Φ3, Ψ4⟩QW , where we replaced the mirror +state Ψ3 by the holomorphic function Φ3 = eiY . The boundary term is controlled by the +function +� +S1dY ˜Ψ123 +��� +ψ=0 = +� +S1 dY µ2(Φ3, 2πKWG−µ2(Ψ1, Ψ2)) +��� +ψ=0 ++ +� +S1 dY µ2(Ψ2, 2πKWG−µ2(Ψ1, Φ3)) +��� +ψ=0 + +� +S1 dY µ2(Ψ1, 2πKWG−µ2(Ψ2, Φ3)) +��� +ψ=0 += 2πqΘ(min(r1, r2) − r) + 2πqΘ(r2 − r)Θ(r − r1) + 2πqΘ(r1 − r)Θ(r − r2). +(8.7) +62 + +which does not have the finite support and the limit is finite and non-zero +lim +r→−∞ +� +S1 dY ˜Ψ123 +��� +ψ=0 = 2πq ̸= 0. +(8.8) +Hence we conclude that the simultaneous replacement of two P1-mirror states Ψ3 and Ψ4 +by the same holomorphic germs qe−iY does not preserve the invariance of the correlation +function, i.e. +⟨Ψ1, Ψ2, Φ3, Ψ4⟩QW − ⟨Ψ1, Ψ2, Φ3, Φ4⟩QW ̸= 0. +(8.9) +Remark: If we replace Ψ4 by the the different holomorphic function Φ4 = qe−iY then the +boundary term is controlled by the different limit +lim +r→+∞ +� +S1 dY ˜Ψ123 +��� +ψ=0 = 0. +(8.10) +Hence we conclude that for the choice of holomorphic functions Φ3 = eiY and Φ4 = qe−iY +preserves the 4-point function, i.e. +⟨Ψ1, Ψ2, Ψ3, Ψ4⟩QW = ⟨Ψ1, Ψ2, Φ3, Φ4⟩QW = ⟨Ψ1, Ψ2, eiY , qe−iY ⟩QW +(8.11) +Note that the equality (8.11) agrees with our heuristic analysis for the possible P1-mirror +states replacement, based on the A-model divisor counting. The GW invariants for P1 are +non-trivial only for degree-1 maps, hence the A-model correlation functions have exactly two +divisor states Ψ+ = eiY and Ψ− = e−iY . +8.2 +n-point function invariance and holomorphic representative +Conjecture: For the mirror states Ψ1, .., Ψn for general toric X the state Ψ1..n, evaluated +on the sum (5.17) over rooted trees with leaves Ψ1, ..., Ψn and χ, such that +QWχ = Ψn+1 − Ψn+1 +��� +ψ=r=0, +G−χ = 0 +(8.12) +satisfy +g(χ, dRΨ1..n) − g(dRχ, Ψ1..n) = 0, +(8.13) +i.e. the boundary term vanishes for all boundary divisors of X. +Proposition: The conjecture holds for X = P1. +63 + +Proof: For the (n + 1)-point correlation function the boundary term is +g(χ∓, dRΨ1..n) − g(dRχ∓, Ψ1..n) = lim +r→±∞ +� +S1 dY Ψ1..n +��� +ψ=0 +(8.14) +In particular, if Y -independent part of Ψ1..n +��� +ψ=0 has finite support for any P1-mirror states +Ψ1, ..., Ψn then the boundary term vanishes. +For any pair of P1 B-model states Ψ1 and Ψ2 we can simplify +KP1G−µ2(Ψ1, Ψ2) = KG−µ2(Ψ1, Ψ2). +(8.15) +This equality follows from the degree counting. The B-model states are at most (1, 1)-forms +in ψ, so is the product. The KG− lowers degree of the state by (1, 1), so there are no higher +order terms in expansion of the B-model propagator KP1G− in A-model propagators KG−. +We can simplify the KG− for product of two P1-mirror states into +Ψ′ +ab = 2πKG−µ2(ΨP1 +a , ΨP1 +b ) = eiY Θ(r − max(ra, rb)) + qe−iY Θ(min(ra, rb) − r) +and observe that it is degree-0 tropical form, hence the product of two such expressions is +also a zero form. The KG−-action on any zero form is zero, i.e. +Ψ′ +abcd = KG−µ2(Ψ′ +ab, Ψ′ +cd) = 0. +(8.16) +This equality allows us to simplify the sum over rooted trees to +Ψ1..n = 1 +4 +n−1 +� +k=1 +µ2(Ψ′ +(1..k, Ψ′ +k+1..n)), Ψ′ +1 = ΨP1 +1 +(8.17) +using the Ψ′ +1..n-expressions defined recursively +• n = 2 +Ψ′ +12 = 2πKG−µ2(ΨP1 +1 , ΨP1 +2 ) = eiY Θ(r − max(r1, r2)) + qe−iY Θ(min(r1, r2) − r); +• n > 2 +Ψ′ +1..n = 2πKG−µ2(ΨP1 +(n, Ψ′ +1..n−1)). +(8.18) +64 + +We can prove by induction that +Ψ′ +1..n = eiY Θ(r − max(r1, .., rn)) + qe−iY Θ(min(r1, .., rn) − r) +(8.19) +and evaluate +Ψ1..n +��� +ψ=0 = +1 +4 · n! +� +σ∈Sn +n−1 +� +k=1 +(eiY Θ(r − max(rσ(1), .., rσ(k))) + qe−iY Θ(min(rσ(1), .., rσ(k)) − r)) +× (eiY Θ(r − max(rσ(k+1), .., rσ(n))) + qe−iY Θ(min(rσ(k+1), .., rσ(n)) − r)). +(8.20) +Each Y -independent term in the sum has finite support so is the whole sum, what completes +the proof of a proposition. +■ +8.3 +3-point functions +Proposition (3-point localization formula): For QW-closed B-model states Ψ1, Ψ2, Ψ3 +the 3-point function equals to the LGS 3-point function for corresponding holomorphic germs +and superpotential W, i.e. +⟨Ψ1, Ψ2, Ψ3⟩QW = ⟨Φ1, Φ2, Φ3⟩W. +(8.21) +Proof: The 3pt function in B-model is +⟨Ψ1, Ψ2, Ψ3⟩QW = g(Ψ1, µ2(Ψ2, Ψ3)). +(8.22) +The product of two QW-closed states is QW-closed, i.e. +QWµ2(Ψ2, Ψ3) = µ2(QWΨ2, Ψ3) + µ2(Ψ2, QWΨ3) = 0, +(8.23) +hence we can replace B-model pairing g by the pairing gΛ +W +⟨Ψ1, Ψ2, Ψ3⟩QW = gΛ +W(Ψ1, µ2(Ψ2, Ψ3)) +(8.24) +and use the localization +⟨Ψ1, Ψ2, Ψ3⟩QW = +� +dW =0 +Ψ1Ψ2Ψ3 +det ∂i∂jW +��� +r=ψΦ=ψR=0 = +� +dW =0 +Φ1Φ2Φ3 +det ∂i∂jW = ⟨Φ1, Φ2, Φ3⟩W. +(8.25) +65 + +Hence we completed the proof of the proposition. +■ +Example: For X = P1 the mirror states for cohomology representatives 1, γP ∈ H∗(QP1) +have the holomorphic germs 1, eiY : +• three 0-forms: The B-model 3-point function vanishes, due to the insufficient degree +of the form +⟨ΨP1 +1 , ΨP1 +1 , ΨP1 +1 ⟩QP1 = +� +dµ 1 = 0 = ⟨1, 1, 1⟩WP1; +(8.26) +• single 1-form: The B-model 3-point function is the integral of the γP over P1 +⟨ΨP1 +1 , ΨP1 +1 , ΨP1 +P ⟩QP1 = +� +dµ ΨP1 +P = +� +P1 γP = 1 = ⟨1, 1, eiY ⟩WP1; +(8.27) +• two 1-forms: The B-model 3-point function vanishes, due to degree selection +⟨ΨP1 +1 , ΨP1 +P , ΨP1 +P ⟩QP1 = 0 = ⟨1, eiY , eiY ⟩WP1; +(8.28) +• three 1-forms: For three 1-forms, Poincare dual to the three distinct points r1, r2, r3 +the B-model 3-point function is +⟨ΨP1 +P , ΨP1 +P , ΨP1 +P ⟩QP1 = +� +dµ ΨP1 +P ΨP1 +P ΨP1 +P = q +� +σ∈S3 +Θ(rσ(1), rσ(2), rσ(3)) += q = ⟨eiY , eiY , eiY ⟩WP1. +(8.29) +8.4 +4-point functions +Proposition (4-point localization formula): If conjecture 8.2 holds then the 4-point +correlation function of mirror states ΨW +a +equals to the 4-point correlation function of the +corresponding holomorphic germs in LGS theory with superpotential W and tropical good +section (7.58), i.e. +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 ⟩QW = ⟨Φ1, Φ2, Φ3, Φ4⟩Strop +W +. +(8.30) +Proof: The invariance theorem from section 5.4, extended by a conjecture 8.2 allows us to +replace the mirror state ΨW +4 +by the holomorphic germ Φ4 in the 4-point function of mirror +66 + +states +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 ⟩QW = ⟨ΨW +1 , ΨW +2 , ΨW +3 , Φ4⟩QW . +(8.31) +The holomorphic germ Φ4 is QW- and G−-closed, hence we can use the recursion relation +theorem from section 5.5 +⟨ΨW +1 , ΨW +2 , ΨW +3 , Φ4⟩QW = d +dǫ +��� +ǫ=0⟨Ψǫ +1, Ψǫ +2, Ψǫ +3⟩QW ǫ +(8.32) +for deformed superpotential +W ǫ = W + ǫΦ4. +(8.33) +The deformed states +Ψǫ +k = ΨW +k + ǫ 2πKWG−µ2(ΨW +k , Φ4) +(8.34) +are QW ǫ-closed hence the 3-point function can be evaluated in terms of LGS theory +⟨Ψǫ +1, Ψǫ +2, Ψǫ +3⟩QW ǫ = +� +dW ǫ=0 +Φǫ +1 · Φǫ +2 · Φǫ +2 +det ∂j∂kW ǫ +��� +Y =Y0 = ⟨Φǫ +1, Φǫ +2, Φǫ +3⟩W ǫ +(8.35) +for holomorphic germs of deformed states +Φǫ +k = Ψǫ +k +��� +ψ=r=0 = Φk + 2πǫ KWG−µ2(ΨW +k , Φ4) +��� +ψ=r=0 = Φk + ǫ Ctrop +W (Φk, Φ4). +(8.36) +The equality (8.36) defines tropical contact term Ctrop +W (Φk, Φ4) and later we will show that +these contact terms match with LGS contact term (3.31) for tropical good section (7.58). +We can substitute the Φǫ +k in terms of Φk and contact term into the 3-point function +derivative +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 ⟩QW = d +dǫ +��� +ǫ=0⟨Φǫ +1, Φǫ +2, Φǫ +3⟩W ǫ += d +dǫ +��� +ǫ=0⟨Φ1, Φ2, Φ3⟩W ǫ + ⟨Ctrop +W (Φ1, Φ4), Φ2, Φ3⟩W ++ ⟨Φ1, Ctrop +W (Φ2, Φ4), Φ3⟩W + ⟨Φ1, Φ2, Ctrop +W (Φ3, Φ4)⟩W += ⟨Φ1, Φ2, Φ3, Φ4⟩Strop +W +(8.37) +The last equality is the the LGS recursion formula (3.18) for the 4pt functions with contact +terms defined by (8.36). +■ +67 + +8.5 +Contact terms +Definition: For a superpotential W, holomorphic function Φ2 and mirror state ΨW +1 +with +holomorphic germ Φ1 we define tropical contact term +Ctrop +W +(Φ1, Φ2) = d +dǫ +��� +ǫ=0 Ψǫ +1 +��� +ψ=r=0 = 2πKWG−µ2(ΨW +1 , Φ2) +��� +ψ=r=0. +(8.38) +Proposition (contact terms for tropical good section): The tropical contact term +equals (as classes in H∗(QW + zG−)) to the LGS contact term for the tropical good section, +i.e. +Ctrop +W (Φ1, Φ2) = CStrop +W +(Φ1, Φ2). +(8.39) +Proof: The product µ2(ΨW +1 , Φ2) is a QW-closed state due to +QWµ2(ΨW +1 , Φ2) = µ2(QWΨW +1 , Φ2) + µ2(ΨW +1 , QWΦ2) = 0. +(8.40) +Hence it represents some class in H∗(QW). Moreover, the same class can be expressed via +µ2(ΨW +1 , Φ2) = µ2(Φ1, Φ2) = Φ1Φ2 ∈ H∗(QW). +(8.41) +Using the map πW : RC∗N → JW we can write the class for the product of two holomorphic +functions Φ1Φ2 in the form πW(Φ1Φ2). Applying an isomorphism J−1 : JW = H∗(QW) → +H∗ +dR(X) : +Ψ �→ J−1(Ψ) from section 2.4 to the class πW(Φ1Φ2) we can construct class +J−1πW(Φ1Φ2) in H∗ +dR(X). +Let us choose representatives γ for each class of cohomology +H∗ +dR(X), so the class J−1πW(Φ1Φ2) is represented by a tropical form γJ−1πW(Φ1Φ2). This +tropical form defines a mirror state ΨW +γJ−1πW (Φ1Φ2), which is G−-closed and represents the +same class to µ2(ΨW +1 , Φ2) in H∗(QW). Hence there exists a tropical form χ such that +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) = QWχ. +(8.42) +We can use χ to evaluate +KWG−µ2(ΨW +1 , Φ2) = KWG−ΨW +γJ−1πW (Φ1Φ2) + KWG−QWχ += −G−χ + (QW + zG−)KWG−χ. +(8.43) +68 + +The tropical contact term is the holomorphic germ of (8.43) +Ctrop +W (Φ1, Φ2) = 2π +� +KWG−µ2(ΨW +1 , Φ2) +� ��� +ψ=r=0 += −2πG−χ +��� +ψ=r=0 + (QW + zG−)(...). +(8.44) +We can represent the QW as a sum of two graded-commuting differentials +QW = 2πiQW + dR +(8.45) +to write a solution to +QWχ = (2πiQW + dR)χ = Ψ +(8.46) +using the homotopy ΣW for QW +χ = +1 +2πiΣWΨ − +1 +(2πi)2ΣWdRΣWΨ + . . . +(8.47) +The radial de Rham dR adds powers in ψR, hence the holomorphic germs for higher terms +in the sum vanish, i.e. +G−ΣWdRΣW +� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψR=0 = 0. +(8.48) +The tropical contact term simplifies to +Ctrop +W (Φ1, Φ2) = −2πG−χ +��� +ψ=r=0 = iG−ΣW +� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψ=r=0 +(8.49) +Let us recall the relation iG− = G− between the B-model G− and LGS G−. Both G− and +QW act trivially on the radial variables r, ψR, hence we can simplify +iG−ΣW +� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψ=r=0 += G−ΣW +�� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψR=r=0 +� ��� +ψΦ=0. +(8.50) +The mirror states are Hodge type tropical forms, i.e. contain the same number of ψΦ and +69 + +ψR for each degree, hence we can further simplify +� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψR=r=0 = +� +µ2(ΨW +1 , Φ2) − ΨW +γJ−1πW (Φ1Φ2) +� ��� +ψR=r=ψΦ=0 += µ2(Φ1, Φ2) − ΦW +γJ−1πW (Φ1Φ2). +(8.51) +For any function Φ we construct another function, defined via +SWπW(Φ) = ΦW +γJ−1πW (Φ) +(8.52) +to express the tropical contact term in the form +Ctrop +W (Φ1, Φ2) = G−ΣW +� +µ2(Φ1, Φ2) − ΦW +γJ−1πW (Φ1Φ2) +� += G−ΣW (µ2(Φ1, Φ2) − SWπW(Φ1Φ2)) +(8.53) +Earlier in a section 3.4 we saw that, in particular, SW defines a section H∗(QW) → H∗(QW + +zG−) hence we can drop exact terms in (8.43). +■ +Example: Let X = P1, ΨW +1 is a mirror state for the point observable at r = r1 +ΨW +1 = ΨP1 +P = δ(r − r1)ψΦψR + eiY Θ(r − r1) + qe−iY Θ(r1 − r) +(8.54) +and holomorphic function +Φ2 = qe−iY . +(8.55) +The tropical contact term is the holomorphic germ of +KWP1G−µ2(ΨW +1 , Φ2) = KG−µ2(ΨW +1 , Φ2) + KG−µ2(WP1, KG−µ2(ΨW +1 , Φ2)) + ... += KG−µ2(Ψ1, Φ2) = KG−(qe−iY δ(r − r1)ψΦψR) += +� ∞ +0 +dt e−tHG+G−(qe−iY δ(r − r1)ψΦψR) += qe−iY +� ∞ +0 +dt δ(r − r1 + t) = qe−iY Θ(r1 − r), +(8.56) +while the germ evaluation gives us +Ctrop +WP1(Φ1, Φ2) = KWP1G−µ2(ΨW +1 , Φ2) +��� +ψ=r=0 = qe−iY Θ(r1). +(8.57) +70 + +The holomorphic representative of the mirror state +Φ1 = ΨW +1 +��� +ψ=r=0 = eiY Θ(−r1) + qe−iY Θ(r1) +(8.58) +gives us contact terms of the form +Ctrop +WP1(qe−iY , qe−iY ) = qe−iY , +Ctrop +WP1(eiY , qe−iY ) = 0. +(8.59) +The tropical contact terms matches with the P1-mirror LGS contact terms for tropical good +section for P1 +Im Strop = C⟨1, eiY ⟩. +(8.60) +8.6 +5-point function +Proposition (5-point localization formula): If conjecture 8.2 holds then the 5-point +correlation function of mirror states ΨW +a +equals to the 5-point correlation function of the +corresponding holomorphic germs in LGS theory with superpotential W and tropical good +section (7.58), i.e. +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 , ΨW +5 ⟩QW = ⟨Φ1, Φ2, Φ3, Φ4, Φ5⟩Strop +W +. +(8.61) +Proof: We can rewrite the 5-point function in B-model using the recursion formula for the +B-model deformation by a holomorphic germ of the mirror state ΨW +5 +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 , ΨW +5 ⟩QW = ⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 , Φ5⟩QW = d +dǫ +��� +ǫ=0⟨Ψǫ +1, Ψǫ +2, Ψǫ +3, Ψǫ +4⟩QW ǫ. +The deformed superpotential is +W ǫ = W + ǫΦ5 +(8.62) +and deformed states are +Ψǫ +k = ΨW +k + ǫKWG−µ2(ΨW +k , Φ5) = ΨW ǫ +k , +(8.63) +while the holomorphic germs are +Φǫ +k = Ψǫ +k +��� +ψ=r=0 = Φk + 2πǫ KWG−µ2(ΨW +k , Φ5) +��� +ψ=r=0 = Φk + ǫ Ctrop +W (Φk, Φ5). +(8.64) +71 + +The deformed states are mirror states for W ǫ hence we can repeat the 4-point function +analysis from previous section +⟨ΨWǫ +1 , ΨWǫ +2 , ΨWǫ +3 , ΨWǫ +4 ⟩QW ǫ = ⟨Ψǫ +1, Ψǫ +2, Ψǫ +3, Φǫ +4⟩QW ǫ = d +dλ +��� +λ=0⟨Ψǫ,λ +1 , Ψǫ,λ +2 , Ψǫ,λ +3 ⟩QW ǫ,λ +(8.65) +with deformed superpotential +W ǫ,λ = W ǫ + λΦǫ +4 = W + ǫΦ5 + λΦ4 + λǫ Ctrop +W (Φ4, Φ5) +(8.66) +and deformed states +Ψǫ,λ +k += Ψǫ +k + λ KW ǫG−µ2(Ψǫ +k, Φǫ +4). +(8.67) +Remark: The appearence of the λǫ Ctrop +W (Φ4, Φ5) is a signature of the non-linear rela- +tion between the linear times ǫ, λ on the image of good section and coordinates T in the +superpotential deformation, introduced by K. Saito. +The states Ψǫ,λ +k +are mirror states hence we can express the 3-point function in terms of +the LGS theory +⟨Ψǫ,λ +1 , Ψǫ,λ +2 , Ψǫ,λ +3 ⟩QW ǫ,λ = +� +dW ǫ,λ=0 +Φǫ,λ +1 +· Φǫ,λ +2 +· Φǫ,λ +2 +det ∂j∂kW ǫ,λ += ⟨Φǫ,λ +1 , Φǫ,λ +2 , Φǫ,λ +3 ⟩W ǫ,λ +(8.68) +for holomorphic germs +Φǫ,λ +k += Ψǫ,λ +k +��� +ψ=r=0 = Φǫ +k + λ KW ǫG−µ2(Ψǫ +k, Φǫ +4) +��� +ψ=r=0 += Φǫ +k + λ Ctrop +W ǫ (Φǫ +k, Φǫ +4). +(8.69) +The Ctrop +W ǫ (Φǫ +k, Φǫ +4) is a contact term for the good section for deformed superpotential Wǫ. +Hence, we can apply the LSG recursion formula (3.18) for the 4-point function +⟨Ψǫ +1, Ψǫ +2, Ψǫ +3, Ψǫ +4⟩QW ǫ = d +dλ +��� +λ=0⟨Ψǫ,λ +1 , Ψǫ,λ +2 , Ψǫ,λ +3 ⟩QW ǫ,λ += d +dλ +��� +λ=0⟨Φǫ +1, Φǫ +2, Φǫ +3⟩W ǫ,λ + ⟨Ctrop +W ǫ (Φǫ +1, Φǫ +4), Φǫ +2, Φǫ +3⟩W ǫ ++ ⟨Φǫ +1, Ctrop +W ǫ (Φǫ +2, Φǫ +4), Φǫ +3⟩W ǫ + ⟨Φǫ +1, Φǫ +2, Ctrop +W ǫ (Φǫ +3, Φǫ +4)⟩W ǫ += ⟨Φǫ +1, Φǫ +2, Φǫ +3, Φǫ +4⟩Strop +W ǫ +(8.70) +72 + +Similarly we can apply the LSG recursion (3.18) for the 5-point function +⟨ΨW +1 , ΨW +2 , ΨW +3 , ΨW +4 , ΨW +5 ⟩QW = d +dǫ +��� +ǫ=0⟨Ψǫ +1, Ψǫ +2, Ψǫ +3, Ψǫ +4⟩QW ǫ = d +dǫ +��� +ǫ=0⟨Φǫ +1, Φǫ +2, Φǫ +3, Φǫ +4⟩Strop +W ǫ += d +dǫ +��� +ǫ=0⟨Φ1, Φ2, Φ3, Φ4⟩Strop +W ǫ ++ ⟨Ctrop +W (Φ1, Φ5), Φ2, Φ3, Φ4⟩Strop +W ++ ⟨Φ1, Ctrop +W (Φ2, Φ5), Φ3, Φ4⟩Strop +W ++ ⟨Φ1, Φ2, Ctrop +W (Φ3, Φ5), Φ4⟩Strop +W ++ ⟨Φ1, Φ2, Φ3, Ctrop +W (Φ4, Φ5)⟩Strop +W += ⟨Φ1, Φ2, Φ3, Φ4, Φ5⟩Strop +W +what completes the proof of the proposition. +■ +8.7 +Parallel transport of a good section +In LGS theory, given a good section SW for superpotential W we can extend it to a good +section SWt for superpotential Wt via the parallel transport. For a given superpotential W +and γ ∈ H∗ +dR(X) we construct a mirror state +ΨW +γ = Ψγ + KG−µ2(W, Ψγ) + KG−µ2(W, KG−µ2(W, Ψγ)) + . . . +(8.71) +which define an image of good section +Im SW = C⟨ΦW +γ |γ ∈ H∗ +dR(X)⟩. +(8.72) +Let W → W + δW, then the change of the mirror state to the leading order is given by +ΨW +δW +γ +− ΨW +γ = KWG−µ2(δW, ΨW +γ ) + O(δW)2. +(8.73) +The corresponding change of holomorphic germ +ΦW +δW +γ +− ΦW +γ = KWG−µ2(δW, ΨW +γ ) +��� +r=ψ=0 = Ctrop +W (δW, ΦW +γ ) = CStrop +W +(δW, ΦW +γ ) +(8.74) +Hence we demonstrated that the tropical good section is parallel with respect to connection +determined by it. +8.8 +Localization of correlation functions +Theorem (localization of correlation functions): The B-model correlation function for +the mirror states constructed for observables γk ∈ H∗ +dR(X) on toric space X equal to the +73 + +LGS correlation function +⟨ΨX +γ1, ..., ΨX +γn⟩QX = ⟨ΦX +γ1, ..., ΦX +γn⟩WX +(8.75) +The mirror LGS theory has the following data +1. The LGS theory is on C∗N, where N is the complex dimension of X. +2. The holomorphic top form on C∗N written in terms of cylindrical coordinates (r, Y ) is +Ω = dY1 ∧ ... ∧ dYN. +(8.76) +3. The LGS superpotential is a mirror superpotential for X, written in terms of the +compactifying divisors for X, i.e. +WX = +� +⃗b∈BX +q⃗b ei⟨⃗b,Y ⟩. +(8.77) +4. The image of a good section +Im Strop +W += C⟨ΦW +γ | γ ∈ H∗ +dR(X)⟩. +(8.78) +5. The LSG observables are holomorphic germs for mirror states +ΦX +γ = ΨX +γ +��� +ψ=r=0. +(8.79) +Proof: We are going to embed the statement of the theorem into more general equality +⟨ΨW +γ1, ..., ΨW +γn⟩QW = ⟨ΦW +γ1, ..., ΦW +γn⟩W. +(8.80) +The theorem is the case when W = WX. Such rewriting allows us to use an induction in +the number of states n. The proof for n = 3 follows from proposition on 3-point localization +from section 8.3. We use the invariance theorem 5.4 under assumption of conjecture 8.2 to +replace the mirror state by its holomorphic germ in the (n + 1)-point correlation function +⟨ΨW +γ1, ..., ΨW +γn, ΨW +γn+1⟩QW = ⟨ΨW +γ1, ..., ΨW +γn, ΦW +γn+1⟩QW . +(8.81) +74 + +We apply the recursion formula theorem 5.5 for deformation of HTQM with differential QW +by a holomorphic germ state ΦW +γn+1 and express the n + 1-point function as a derivative of +n-point function in deformed theory, i.e. +⟨ΨW +γ1, ..., ΨW +γn, ΦW +γn+1⟩QW = d +dǫ +��� +ǫ=0⟨ΨW ǫ +γ1 , ..., ΨW ǫ +γn ⟩Qǫ +W . +(8.82) +Using our expression (7.55) for the B-model deformation by a holomorphic function we +replace Qǫ +W by QW ǫ for superpotential, deformed by the holomorphich germ of ΨW +n+1 +W ǫ = W + ǫΦW +γn+1. +(8.83) +Using an assumption of induction that the equality holds for n-point correlation functions +⟨ΨW +γ1, ..., ΨW +γn⟩QW = ⟨ΦW +γ1, ..., ΦW +γn⟩W +(8.84) +we can express the (n+1)-point correlation function in B-model in terms of n-point functions +in LSG theory +⟨ΨW +γ1, ..., ΨW +γn, ΨW +γn+1⟩QW = d +dǫ +��� +ǫ=0⟨ΦW ǫ +γ1 , ..., ΦW ǫ +γn ⟩W ǫ +(8.85) +for holomorphic functions given by the holomorphic germs of deformed states +ΦW ǫ +γk = ΨW ǫ +γk +��� +ψ=r=0 = ΦW +γk + 2πKWG−µ2(ΦW +γn+1, ΨW +γk) +��� +ψ=r=0. +(8.86) +According to the tropical good section proposition 8.5 we can identify the second term in +the expression above with tropical contact term +2πKWG−µ2(ΦW +γn+1, ΨW +γk) +��� +ψ=r=0 = CStrop +W +(ΦW +γk, ΦW +γn+1). +(8.87) +The recursion formula (3.18) for the LGS correlation functions allows us to rewrite the +derivative of n-point function as (n + 1)-point correlation function in LSG theory with +superpotential W, i.e. +⟨ΨW +γ1, ..., ΨW +γn, ΨW +γn+1⟩QW = d +dǫ +��� +ǫ=0⟨ΦW ǫ +γ1 , ..., ΦW ǫ +γn ⟩QW ǫ = ⟨ΦW +γ1, ...ΦW +γn, ΦW +γn+1⟩W +(8.88) +and complete the proof of the theorem. +■ +75 + +Acknowledgments +We are grateful to Yasha Neiman for many discussions on the topics presented in this paper. +The work A.L. is supported by Wu Wen-Tsun Key Lab of Mathematics. The work of V.L. is +supported by the Quantum Gravity Unit of the Okinawa Institute of Science and Technology +Graduate University (OIST). +References +[1] G. Mikhalkin, “Introduction to Tropical Geometry (notes from the IMPA lectures in +Summer 2007),” 2007. +[2] G. Mikhalkin, Amoebas of Algebraic Varieties and Tropical Geometry, pp. 257–300. +Springer US, Boston, MA, 2004. +[3] G. Mikhalkin and J. Rau, Tropical geometry, vol. 8. MPI for Mathematics, 2009. +[4] A. Gathmann and H. Markwig, “Kontsevich’s formula and the WDVV equations in +tropical geometry,”. https://arxiv.org/abs/math/0509628. +[5] A. Givental and B. Kim, “Quantum cohomology of flag manifolds and Toda lattices,” +Communications in mathematical physics 168 no. 3, (1995) 609–641. +[6] C. Vafa and E. Zaslow, Mirror Symmetry: Clay Mathematics Monographs, Vol. 1. +AMS-CMI, 2003. +[7] E. Witten, “Topological Sigma Models,” Commun. Math. Phys. 118 (1988) 411. +[8] A. Losev and V. 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Phys. 165 (1994) 311–428, arXiv:hep-th/9309140. +77 + diff --git a/2NAzT4oBgHgl3EQfuP1K/content/tmp_files/load_file.txt b/2NAzT4oBgHgl3EQfuP1K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2e4d156a401e143bfa259270f57b9165224d1b9 --- /dev/null +++ b/2NAzT4oBgHgl3EQfuP1K/content/tmp_files/load_file.txt @@ -0,0 +1,2753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf,len=2752 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='01687v1 [hep-th] 4 Jan 2023 Tropical Mirror Symmetry: Correlation functions Andrey Losev Wu Wen-Tsun Key Lab of Mathematics, Chinese Academy of Sciences, USTC, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='96, JinZhai Road Baohe District, Hefei, Anhui, 230026, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='China National Research University Higher School of Economics, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Moscow, Russia, 119048 Vyacheslav Lysov Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan Abstract We formulate the mirror symmetry for correlation functions of tropical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We prove the tropical mirror correspondence for correlation functions of evaluation ob- servables on toric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The key point of the proof is the localization of correlation functions for mirror states in type-B higher topological quantum mechanics on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The correlation functions localize to the correlation functions of holomorphic func- tions, defined recursively in Landau-Ginzburg-Saito theory with exponential mirror superpotential and tropical good section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Contents 1 Introduction 3 2 Mirror Correspondence 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 Toric varieties .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 73 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8 Localization of correlation functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 73 2 1 Introduction The real tropical numbers is a set of real numbers, extended by {−∞} with arithmetic operations: tropical addition x +T y = max(x, y) and tropical multiplication x ∗T y = x + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The set of tropical numbers is a semigroup with respect to tropical addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The tropical numbers appeared at different times at several branches of mathematics: Maslov introduced notion of tropical integration, while computer scientist Imre Simon introduced and adjective tropical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Mikhalkin [1], [2], [3] used tropical numbers to study geometric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The collection of problems and methods become the Tropical Geometry research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In many cases the enumerative problems in algebraic geometry over complex numbers have tropical counter- parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular, we can define the tropical Gromov-Witten invariants by counting graphs passing through some cycles, see Mikhalkin [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Mikhalkin [2] observed that the number of tropical curves in P2 of degree 3, passing through 8 points in general position is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' This number, matches with counting of degree-3 algebraic curves passing through 8 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Gathmann and Markwig [4] showed that equality generalizes for all Gromov-Witten invariants of P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The matching of tropical and complex invariants was formalized into: Theorem (tropical correspondence): Gromov–Witten invariant coincides with its trop- ical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We propose to analyze the tropical correspondence theorem in context of mirror symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Theorem (mirror symmetry): The Gromov–Witten invariants for X equal to the corre- lation functions in Landau-Ginzburg-Saito theory on mirror space X∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The mirror symmetry for toric space X relates [5] the GW invariants to Landau-Ginzburg- Saito theory with exponential superpotentials, see [6] for review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Theorem (tropical mirror symmetry): The tropical Gromov–Witten invariants for toric space X of complex dimension N equal to the correlation functions in Landau-Ginzburg- Saito theory on mirror space X∨ = C∗N with certain exponential superpotential, canonical holomorphic top form and mirror K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito’s good section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3 Our main result is the proof of the tropical mirror symmetry theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' There are two possible application of our result: We can assume the tropical correspondence theorem and use our proof as a new proof of mirror symmetry for the toric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Alternatively, we can assume the mirror symmetry theorem and use our proof as a proof for the tropical correspondence for toric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our proof we use several key ideas: tropical numbers as a scaling of the exponential map, quantum mechanics representation of the tropical GW invariants, mirror relation for quantum mechanics as summation over divisor states and localization for HTQM correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Real tropical numbers can be constructed from the field of real numbers with usual addition and multiplication by performing the exponential map X = e x ǫ followed by the limit ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The exponential map turns multiplication into addition for any value of ǫ, while the tropical addition requires the limit lim ǫ→0 ǫ ln(e x ǫ + e y ǫ ) = max(x, y) = x +T y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The scaling of exponential map generalizes to the complex geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The scaling construc- tion also provides an heuristic proof of the Tropical Correspondence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Gromov-Witten invariants over complex numbers can be described using A-type twisted topological string theory [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our work [8] we proved that the tropical GW in- variants can be described using the higher topological quantum mechanics (HTQM) with circle action on graphs, introduced in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Certainly, the HTQM can be constructed as the ǫ → 0 limit of maximally degenerate complex structure (very long strings) in topological string theory, but the fact that it captures all tree-level tropical GW invariants is a novel result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Moreover in [10] (see also [9]) it was shown that the quantum mechanics, similar to the one we described in previous paper, provides a solution to the WDVV equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Authors of [11] used the 2d CFT to relate the sum over holomortex insertions in A-type topological string with superpotential deformation of the 2d CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our paper [8] we showed that the sum over divisor states in A-type HTQM amplitudes equals to the amplitudes in B- type HTQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-type HTQM is the deformation of the A-type HTQM by the boundary divisor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular we showed that for the case of toric spaces the B-type HTQM has exponential superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The topological nature of the B-type topological string allows for the drastic simplification for correlation functions of certain observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular, the correlation functions of the 4 mirrored evaluation observables can be written using the recursive construction, discussed in section 8 of present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The base of the recursion, the 3-points functions are evaluated in terms of the residue formula, constructed from superpotential and holomorphic top form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such simplification is a reflection of a localization-like phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Typically, localization in field theory and quantum mechanics requires path integral formulation, while we showed that the localization construction can be realized using the operator formalism in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The recursive construction for implies that the superpotential alone is not enough to evaluate the correlation functions for more than 3 observables!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The additional data is given in the form of good section, introduced and studied by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The correlation functions in Landau-Ginzburg-Saito theory have non-trivial dependence on the choice of good section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' As a part of the proof for the main theorem we derive the good section for mirror superpotentials from the tropical mirror correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The structure of our paper is as follows: In section 2 we briefly review the mirror symme- try and formulate it in the form of the equality for correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 we review the recursive construction for correlation functions in Landau-Ginzburg-Saito theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In section 4 we define the Higher Topological Quantum Mechanics on trees, describe the amplitudes and describe a deformation of HTQM by special state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In section 5 we introduce the notion of correlation functions for HTQM and discuss their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In section 6 we briefly review the HTQM representation for tropical GW invariants and mirror relation between A- and B- types HTQMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In section 7 we will introduce a notion of localization for mirror states, while in the last section we will use the localized states to evaluate the correlation functions in mirror HTQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2 Mirror Correspondence The mirror symmetry describes a relation between the A- and B-models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The A-model of our interest is the theory of the Gromov-Witten invariants for a Kahler manifold X of dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model is the theory of complex structure deformations on the dual complex manifold X∨ of same dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In well-known examples both X and X∨ are compact Calabi-Yau 3-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can generalize the correspondence away from three-dimensional Calabi-Yau spaces by relaxing the compactness condition on X∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The complex structure deformations in that case are also parametrized by a holomorphic a function W and the corresponding B-model in commonly referred to as the Landau-Ginzburg (LG) theory with holomorphic superpotential 5 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our paper we will discuss the A-model on a toric space X of dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Furthermore, we will perform a certain scaling procedure to the geometry, which transforms the usual GW invariants into the tropical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The corresponding mirror B-model becomes the LG theory on X∨ = C∗N with exponential superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In the rest of this section we will briefly review the definition of toric manifolds, tropical GW invariants and give a detailed formulation of the mirror relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 Toric varieties Toric manifold X is a compactification of C∗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can represent C∗N = RN × TN with the radial part RN, equipped with standard coordinates ri, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., N and angular part, N-dimensional torus TN = (S1)N, with standard angular coordinates φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Equivalently, we can say that the C∗N is a trivial N-dimensional toric fibration over RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We describe the compactification of C∗N using the fibraton data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The radial part is compactified by the hyperplanes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Each hyperplane is given in terms of the inside-pointing N-dimensional normal vector ⃗ba with components bi a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Each normal vector has integer components i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' bi a ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The two normal vectors with proportional components describe the same hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can fix this ambiguity by choosing the primitive normal vector, such that gcd(b1 a, b2 a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', bN a ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For toric space X we will denote the set of such normal vectors by BX = {⃗ba}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In order to get a compactification of a complex manifold, we require that one of the circles S1 ⊂ TN inside the toric fibration shrinks to zero when we approach each of the compactifying hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The choice of a circle is given by a class in π1(TN) = H1(TN, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For the hyperplane with normal vector ⃗ba the corresponding class is the class of � bi adφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will refer to the compacting hyperplanes as compactifying divisors, and to BX as the set of all compactifying divisors for toric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 Gromov-Witten theory The Gromov-Witten invariant NX β (C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Cn) counts the number of algebraic curves of degree-β, genus-0 in complex space X, passing through the cycles C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The integral 6 representation NX β (C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Cn) = � M0,n(X,β) n� α=1 ev∗ αγα, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) uses M0,n(X, β), the moduli space of degree β curves in X with n marked points, compact- ified by quasi-maps, equipped with the evaluation map evα : M0,n(X, β) → X : (φ : P1 → X, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', zn) �→ φ(zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) The γα are special representatives (smoothed out delta functions on cycles) of the Poincare dual to the cycles Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In mathphysics literature the commonly used notation for the same GW invariant is ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn⟩X β,0 and we will use it in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For a given cycles γk ∈ H∗(X) we can organize the genus-0 GW invariants of different degrees into a single expression ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn⟩X = � β∈H1,1(X) qβ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn⟩X β,0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) where q, describes the Kahler moduli of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For generic X the GW invariants are formal series in q, but for toric X they simplify to the polynomials in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: In present paper we will restrict our attention to the GW invariants for 3 or more observables i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' There are interesting geometric invariants of X, with natural description in the form of GW invariants for two observables, but most of them can be reformulated as invariants with 3 or more observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Tropical GW invariants On toric manifold X we can perform a tropical limit: a coordinate transformation in the form of scaling (rk, φk) → (rk/ǫ, φk) followed by the ǫ → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For more details see our previous paper [8] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The limit of a smooth algebraic curve of genus zero in toric space X is a circle bundle over a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The tree is embedded by a piece-wise linear map into radial part of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The limiting curve is known as a tropical curve and was extensively studied by Mikhalkin and collaborators [2] in context of Tropical Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The moduli of a tropical curve are position of a root vertex, lengths and twists of internal edges of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We discussed the moduli space in our work [8], while for more detailed review see Mikhalkin’s book [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can take the tropical limit for the differential forms γα to define the tropical A-model 7 observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Note that the tropical limit turns smooth forms on toric space X into singular forms on radial part of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can define the tropical GW invariants as the integral over tropical moduli space of tropical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our work [8] we showed that the tropical GW invariants can be written as a sum of amplitudes in Higher Topological Quantum Mechanics (HTQM) on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will provide the detailed definition of HTQM and describe the amplitudes in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In many examples it was observed that the tropical GW invariants match with the con- ventional GW invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' This observation was formalized into: Theorem (tropical correspondence): For the toric space X and smooth differential forms γk ∈ H∗ dR(X) the Gromov-Witten invariant ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn⟩X β matches with tropical Gromov- Witten invariant ⟨γtrop 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γtrop n ⟩X β for the tropical limit γtrop k of the forms γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Evidence: There are several different types of evidence for the theorem: Mikhalkin [2] counted the number degree-3 tropical curves in P2 passing through the 8 points in general position, by presenting the corresponding graphs, counted with proper multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The total number he obtained was 12, what matched with the well known N3 = 12 answer for the same problem for algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Gathmann and Markwig [13] derived the recursion formula for number of tropical curves of degree-d, passing through the 3d − 1 points on P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Their result matches with the Kontsevich-Manin recursion formula [14] for algebraic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such matching essentially gives a prof of tropical correspondence for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' There are several results on match for the descendant GW invariants: The case of P1 was discussed by B¨ohm, Goldner and Markwig [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Markiw and Rau [16] proved the equality for descendant invariants for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Our construction of tropical numbers and geometric objects as a scaling limit ǫ → 0 in cylindrical coordinates serves as an heuristic proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The GW invariants do not depend on the choice of coordinates, hence remain the same for any non-zero value of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In [8] we derived the tropical mirror superpotential for toric X and it matches with exponential mirror superpotential derived in [5], [17] for the same toric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Given 8 that the mirror symmetry relation holds for toric spaces we can use our result as an evidence in favor of the tropical correspondence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 Mirror relation The mirror of the N-dimensional toric manifold X is a non-compact N-dimensional Calabi- Yau X∨ = C∗N with holomorphic superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will use the toric representation C∗N = RN × TN with radial coordinates rj and angular (holomorphic) coordinates Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The holomorphic top form in these coordinates is Ω = dY 1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. ∧ dY N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Let us formulate several relations, implied by the mirror correspondence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The Jacobi ring for superpotential W is JW = RC∗N/IW, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) where RC∗N is the ring of holomorphic functions on C∗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In our coordinates RC∗N is the ring of periodic functions of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The IW is the ideal generated by the partial derivatives of W IW = �∂W ∂Yj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) Remark: If W has isolated critical points then JW is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition (mirror for observables): The de Rahm cohomology of toric space X iso- morphic (as a vector spaces !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=') to the Jacobi ring J : H∗ dR(X) → JWX : γ �→ Jγ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) for mirror superpotential WX = � ⃗b∈BX q⃗b ei⟨⃗b,⃗Y ⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) where the sum is taken over the compactifying divisors BX of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can refine the mirror relation for observables to the isomorphism of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition (mirror for Frobenius rings): The quantum cohomology of toric space X isomorphic (as graded-commutative Frobenius rings) to the Jacobi ring for mirror super- 9 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Frobenius ring structure (CA, gA) on quantum cohomology ring of X is determined by the 3-point GW invariants of X CA αβδ = ⟨γα, γβ, γδ⟩X, gA αβ = ⟨γα, γβ, 1⟩X = � X γα ∧ γβ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) while the Frobenius ring structure (CB, gB) for Jacobi ring can be formulated using the residue formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence the mirror relation for rings can be formulated as equality CB αβδ = � dWX=0 Φγα Φγβ Φγδ det ∂k∂lWX , gB αβ = � dWX=0 Φγα Φγβ det ∂k∂lWX , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) where Φγ is a representative of a class Jγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' One can show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) is independent on the choice of representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' There is a further generalization of the mirror relation for rings, which includes all GW invariants of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Let γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn be a basis in H∗ dR(X) and T k − linear coordinates on this space in this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can organize all genus-0 GW invariants for X into the generating function FA(T, q) = ∞ � j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.,jn=0 ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γ1 � �� � j1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', γn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., γn � �� � jn ⟩X n � k=1 (T k)jk jk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) Parameters q represent the Kahler moduli dependence for X in A-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model generating function was defined in works of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='Saito [12], Blok-Varchenko [18] and Dijkgraaf-Verlinde-Verlinde [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Namely ∂3FB(T, q) ∂T α ∂T β ∂T δ = � dW =0 Φα(T) · Φβ(T) · Φδ(T) det ∂k∂lW(T) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) where W(T) is the deformation of mirror superpotential in special coordinates, introduced by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito, the holomorphic functions Φα(T) are partial derivatives of superpotential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Φα(T) = ∂W ∂T α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) The partial derivatives ∂k∂lW of W are taken in coordinates Y , where the holomorphic top form is dY1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. ∧ dYN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 10 Theorem (mirror correspondence): For toric space X the generating function of the GW invariants equals to the B-model generating function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) for deformations of mirror superpotential WX i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' FA(T, q) = FB(T, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) Remark: The equality above is an equality for the formal series in q and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For the toric GW invariant each coefficient in T-expansion is a polynomial in q, rather than the formal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Therefore for the case of toric A-model the corresponding mirror B-model expression should also be a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence we will focus on proving the equality of two polynomials for the coefficients of formal series in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In the next section we will give a description for the B-model coefficients in the generating function expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3 Mirror for correlation functions For arbitrary three holomorphic functions we define the 3-point correlation function ⟨Φα, Φβ, Φγ⟩W = � dW =0 Φα · Φβ · Φγ det ∂k∂lW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) Then the generating function of B-model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) can be written in the form ∂3FB(T, q) ∂T α ∂T β ∂T γ = ⟨Φα(T), Φβ(T), Φγ(T)⟩W (T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) In particular, the relation above implies that the cubical term in T-expansion is given by FB(T, q) = 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='T αT βT γ ⟨Φα, Φβ, Φγ⟩W + O(T 4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) where Φα = Φα(0) are representatives for the classes in Jacobi ring JW for W = W(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The quartic term in the expansion for FB can be expressed in the form ∂4FB ∂T α ∂T β ∂T γ ∂T δ ��� T=0 = ∂ ∂T δ ��� T δ=0 ⟨Φα, Φβ, Φγ⟩W +T δΦδ + � ∂ ∂T δ ��� T δ=0Φα(T), Φβ, Φγ � W + � Φα, ∂ ∂T δ ��� T δ=0Φβ(T), Φγ � W + � Φα, Φβ, ∂ ∂T δ ��� T δ=0Φγ(T) � W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) 11 The expression above has the following interpretation: The first term is the change in 3- point correlation function under the change of W in the direction of Φδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The last three terms describe the change of the functions Φα(T) under the transport in direction of Φδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can introduce connection CW, so the change of a function Φα(T) in direction Φδ is given by ∂ ∂T δ ��� T δ=0Φα(T) = CW(Φδ, Φα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) The combination of four terms in the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) is known as the recursion formula for the 4-point function in 2-dimensional topological theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The commonly used form for the recursion relation is (see also [20]) ⟨Φα, Φβ, Φγ, Φδ⟩W = d dǫ ��� ǫ=0⟨Φα, Φβ, Φγ⟩W +ǫΦδ + ⟨CW(Φδ, Φα), Φβ, Φγ⟩W + ⟨Φα, CW(Φδ, Φβ), Φγ⟩W + ⟨Φα, Φβ, CW(Φδ, Φγ)⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) Note that the 4-point function depends on a choice of holomorphic function representatives for the classes in Jacobi ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The connection coefficients CW(Φδ, Φα) are referred to in [20] as the contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The fourth derivative of the generating function can be written in terms of the 4-point function, defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) ∂4FB ∂T α ∂T β ∂T γ ∂T δ ��� T=0 = ⟨Φα, Φβ, Φγ, Φδ⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) The recursion relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) can be further generalized to the case of n-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In physics literature the recursion relations of that type are commonly discussed in context of the Landau-Ginzburg theory [19], so we will adopt the same name for our review of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will give a detailed version of the recursion formula and contact terms later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The expansion coefficients of the generating function become the n-point functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ∂nFB(T, q) ∂T α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='∂T αn ��� T=0 = ⟨Φα1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Φαn⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) The mirror relation for generating functions implies the mirror relation for the correlation functions: Proposition (mirror for correlation functions): For toric space X the GW invari- ants fo cycles γα equal to the correlation function of special representatives SJγα of the 12 corresponding the Jacobi ring classes Jγα for the mirror superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', γn⟩X = ⟨SJγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', SJγn⟩W X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 Landau-Ginzburg-Saito theory The study of Landau-Ginzburg theory was motivated by the theory of critical phenomena, which later grown into 2D CFT and eventially become part of the topological string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For review see [17], namely the (2, 2)-supersymmetric sigma models on non-compact spaces in B-type twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' After the B-type twisting, the theory is not superconformal and further requires setting the anti-holomorphic superpotential to zero, while keeping the holomorphic superpotential W fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' B-twisting is anomalous that results in appearence of the holomor- phic top form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The Landau-Ginzburg-Saito theory on complex space X with superpotential W is a collection of the following data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' non-compact complex manifold X of dimension N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' non-degenerate holomorphic top form Ω ∈ ΩN,0(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) Note that the pair (X, Ω) can be constructed from non-compact Calabi-Yau manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' holomorphic function W : X → C, called superpotential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Good section S : JW → RX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The readers, familiar with mathphysics literature on the Landau-Ginzburg models, might not be familiar with the notion of good section and its significance for the LGS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed all of the LGS correlation functions can be obtained from the T-expansion of the generating function FLG(T) defined via the 3-point function ∂3FLG(T) ∂T α ∂T β ∂T γ = � ∂W ∂T α, ∂W ∂T β , ∂W ∂T γ � W (T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) However, in order to use the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) we need to define the good times T, which typ- ically have complicated functional relation to the deformations of W, linear on Jacobi ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 13 The good section data is equivalent to the choice of good times T, but is has more straight- forward meaning for the recursive definition of correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In some simple cases (polynomial superpotentials) the good section can be constructed from the superpotential if we impose an extra requirement of homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For more information about the good times and good section relation see [12] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The most studied LGS theories have complex manifold CN, with coordinates xj, canonical holomorphic top form Ω = dx1 ∧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.∧dxN and polynomial superpotential W(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The simplest LGS model of this class has single variable (N = 1) and polynomial superpotential of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The image of good section in this case is well known and consists of monomials of degree up to k − 2 Im S = C⟨1, x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., xk−2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) Note that in case of polynomial superpotential with two variables, the good section is known only for limited classes of superpotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Our main interest is the mirror of the A-model for toric manifold X of complex dimension N, given in terms of compactifying divisors BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The corresponding LGS theory has complex manifold C∗N equipped with cylindrical coordinates: the radial coordinates rj ∈ R and holomorphic angular coordinates Yj ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The holomorphic top form in this coordinates is Ω = dY1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ∧ dYN, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) while the superpotential is the exponential function, written using the primitive normal vectors WX = � ⃗b∈BX q⃗b ei bkYk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) The good sections has not been constructed for all LGS theories with exponential superpo- tentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The simplest exponential superpotential is the mirror superpotential for X = P1 WP1 = eiY + qe−iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) The image of good section is Im SP1 = C⟨1, eiY ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 Correlation functions in Landau-Ginzburg-Saito theory Definition: For holomorphic functions Φα, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., n > 2 in LGS theory on C∗N with 14 superpotential W the n-point correlation function ⟨Φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Φn⟩W is defined recursively: the 3-point function is given by the residue formula ⟨Φ1,Φ2, Φ3⟩W = � dW =0 Φ1Φ2Φ3 det ∂j∂kW (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) The (n+1)-point function is defined recursively via n-point functions and their deriva- tives according to formula below ⟨Φ1, Φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Φn, Φn+1⟩W = d dǫ ��� ǫ=0⟨Φ1, Φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Φn⟩W +ǫΦn+1 + ⟨CW(Φ1, Φn+1), Φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Φn⟩W + ⟨Φ1, CW(Φ2, Φn+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Φn⟩W + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. + ⟨Φ1, Φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., CW(Φn, Φn+1)⟩W (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) Earlier we saw that the n-point correlation functions represent the coefficient in the generat- ing function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) hence they are symmetric under permutation of all arguments Φ1, Φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The 3-point function in our definition is manifestly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The symmetry of higher point functions is rather obscure from the recursive definition and require certain properties of the contact terms CW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Following the literature [12] and [20], we will formulate this properties in terms of the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito’s connection on Brieskorn cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition: The correlation functions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) are symmetric if connection is symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' connection is flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' connection preserves the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In [20] was proposed a construction of CW in terms of the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito’s good section S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such connection is manifestly symmetric, while the flatness and metric preservation are derived from the properties of a good section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Cohomology and pairing In order to give a definition of good section and contact terms we will introduce a cohomology theory, motivated by topological string theory of type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 15 Let us consider a graded vector space VLGS = RC∗N ⊗ C[ψi Φ] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) for parity-odd variables ψi Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' On VLGS there is a pair of graded-commuting differentials QW = ∂W ∂Yj ∂ ∂ψj Φ , G− = ∂ ∂Yj ∂ ∂ψj Φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) Remark: VLGS is isomorphic to the space of polyvector fields on C∗N, hence it is equipped with parity-odd symplectic structure and holomorphic top form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The G− is a Batalin–Vilkovisky (BV) operator on VLGS, which generalizes the divergence on vector fields to polyvector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The local holomorphic observables of dimension-0 in topological string theory of type B can be identified with polyvector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The G− is the action of the superpartner to certain U(1)-rotation, which preserves insertion positions for these observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: On VLGS there is C[[z]]-valued Saito’s higher residue pairing K(v1, v2) = � S1N dNY � RN dNr � dNψΦdNψR v1 ∧ e−iΛ{QW +zG−+dR,L}v2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) where Λ is a real parameter, L = N � k=1 rkψk Φ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) is the localization function and dR = ψj R ∂ ∂rj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) is the radial de Rham operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The integral form of the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito’s pairing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) was proposed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The C-valued higher pairings K(k) are defined as expansion coefficients in z- expansion of K i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K(v1, v2) = ∞ � k=0 zk K(k)(v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 we will discuss a similar pairing in details, so for now let us list some properties 16 of the pairing without the proof: The operators QW − zG− and QW + zG− are conjugated with respect to the pairing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K((QW − zG−)v1, v2) = −(−1)|v1|K(v1, (QW + zG−)v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) Hence we can descend the pairing to the pairing on cohomology H∗(QW − zG−) ⊗ H∗(QW + zG−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' One can show that all cohomology for QW ± zG− are holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing on H∗(QW − zG−) ⊗ H∗(QW + zG−) is independent of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence we can choose Λ → ∞, what localizes the pairing on a sum over critical points of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular, the first two pairings on holomorphic functions K(0)(Φ1, Φ2) = (2πi)N � dW =0 Φ1Φ2 det ∂i∂jW (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) and K(1)(Φ1, Φ2) = (2πi)N 1 2 � dW =0 (∂k∂lW)−1(Φ1 ∂k∂lΦ2 − Φ2 ∂k∂lΦ1) det ∂m∂nW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) We can use the pairing K(0) to establish an isomorphism between the Jacobi ring JW and H∗(QW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The cohomology of QW + zG− were introduced by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito under the name of Brieskorn lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 Contact terms from good section The construction of Jacobi ring comes with canonical projection πW : RC∗N → JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Given a pair of homolorphic functions Φ1 and Φ2 we can project their product Φ1Φ2 to the class πW(Φ1Φ2) in Jacobi ring JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The section (which inverts πW) SW : JW → RC∗N turns this class into holomorphic function SW πW(Φ1Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The difference Φ1Φ2 − SW πW(Φ1Φ2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) 17 is trivial in Jacobi ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' An isomorphism between the JW and H∗(QW) means that there exists a map ΣW : RC∗N → VLGS such that Φ1Φ2 − SWπW(Φ1Φ2) = QWΣW(Φ1Φ2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) and ΣWSW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) The choice of such ΣW is known as the choice of homotopy for QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: We define a contact term fo Φ1 and Φ2 in LGS theory with section SW CS W(Φ1, Φ2) = ±G−ΣW(Φ1Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) In other terms the product of two functions Φ1Φ2 can be decomposed into the sum of the image of SW and a linear combination of ∂1W, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., ∂NW, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Φ1Φ2 = SWπW(Φ1Φ2) + σk∂kW (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) The ΣW(Φ1Φ2) has the form σk(Y )ψk Φ, so G−-action on it is G−ΣW(Φ1Φ2) = ∂σk(Y ) ∂Yk , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' just a divergence of the vector field σk(Y )∂Yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Note that for a given SW the decomposition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) does not uniquely fixes the σk(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The freedom of choice σ is fixed by the choice of homotopy ΣW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Note that the dependence of contact term CW on the choice of homotopy ΣW is (QW + zG−)-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' It was shown that the correlation functions are well-defined in H∗(QW +zG−), so the choice of homotopy does not affect the recursion formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can define a natural projection π : VLGS ⊗ C[[z]] → VLGS, given by an evaluation at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The projection π is a chain map, hence it induces projection on cohomology π : H∗(QW + zG−) → H∗(QW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) The section SW induces a section SSaito : JW = H∗(QW) → H∗(QW + zG−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, every holomorphic function is both QW- and G−-closed, hence it describes a class in H∗(QW + 18 zG−), which we take as an image of the SSaito-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The good section SSaito : H∗(QW) → H∗(QW + zG−) is a section for π i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' π ◦ SSaito = IdH∗(QW );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) the higher pairings (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) vanish for all pairs Φ1, Φ2 ∈ Im SSaito i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e K(k)(Φ1, Φ2) = 0, ∀ k > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) For a given section SSaito we can construct the corresponding contact term and con- nection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The good section SSaito is preserved under the parallel transport respect to this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4 HTQM on trees In previous work [8] we showed that the tropical Gromow-Witten invariants can be described using the higher topological quantum mechanics (HTQM) on tree graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In this section we will briefly review the definition on the HTQM on trees, describe the amplitudes and construct a family of HTQMs as a deformation of HTQM by a certain type of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 Higher topological quantum mechanics Definition: The higher topological quantum mechanics, HTQM (with the circle action) (V, Q, G±) is a collection of the following data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Z2-bi-complex (V, Q, G−), namely: Z2-graded vector space V , can be infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' There is a decomposition of V = V0 ⊕ V1 into even V0 and odd V1 under the grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will use the notation |v| ∈ Z2 to describe the grading of a vector v ∈ V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' pair of differentials Q, G− : V → V , such that – grading-odd operators: |Qv| = |G−v| = |v| + 1, – square to zero: Q2 = G2 − = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' two differentials graded-commute, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' {Q, G−} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Unnormalized homotopy G+ : V → V , such that grading-odd operator: |G+v| = |v| + 1, squares to zero G2 + = 0, {G+, G−} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In case V is infinite-dimensional we impose certain consistency conditions on HTQM data (V, Q, G±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We define the Hamiltonian operator H = {Q, G+} : V → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The consistency conditions are formulated in terms of Hamiltonian: The hamiltonian H is such that the evolution operator e−tH is well defined for t ≥ 0 in the following sense: – it is a solution to the ODE (∂t + H)e−tH = 0, e−0·H = 1, t ∈ R+ ∪ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) – forms a 1-parameter semi-group with multiplication e−t1He−t2H = e−(t1+t2)H, ∀ t1, t2 ∈ R+ ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) We require that the t → ∞ limit of the evolution operator exists and is the projector on ker H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' lim t→+∞ e−tH = Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) The projector Π0 obeys Π0G± = G±Π0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 HTQM on trees Definition: The HTQM (V, Q, G±, µ2, g) on a connected tree Γ with distinct root vertex is the collection of the following data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 1-valent vertices are assigned the HTQM states i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' va ∈ V, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., n1 = |V1(Γ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2-valent vertices are assigned observables Oα ∈ V ⊗ V ∗ α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., n2 = |V2(Γ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3-valent vertices are assigned the multiplication µ2 : V ⊗ V → V such that the triple (Q, G−, µ2) obeys 20 µ2 is graded commutative µ2(v, w) = (−1)|v||w|µ2(w, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) µ2 is associative µ2(µ2(v, w), u) = µ2(v, µ2(w, u));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) Leibniz rule for (µ2, Q) Qµ2(v, w) = µ2(Qv, w) + (−1)|v|µ2(v, Qw);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) the pair (G−, µ2) obeys the 7-term relation for all v, u, w ∈ V G−µ2(µ2(v, w), u) = µ2(G−µ2(v, w), u) + (−1)|w|(|v|−1)µ2(w, G−µ2(v, u)) + (−1)|v|µ2(v, G−µ2(w, u)) − µ2(G−v, µ2(w, u)) − (−1)|v|µ2(v, µ2(G−w, u)) − (−1)|u|+|v|µ2(v, µ2(w, G−u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' the root 3-valent vertex assigned the multiplication µ0 3 = g ◦ µ2 : V ⊗3 → R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) constructed from Frobenius structure (g, µ2, Q), where the scalar product, commonly referred to as the pairing, g : V ⊗ V → R obeys the following properties: non-degeneracy on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' the graded-symmetry g(v, w) = (−1)|v||w|g(w, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) Q-invariance g(Qv, w) + (−1)|v|g(v, Qw) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) G±-invariance g(G±v, w) = (−1)|v|g(v, G±w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) evolution invariance g(e−tHv, w) = g(v, e−tHw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Amplitudes on trees Definition: The evolution operator in HTQM (V, Q, G±) is U(t, dt, dϕ) = e−tH+G+dt+G−dϕ ∈ Ω∗(R+ × S1) ⊗ End(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) Definition: For each tree Γ we define pre-amplitude PAΓ : V ⊗n1(Γ) ⊗ (V ⊗ V ∗)⊗n2(Γ) → Ω∗(M(Γ)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) where M(Γ) is the moduli space of trees Γ, defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Each connected tree Γ defines a contraction in tensor algebra ⟨ ⟩Γ : (V ⊗ V ∗)⊗E ⊗ V ⊗n1 ⊗ (V ∗ ⊗ V )⊗n2 ⊗ (V ∗⊗2 ⊗ V )⊗(n3−1) ⊗ V ∗⊗3 → R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) where n3 is the number of 3-valent vertices in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pre-amplitude on a tree Γ with states va and operators Oα is PAΓ(va;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Oα) = � U⊗I(Γ) ⊗ 1⊗n1 n1 � a=1 va n2 � α=1 Oα ⊗ µ⊗(n3−1) 2 ⊗ µ0 3 � Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) Note that the pre-amplitude on tree Γ has no evolution operators on external edges (edges attached to leaves of a tree Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The amplitude on connected tree Γ is an integral AΓ = � M0(Γ) PAΓ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) over moduli space of connected tree M0(Γ) = � R+ × S1�I(Γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 Amplitudes in homotopy notation Definition: The propagator K : V → V for HTQM (V, Q, G±) is K = lim T→∞ � T 0 dt e−tH G+ = � ∞ 0 dt e−tH G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) 22 Note, that the integral has potential divergence, when the exponent vanishes for states from ker H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The G+ in the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) and the HTQM property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) evaluates G+v = 0 on all v ∈ ker H, hence Kv = 0 for such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The propagator K is a homotopy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' {Q, K} = � ∞ 0 dt e−tH {Q, G+} = − � ∞ 0 d � e−tH� = e−tH��� 0 − e−tH��� ∞ = 1 − Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) We can perform the moduli space integral in the amplitude definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) and express the amplitude using propagators AΓ(va;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Oα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K) = � (2πKG−)⊗I(Γ) n1(Γ) � a=1 va n2(Γ) � α=1 Oα ⊗ µ⊗(n3(Γ)−1) 2 ⊗ µ0 3 � Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) Note that the factors 2πG− originate from the angular parts of the moduli space integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' It is very convenient to introduce a graphical representation for amplitudes: we use solid lines for edges, equipped with propagator 2πKG− and dashed lines for edges without the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We label 1-valent vertices by the corresponding states va, 2-valent vertices by the corresponding operators Oα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For each tree there is a single special vertex, responsible for the pairing in HTQM, which can be either the 3-valent special vertex equipped with µ0 3-multiplication, or the 2-valent vertex, equipped with the pairing g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The graphical rep- resentation is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Below we present three graphical representations for the same amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' µ2 µ0 3 v1 v2 v3 v4 O1 O2 µ2 µ2 v1 v2 v3 v4 O1 O2 g µ0 3 µ2 v1 v2 v3 v4 O1 O2 The amplitude, evaluated from the left representation is AΓ = (2π)3 µ0 3(v4, KG−O1v3, KG−µ2(v1, KG−O2v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) We can use the Frobenius structure for (g, µ2) to evaluate the same amplitude, while moving the 3-valent special vertex with µ0 3 to the edge with v4-state and turning it into 2-valent vertex with pairing g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' AΓ = (2π)3 g(v4, µ2(KG−O1v3, KG−µ2(v1, KG−O2v2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) 23 We can revaluate the same amplitude by moving the µ0 3 to the different 3-vertex AΓ = (2π)3 µ0 3(v1, KG−O2v2, KG−µ2(v4, KG−O1v3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) The two representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) are related by the KG−-flip g(KG−v, w) = g(v, KG−w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) We can derive the flip formula using the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) of propagator g(KG−v, w) = lim T→∞ � T 0 dt g � e−tHG+G−v, w � = lim T→∞ � T 0 dt g � G+G−v, e−tHw � = lim T→∞ � T 0 dt g � v, G+G−e−tHw � = g(v, KG−w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) In the equality we used the integral representation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) for the propagator, the G±- invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) and the evolution invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 Deformation of HTQM by a state In our work [8] on tropical mirror we argued that the HTQM on trees admits a “state-operator correspondence”-type relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can turn a HTQM state Ψ ∈ V into an operator OΨ = µ2(Ψ, ·) : V → V (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) acting as the µ2-multiplication by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such relation allowed us to indirectly study the HTQM deformation by a state, by the means of turning state into an operator first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For the discus- sion in later parts of the paper we introduce the notion of the HTQM, deformed by a state below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition (HTQM deformation by a state): Given HTQM (V, Q, G±, µ2, g) on trees and an even state ǫΨ such that QΨ = G−Ψ = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) there is an one-parameter family (V, Qǫ, G±, µ2, g) of HTQMs on trees with differential Qǫ = Q − [G−, µ2(ǫΨ, ·)] + O(ǫ2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) 24 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' the action on states is given by Qǫv = Qv − G−µ2(ǫΨ, v) + µ2(ǫΨ, G−v) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) Proof: For a proof we need to check that the family (V, Qǫ, G±, µ2, g) satisfies the definitions from sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The gradings of G− and Q are odd, while the grading of ǫΨ is even hence the grading of Qǫ is odd i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' |Qǫ| = |[G−, µ2(ǫΨ, ·)]| = 1 + |ǫΨ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) The Qǫ is a differential to the leading order in ǫ, what follows from the square-evaluation QǫQǫv = Q2v − QG−µ2(ǫΨ, v) + Qµ2(ǫΨ, G−v) − G−µ2(ǫΨ, Qv) + µ2(ǫΨ, G−Qv) + G−µ2(ǫΨ, G−µ2(ǫΨ, v)) − G−µ2(ǫΨ, µ2(ǫΨ, G−v)) − µ2(ǫΨ, G−µ2(ǫΨ, G−v)) = 1 2µ2(G−µ2(ǫΨ, ǫΨ), G−v) − 1 2G−µ2(G−µ2(ǫΨ, ǫΨ), v) = O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) In the equality we used Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7), associativity of the multiplication µ2 and the 7-term relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The QǫQǫ = 0 holds for all orders in ǫ for the states ǫΨ, such that G−µ2(ǫΨ, ǫΨ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) The condition above does not follow from the G−Ψ = 0, since the pair (G−, µ2) does not obey the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Qǫ and G− form a pair of graded-commuting differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, we can simplify the graded commutator {Qǫ, G−}v = QG−v − G−µ2(ǫΨ, G−v) + µ2(ǫΨ, G−G−v) + G−Qv − G−G−µ2(ǫΨ, v) + G−µ2(ǫΨ, G−v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) The pair (Qǫ, µ2) obey DGA: By construction µ2 is an associative, graded commutative multiplication and Qǫ is the differential, so we just need to check the Leibniz rule for (µ2, Qǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 25 Indeed, we can evaluate µ2(Qǫv, w) + (−1)|v|µ2(v, Qǫw) − Qǫµ2(v, w) = −µ2(G−µ2(ǫΨ, v), w) + µ2(µ2(ǫΨ, G−v), w) − (−1)|v|µ2(v, G−µ2(ǫΨ, w)) + (−1)|v|µ2(v, µ2(ǫΨ, G−w)) − µ2(ǫΨ, G−µ2(v, w)) + µ2(G−µ2(v, w), ǫΨ) + (−1)|w|(|v|−1)µ2(w, G−µ2(v, ǫΨ)) + (−1)|v|µ2(v, G−µ2(w, ǫΨ)) − µ2(G−v, µ2(w, ǫΨ)) − (−1)|v|µ2(v, µ2(G−w, ǫΨ)) = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) In the equality we used the Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) and 7-term relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Qǫ-invariance of the pairing follows from g(Qǫv, w) + (−1)|v|g(v, Qǫw) = −g(G−µ2(ǫΨ, v), w) + g(µ2(ǫΨ, G−v), w) + (−1)|v|g(v, µ2(ǫΨ, G−w)) − (−1)|v|g(v, G−µ2(ǫΨ, w)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) We used the Q-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) and G± invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) and Frobenius structure for (g, µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Hamiltonian in deformed HTQM is given by Hǫ = {Qǫ, G+} = H − {G+, [G−, µ2(ǫΨ, ·)]} = H − ǫ VΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='38) The last equality introduces VΨ, the linear in ǫ correction to the Hamiltonian in deformed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Cauchy problem for the evolution operator (∂t + H − ǫVΨ)e−tHǫ = 0, e−0·Hǫ = 1, t ∈ R+ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='39) can be solved in power series in ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The evolution operator for deformed theory is e−tHǫ = e−tH + � t 0 ds e(s−t)H ǫVΨ e−sH + O(ǫ2) = e−tH + � t 0 ds e(s−t)H{G+, [G−, µ2(ǫΨ, ·)]}e−sH + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) We can rewrite the integral in more symmetric form � t 0 ds esH−tH ǫVΨ e−sH = � t1>0, t2>0, t1+t2=t dt1dt2 e−t2H ǫVΨ e−t1H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) The symmetric formula is a common formula for perturbation theory in quantum mechanics 26 and has the following interpretation: The t1 and t2 describe the decomposition of the interval [0, t] into two sub-intervals: [0, t1] of length t1 and [t1, t] of length t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Each interval is equipped with the evolution operator e−t1H and e−t2H, while the splitting point carries the insertion of the deformation ǫVΨ of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The evolution operators in deformed theory form a semi-group, what follows from the analysis of the composition of two perturbative solutions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) with times t and s e−sHǫe−tHǫ = e−(t+s)H + � t 0 dt1 e−(t+s−t1)H ǫVΨ e−t1H + � s+t t dt1 e−(s+t−t1)H ǫVΨ e−t1H + O(ǫ2) = e−(t+s)Hǫ + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The evolution with respect to Hǫ preserves the pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, we can evaluate g(e−tHǫv, w) = g(v, e−tHw) + � t1+t2=t dt1dt2 g � e−t2H ǫVΨ e−t1Hv, w � = g(v, e−tHw) + � t1+t2=t dt1dt2 g � v, e−t1H ǫVΨ e−t2Hw � = g(v, e−tHǫw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='42) We used the evolution invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) and the invariance of the pairing with respect to the ǫVΨ-action, what follows from the Qǫ-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) and G±-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) and the following relation g(ǫVΨ v, w) = g({G+, Qǫ − Q}v, w) = g(G+(Qǫ − Q)v, w) + g((Qǫ − Q)G+v, w) = (−1)2|v|+2g(v, (Qǫ − Q)G+w) + (−1)2|v|+2g(v, G+(Qǫ − Q)w) = g(v, ǫVΨ w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='43) We use the semi-continuity of the kernel: The kernel of H can only decrease under the small deformation, hence dim ker Hǫ ≤ dim ker H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The possible decrease of the kernel is due to the obstruction for deformation of v0 ∈ ker H into v0ǫ ∈ ker Hǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Given a state v0 ∈ ker H we can deform it by a O(ǫ)-term to construct a state v0ǫ = v0 + ǫv1 + O(ǫ2) ∈ ker Hǫ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='44) 27 where the deformation ǫv1 is a solution to Hǫv0ǫ = −ǫVΨv0 + ǫHv1 + O(ǫ2) = O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='45) The solution for ǫv1 can be written in the form ǫv1 = � ∞ 0 dt e−tHG+G−µ2(ǫΨ, v0) = KG−µ2(ǫΨ, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) Indeed, we can check Hǫv1 = � ∞ 0 dt H e−tHG+G−µ2(ǫΨ, v0) = − � ∞ 0 d � e−tHG+G−µ2(ǫΨ, v0) � = −e−tHG+G−µ2(ǫΨ, v0) ��� t=∞ t=0 = G+G−µ2(ǫΨ, v0) − Π0G+G−µ2(ǫΨ, v0) = G+G−µ2(ǫΨ, v0) = ǫ VΨv0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='47) where we replaced the t → ∞-limit by a projector Π0 and used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) to eliminate the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The deformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) exists for all v0 ∈ ker H, hence dim ker Hǫ = dim ker H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) acquires most of its value near t = 0, since for large t, the exponential operator e−tH is close to the projector Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence we can write an approxi- mation for the integral in the form of the finite region integral ǫv1 = � ∞ 0 dt e−tHG+G−µ2(ǫΨ, v0) ≈ � T 0 dt e−tHG+G−µ2(ǫΨ, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='48) Let us choose a basis v0 k in ker H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Since ker H and ker Hǫ are of the same dimension then the corresponding deformed states v0ǫ k form a basis in ker Hǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We use the non-degeneracy of the pairing g to identify the vector space V and its dual V ∗ to express the projector Π0 = � v0 kv0 k ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='49) The projector Πǫ 0 on ker Hǫ is written using the deformed states v0ǫ k Πǫ 0 = � v0ǫ k v0ǫ k ∗ = Π0 + � ǫv1 k v0 k ∗ + ǫ � v0 k ǫv1 k ∗ + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='50) 28 The t → ∞ limit of the deformed evolution operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) is lim t→∞e−tHǫ = Π0 + lim t→∞ � t 0 ds esH−tH ǫVΨ e−sH + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='51) We decompose the integration interval [0, t] into three regions and evaluate the limit for the integration over each region: Right side of the interval: The t2 is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e t2 ∈ [0, T] for some finite T, while t1 ≈ t is very large and we can replace the corresponding exponential factor by the projector Π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) in this region evaluates into lim t→∞ � T 0 dt2 e−t2H ǫVΨ e−tHet2H = � T 0 dt2 e−t2H ǫVΨ · Π0 = � T 0 dt2 e−t2HǫVΨ � v0 kv0 k ∗ = � � T 0 dt2 e−t2HG+G−µ2(ǫΨ, v0 k)v0 k ∗ ≈ � � ∞ 0 dt2 e−t2HG+G−µ2(ǫΨ, v0 k)v0 k ∗ = � KG−µ2(ǫΨ, v0 k)v0 k ∗ = � ǫv1 k v0 k ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='52) Left side of the interval: The t1 is such that t1 < T, while t2 ≈ t is very large and we can replace the corresponding exponential factor by the projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) in this region evaluates into lim t→∞ � T 0 dt1 e−tHet1H ǫVΨ e−t1H = � T 0 dt1 Π0 ǫVΨ e−t1H = � T 0 dt1 � v0 kv0 k ∗ ǫVΨe−t1H = � v0 k �� T 0 dt1 e−t1HG+G−µ2(ǫΨ, v0 k) �∗ ≈ � v0 k �� ∞ 0 dt1 e−t1HG+G−µ2(ǫΨ, v0 k) �∗ = � v0 k � KG−µ2(ǫΨ, v0 k) �∗ = � v0 k ǫv1 k ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='53) Middle of the interval: For the middle region t1 ∈ [t/2 − T, t/2 + T], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' both t1 and t2 are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) in this region evaluates into lim t→∞ � t/2+T t/2−T dt2 e−t2H ǫVΨ e−t1H = � T −T dt1 Π0 ǫVΨ Π0 = � T −T dt1 0 = O(T) · 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='54) 29 The sum over three contributions lim t→∞ e−tHǫ = Π0 + � ǫv1 k v0 k ∗ + � v0 k ǫv1 k ∗ = Πǫ 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='55) matches with our expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='50) for projector on ker Hǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can use the (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='50)-representation to check the properties of projector in deformed theory Πǫ 0G±v = Π0G±v + � � ǫv1 k v0 k ∗ + v0 k ǫv1 k ∗� G±v = � ǫv1 k g(v0 k, G±v) + � v0 k g(ǫv1 k, G±v) = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='56) and G±Πǫ 0v = G±Π0v + G± � � ǫv1 k v0 k ∗ + v0 k ǫv1 k ∗� v = � (G± ǫv1 k) v0 k ∗ + � (G±v0 k) ǫv1 k ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='57) In equalities we used the G±-invariance of the pairing, G±v0 k = 0 and expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) for v1 k to evaluate G± ǫv1 k = G± � ∞ 0 dt e−tHG+G−µ2(ǫΨ, v0 k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58) ■ Definition: For any state v in HTQM (V, Q, G±) its leading order deformation by Q-, G−-closed state Ψ is vǫ = v + KG−µ2(ǫΨ, v) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) Proposition (preservation of closeness): If v is Q- and G−-closed, then the deformed state vǫ is Qǫ- and G−- closed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Qǫvǫ = G−vǫ = O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='60) Proof: The G−-closeness if fairly straightforward G−vǫ = G−v + G−KG−µ2(ǫΨ, v) = −G2 −Kµ2(ǫΨ, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='61) 30 The Qǫ-action on the deformed state Qǫvǫ = Qǫv + QKG−µ2(ǫΨ, v) + O(ǫ2) = Qv − G−µ2(ǫΨ, v) + QKG−µ2(ǫΨ, v) + O(ǫ2) = −G−µ2(ǫΨ, v) + (1 − Π0)G−µ2(ǫΨ, v) + O(ǫ2) = O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='62) In the equality we used the homotopy formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21), Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) for even state ǫΨ and the projector Π0 property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) from the HTQM definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ Remark: The formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) for deformation of a state v, describes a connection on Q+zG−- cohomology, fibered over the space of HTQM deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6 Diagrammatic representation of deformed theory We can express the propagator for deformed theory as expansion in ǫ in terms of propagators in original theory Kǫv = � ∞ 0 dt e−tHǫG+v = � ∞ 0 dt e−tHG+v + � ∞ 0 dt � t1+t2=t, t1, t2>0 dt1dt2 e−t1H{G+, [G−, µ2(ǫΨ, ·)]}e−t2HG+v + O(ǫ2) = Kv + � (R+)2 dt1dt2 e−t1HG+[G−, µ2(ǫΨ, ·)]e−t2HG+v + O(ǫ2) = Kv + K[G−, µ2(ǫΨ, ·)]Kv + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The propagator for deformed theory further simplifies if we use it in KG−-combination i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' KǫG−v = KG−v + KG−µ2(ǫΨ, KG−v) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='63) We can give a graphical representation for a propagator in deformed theory (denoted as thick solid line) in terms of the diagrams in the original theory = + ǫΨ In case G−µ2(ǫΨ, ǫΨ) = 0 the KG− in deformed theory can be written to all orders in 31 ǫ in the form KǫG−v = KG− + KG−µ2(ǫΨ, KG−v) + KG−µ2(ǫΨ, KG−µ2(ǫΨ, KG−v)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='64) with graphical representation = + ǫΨ + ǫΨ ǫΨ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The diagrammatic expression for the deformed state (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59), denoted as the thick dashed line, is the sum of two terms vǫ = v + ǫΨ v In case G−µ2(ǫΨ, ǫΨ) = 0 the higher order terms of the state deformation take the form vǫ = v + ǫΨ v + ǫΨ ǫΨ v + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The diagrammatic expression above describes the sum vǫ = v + KG−µ2(ǫΨ, v) + KG−µ2(ǫΨ, KG−µ2(ǫΨ, v)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='65) 5 Correlation functions in HTQM In this section we introduce the correlation functions for the states in HTQM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The correla- tion functions obey certain nice properties such as symmetry in all arguments, Q-invariance and recursion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: The tree-level connected n-point correlation function for states Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn in HTQM (V, Q, G±, g, µ2) is ⟨Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn⟩Q = � Γ,σ∈Sn A0 Γ(Ψσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψσ(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K) |Aut(Γ)| , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) 32 where |Aut(Γ)| is the symmetry factor for the tree Γ and K is a homotopy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The sum- mation is taken over all distinct 3-valent connected trees Γ with n leaves and over possible assignment of states Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn on leaves of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The sum over permutations in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) makes the correlation function ⟨Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn⟩Q manifestly symmetric in all arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The sum over trees, weighted with the symmetry factors, in correlation func- tions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) is a signature of their relation to amplitudes in certain Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We conjecture that the QFT is a BCOV-like theory [21], see also [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We are working on further investigation of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The HTQM states Ψa, relevant for the mirror symmetry, are even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' |Ψa| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence, for simplicity, we will assume that |Ψa| = 0 for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such assump- tion will drastically reduce the complexity of the sign factors in various expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 3-point function There is a single tree with 3-valent vertices and three leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence, the 3-point correlation function is just a sum over 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' possible permutations of states Ψ1, Ψ2, Ψ3 on the leaves of a tree below Ψ1 Ψ2 Ψ3 The amplitudes for each permutation are identical, hence we have a sum of 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' = 6 identi- cal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The symmetry factor of the graph above is |Aut Γ3| = 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' = 6, so the 3-point correlation function simplifies to ⟨Ψ1, Ψ2, Ψ3⟩Q = 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' µ0 3(Ψ1, Ψ2, Ψ3) = µ0 3(Ψ1, Ψ2, Ψ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) 33 The 3-point function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) is invariant under the shift Ψ3 → Ψ3 + Qχ, given that QΨ1 = QΨ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, we can evaluate the difference ⟨Ψ1, Ψ2, Qχ⟩Q = g(Qχ, µ2(Ψ1, Ψ2)) = g(χ, Qµ2(Ψ1, Ψ2)) = g(χ, µ2(QΨ1, Ψ2)) + g(χ, µ2(Ψ1, QΨ2)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) We used the Q-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) and Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) to simplify the expres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 4-point function There is a single tree with 3-valent vertices and four leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence the 4-point correlation function is just a sum over 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' possible distributions of states Ψ1, Ψ2, Ψ3, Ψ4 on the leaves of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The 24 terms in a sum of three groups of 8, with equal amplitudes in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The 3 (independent) amplitudes are depicted below and are commonly referred to as the s−, t−, u−diagrams Ψ4 Ψ1 Ψ3 Ψ2 Ψ4 Ψ1 Ψ2 Ψ3 Ψ3 Ψ1 Ψ2 Ψ4 The 4-point correlation function is the sum of three contributions ⟨Ψ1, Ψ2, Ψ3, Ψ4⟩Q = AΓ4(Ψ1, Ψ2, Ψ3, Ψ4) + AΓ4(Ψ1, Ψ3, Ψ2, Ψ4) + AΓ4(Ψ1, Ψ4, Ψ2, Ψ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) The number 8 equals to the symmetry factor |Aut Γ4| = 8 of a tree and is constructed as 2·2·2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The amplitudes on graphs, related by symmetry, are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, the amplitude for the first tree AΓ4(Ψ1, Ψ2, Ψ3, Ψ4) = g(µ2(Ψ3, Ψ4), 2πKG−µ2(Ψ1, Ψ2)) = g(µ2(Ψ3, Ψ4), 2πKG−µ2(Ψ2, Ψ1)) = g(µ2(Ψ4, Ψ3), 2πKG−µ2(Ψ1, Ψ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) is invariant under the exchange of two pairs of states Ψ1 ↔ Ψ2 and Ψ3 ↔ Ψ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, the Ψa are even states and µ2 is graded-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The last factor of 2 is related to the reflection 34 of the tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' exchange of two pairs Ψ1, Ψ2 and Ψ3, Ψ4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' g(µ2(Ψ4, Ψ3), KG−µ2(Ψ1, Ψ2)) = g(KG−µ2(Ψ4, Ψ3), µ2(Ψ1, Ψ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) The invariance of the amplitude follows from the KG−-flip relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The 4-point amplitude is invariant under the shift Ψ4 → Ψ4 + Qχ, given that QΨa = G−Ψa = G−χ = 0, a = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) Indeed, we can evaluate 2 2π⟨Ψ1, Ψ2, Ψ3, Qχ⟩Q = � σ∈S3 g(Qχ, µ2(Ψσ(3), KG−µ2(Ψσ(1), Ψσ(2)))) = � σ∈S3 g(χ, Qµ2(Ψσ(3), KG−µ2(Ψσ(1), Ψσ(2)))) = � σ∈S3 g(χ, µ2(Ψσ(3), {Q, K}G−µ2(Ψσ(1), Ψσ(2)))) = � σ∈S3 g(χ, µ2(Ψσ(3), G−µ2(Ψσ(1), Ψσ(2)))) = 2g(χ, G−µ2(µ2(Ψ1, Ψ2), Ψ3)) = −2g(G−χ, µ2(µ2(Ψ1, Ψ2), Ψ3)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) In rewriting the equality we used the Q-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11), Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7), the homotopy formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21), the projector property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4), the 7-term relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) for even G−-closed states in the form G−µ2(µ2(Ψ1, Ψ2), Ψ3) = µ2(G−µ2(Ψ1, Ψ2), Ψ3) + µ2(Ψ2, G−µ2(Ψ1, Ψ3)) + µ2(Ψ1, G−µ2(Ψ2, Ψ3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) The last equality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) uses the G±-invariance of pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) and completes the proof of invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Generating function We introduce formal parameters tk, such that t2 k = 0 and combine Ψk into even state Ψ = Ψ(t) = � tkΨk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) 35 The n-point correlation function of Ψ has t-expansion with coefficients being the n-point correlation functions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ � �� � n ⟩Q = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' · ⟨t1Ψ1, t2Ψ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., tnΨn⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) Definition: The n-point correlation function of Ψ can be organized into generating function F0(Ψ, K) = ∞ � k=3 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ � �� � k ⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) The generating function equals to the the sum over connected 3-valent trees F0(Ψ, K) = � Γ AΓ(Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K) |Aut(Γ)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) The diagrammatic expression for generating function is Ψ Ψ Ψ + 1 8 1 6 Ψ Ψ Ψ Ψ + 1 8 Ψ Ψ Ψ Ψ Ψ + 1 8 Ψ Ψ Ψ Ψ Ψ Ψ + 1 48 Ψ Ψ Ψ Ψ Ψ Ψ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 Invariance theorem In our analysis for the 3- and 4-point functions we observed the invariance under the shift by a Q-exact term, given that the states were Q- and G-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The invariance generalizes to the n-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Theorem (invariance): Given the states Ψ1, Ψ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn, χ of HTQM on trees (V, Q, G±, µ2, g) such that QΨα = G−Ψα = G−χ = 0, α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) 36 the (n + 1)-point correlation function vanishes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn, Qχ⟩Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) Proof: We use the generating function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) to prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The change in generating function can be expressed δF0(Ψ) = F0(Ψ + δΨ) − F0(Ψ) = g(δΨ, ˜γ) + O(δΨ)2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) via the state ˜γ, defined as a sum over 3-valent rooted trees, weighted with symmetry factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The first several trees of the sum ˜γ are presented below 1 2 Ψ Ψ + 1 2 Ψ Ψ Ψ + 1 2 Ψ Ψ Ψ Ψ + 1 8 Ψ Ψ Ψ Ψ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The diagrammatic sum for ˜γ takes the form ˜γ = 1 2µ2(Ψ, Ψ) + 1 2µ2(Ψ, 2πKG−µ2(Ψ, Ψ)) + 1 2µ2(Ψ, 2πKG−µ2(Ψ, 2πKG−µ2(Ψ, Ψ))) + 1 8µ2(2πKG−µ2(Ψ, Ψ), 2πKG−µ2(Ψ, Ψ)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) Note that ˜γ is an even state since Ψ is even and KG− is even operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We also introduce a related even state γ = Ψ + 2πKG−˜γ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) used in [10], which is G−-closed G−γ = G−Ψ − 2πKG2 −˜γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) The ˜γ and γ obey the “root-cutting” relation ˜γ = 1 2µ2(γ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) 37 The representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) for ˜γ is very convenient for deriving the recursive formula Q˜γ = 1 2Qµ2(γ, γ) = µ2(γ, Qγ) = µ2(γ, QΨ + 2πQKG−˜γ) = µ2(γ, 2πG−˜γ) + µ2(γ, 2πKG−Q˜γ) = πµ2(γ, G−µ2(γ, γ)) + µ2(γ, 2πKG−Q˜γ) = π 3 G−µ2(γ, µ2(γ, γ)) + µ2(γ, 2πKG−Q˜γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) In our derivation we used Leibniz rule (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7), homotopy formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21), the projector property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) and the 7-term relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) for even, G−-closed state γ in the form G−µ2(γ, µ2(γ, γ)) = 3µ2(γ, G−µ2(γ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) We can replace the Q˜γ in the last expression and use G2 − = 0 to get Q˜γ = π 3 G−µ2(γ, µ2(γ, γ)) + µ2(γ, 2πKG−µ2(γ, 2πKG−Q˜γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) We can iterate the process for of the Q˜γ-replacement to arrive into Q˜γ = π 3G−µ2(γ, µ2(γ, γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) The invariance of the correlation functions follows from F0(Ψ + Qχ) − F0(Ψ) = g(Qχ, ˜γ) = g(χ, Q˜γ) = π 3 g(χ, G−µ2(γ, µ2(γ, γ))) = −π 3 g(G−χ, µ2(γ, µ2(γ, γ))) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) We used the Q-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11), the G±-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) and the assumption of the theorem that G−χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 Recursion relation for correlation functions Let us consider the HTQM (V, Q, G±, µ2, g) and four Q-, G−-closed states Ψa, a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The 4-point correlation function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) of such states can be written in the following form ⟨Ψ1, Ψ2,Ψ3, Ψ4⟩Q = µ0 3 (2πKG−µ2(Ψ4, Ψ1), Ψ2, Ψ3) + µ0 3 (Ψ1, 2πKG−µ2(Ψ4, Ψ2), Ψ3) + µ0 3 (Ψ1, Ψ2, 2πKG−µ2(Ψ4, Ψ3)) = d dǫ ��� ǫ=0µ0 3(Ψǫ 1, Ψǫ 2, Ψǫ 3) = d dǫ ��� ǫ=0⟨Ψǫ 1, Ψǫ 2, Ψǫ 3⟩Qǫ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) 38 where we used the leading order deformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) of states Ψ1, Ψ2, Ψ3 by the state Ψ4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ψǫ a = Ψa + 2πKG−µ2(ǫΨ4, Ψa) + O(ǫ2), a = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) We can describe the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) using diagrammatic representation Ψǫ 1 Ψǫ 2 Ψǫ 3 d dǫ|ǫ=0 = Ψ1 Ψ2 Ψ3 Ψ4 + Ψ1 Ψ2 Ψ3 Ψ4 + Ψ1 Ψ2 Ψ3 Ψ4 Theorem (recursion relation): The (n + 1)-point correlation function for the Q-, G−- closed states Ψ0, Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψn in HTQM on trees (V, Q, G±, µ2, g) can be expressed as a derivative of n-point correlation function in HTQM, deformed by the state Ψ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn, Ψ0⟩Q = d dǫ ��� ǫ=0⟨Ψǫ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψǫ n⟩Qǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) The deformed HTQM (V, Qǫ, G±, µ2, g) has differential Qǫ = Q − 2π[G−, µ2(ǫΨ0, ·)] + O(ǫ2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) while deformed states are Ψǫ a = Ψa + 2πKG−µ2(ǫΨ0, Ψa) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) Proof: The key idea in our proof is to use the generating function for the amplitudes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The generating function in deformed theory is F0(Ψǫ, Kǫ) = ∞ � k=3 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψǫ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψǫ � �� � k ⟩Qǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) Let us recall the formula for the change of generating function under variation of the external state Ψ F0(Ψ + ǫδΨ, K) = F0(Ψ, K) + g(˜γ, ǫδΨ) + O(ǫ2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) 39 where the state ˜γ is a sum over rooted trees (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Similarly we can derive the change of generating function under the change of propagator K F0(Ψ, K + ǫδK) = F0(Ψ, K) + πg(˜γ, ǫδK˜γ) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) The full change of the generating function is presented on the picture below Ψ Ψ Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' δ = δΨ g Ψ Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' + g δK Ψ Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ψ Ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We use the state deformation formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) for δΨ and the propagator in deformed theory (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='63) for δK in terms of the state Ψ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ǫδKG− = KǫG− − KG− = 2πKG−µ2(ǫΨ0, KG−·), ǫδΨ = Ψǫ − Ψ = 2πKG−µ2(ǫΨ0, Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) Using the KG−-flip (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) and relations for the sum over rooted trees (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) we can rewrite the generating function in deformed theory F0(Ψǫ, Kǫ) − F0(Ψ, K) = g(˜γ, ǫδΨ) + πg(˜γ, ǫδK˜γ) + O(ǫ2) = g(ǫΨ0, µ2(2πKG−˜γ, Ψ)) + πg(ǫΨ0, µ2(2πKG−˜γ, KG−˜γ)) + O(ǫ2) = g(ǫΨ0, ˜γ) − 1 2g(ǫΨ0, µ2(Ψ, Ψ)) + O(ǫ2) = F0(Ψ + ǫΨ0, K) − F0(Ψ, K) − 1 2g(ǫΨ0, µ2(Ψ, Ψ)) + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) The derivative of the relation becomes ∞ � k=3 d dǫ ��� ǫ=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψǫ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψǫ � �� � k ⟩Qǫ = d dǫ ��� ǫ=0F0(Ψ + ǫΨ0, K) − 1 2 g(Ψ0, µ2(Ψ, Ψ)) = 3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψ, Ψ, Ψ0⟩Q + ∞ � k=4 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='k⟨Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ � �� � k−1 , Ψ0⟩Q − 1 2⟨Ψ, Ψ, Ψ0⟩Q = ∞ � k=3 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ � �� � k , Ψ0⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) 40 The equality holds at each order in Ψ so d dǫ ��� ǫ=0⟨Ψǫ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψǫ � �� � k ⟩Qǫ = ⟨Ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψ � �� � k , Ψ0⟩Q (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) what completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 6 Mirror for HTQM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 A-model Tropical Gromov-Witten theory on toric manifold X of complex dimension N defines the A-type HTQM on trees, also denoted as the A-model in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In this section we will describe the HTQM data (V, Q, G±, g, µ2) for A-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Let us consider pair |ω, ⃗m⟩, of tropical form ω on C∗N, and N-dimensional integer-valued vector ⃗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing on |ω, ⃗m⟩ is the integration of corresponding form forms g(|ω1, ⃗m1⟩, |ω2, ⃗m2⟩) = δ⃗m1+⃗m2,⃗0 � C∗N ω1 ∧ ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) The vector space VA is the space of tropical differential forms on C∗N, equipped with the integer vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' VA = Ωtrop(C∗N) ⊗ R⟨⃗m | ⃗m ∈ ZN⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) The Z2-grading of the state |ω, ⃗m⟩ is the grading of the form ω, a degree of the differential form mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The differential Q is the de Rham operator on C∗N Q|ω, ⃗m⟩ = |dω, ⃗m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) The G± on states is a contraction with the constant radial (angular) vector field ⃗m G+|ω, ⃗m⟩ = |ιR ⃗mω, ⃗m⟩ = |ιmi∂riω, ⃗m⟩, G−|ω, ⃗m⟩ = |ιΦ ⃗mω, ⃗m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) The multiplication µ2 : V ⊗V → V is the wedge product on differential forms supplemented with addition of corresponding vectors µ2(|ω1, ⃗m1⟩, |ω2, ⃗m2⟩) = |ω1 ∧ ω2, ⃗m1 + ⃗m2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) 41 The pair (Q, µ2) is essentially an external derivative and the wedge product, hence obeys the DGA properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pair (G−, µ2) obeys the 7-term relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The Hamiltonian H = {Q, G+} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) is the Lie derivative along the constant radial vector field ⃗m H|ω, ⃗m⟩ = {Q, G+}|ω, ⃗m⟩ = |{d, ιR ⃗m}ω, ⃗m⟩ = |LR ⃗mω, ⃗m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) The evolution operator e−tH, defined as solution to (∂t + H)e−tH = (∂t + LR ⃗m)e−tH = 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) is a 1-parameter family of diffeomorphisms Φt ⃗m : ri �→ ri − mit e−tH|ω, ⃗m⟩ = |(Φt ⃗m)∗ω, ⃗m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) The composition property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) naturally holds for diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Since the vector fields are constant vector fields the corresponding flows do not develop any singularities, hence composition is valid for all values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 Correlation functions in A-model In our paper [8] we showed that the tropical GW invariant of genus-0 and degree-β on toric space X is the sum of the A-type HTQM amplitudes ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', γn⟩X β = � Γ AΓ(Ψγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψγn, Ψ⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψ⃗bB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' K) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) with two types of states: Evaluation states Ψγk = |γk,⃗0⟩, constructed from the tropical forms γk on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The space Ωtrop(X) of tropical forms on X is a subspace of tropical forms on C∗N with good behaviour at compactifying divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Divisor states Ψ⃗ba = |1,⃗ba⟩, where ⃗ba are primitive normal vectors for compactifying divisors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Using the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) we can replace the sum over the amplitudes by the correlation function and formulate the HTQM representation for the tropical GW invariants in the form: 42 Theorem (HTQM representation of tropical GW): For toric space X, given in terms of boundary divisors BX, and tropical cycles γk, the tropical GW invariant matches with the HTQM correlation function i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', γn⟩X β = 1 d1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. · dB!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='⟨Ψγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψγn, Ψ⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψ⃗b1 � �� � d1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψ⃗bB, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψ⃗bB � �� � dB ⟩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) Here Ψγa are the evaluation states for tropical cycles γa, Ψ⃗b are divisor states for boundary divisors from BX of dimension B = dim BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The degree of the map β ∈ H1,1(X) determines the number da of divisor states Ψ⃗ba of a given type via the corresponding tropical intersection number of β and boundary divisor with normal vector ⃗ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The GW theory is defined on classes of cycles Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The change of a cycle within the same class leads to the shift of an evaluation observable (Poincare-dual form) γ by an exact form γ → γ + dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) The shift of evaluation observable γ changes the corresponding A-model state Ψγ by a Q- exact term Ψγ → Ψγ+dλ = Ψγ + QΨλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) Furthermore, since the states Ψγ and Ψλ carry trivial integer vector, they are G−-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' G−Ψλ = G−|λ,⃗0⟩ = 0, G−Ψγ = G−|γ,⃗0⟩ = 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) The forms γa are closed forms so is the corresponding HTQM states Ψγ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence we can use the invariance theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) to verify that the tropical GW invariants in HTQM represen- tation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) are defined on cohomology classes of γa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Dual variables It is convenient to introduce angular variables Yj ∈ S1, dual to the integer vector components mi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We introduce a Fourier transform of a state � ⃗m∈ZN c⃗m|ω⃗m, ⃗m⟩ �→ Ψ = � ⃗m∈ZN ei⟨⃗m,⃗Y ⟩c⃗m|ω⃗m, ⃗m⟩ ∈ VB = Ωtrop(C∗N) ⊗ C∞(TN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) 43 For convenience we describe the differential forms on X using Grassmann variables: drj = ψj R, dφj = ψj Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) The differentials on mirror states in new notations become first and second order differential operators QΨ = � ⃗m∈ZN ei⟨⃗m,⃗Y ⟩c⃗m|dω⃗m, ⃗m⟩ = dRΨ = ψk R ∂ ∂rk Ψ, G−Ψ = � ⃗m∈ZN ei⟨⃗m,⃗Y ⟩c⃗m|ιΦ ⃗mω, ⃗m⟩ = −i ∂ ∂Yk ∂ ∂ψk Φ Ψ, G+Ψ = � ⃗m∈ZN ei⟨⃗m,⃗Y ⟩c⃗m|ιR ⃗mω, ⃗m⟩ = −i ∂ ∂Yk ∂ ∂ψk R Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) The multiplication µ2, on forms becomes multiplication of functions on superspace with coordinates r, Y, ψR, ψΦ µ2(Ψ1, Ψ2) = Ψ1 · Ψ2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) while the pairing is the integration over superspace g(Ψ1, Ψ2) = � dµ Ψ1Ψ2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) where Berezin integration measure for dimC X = N is dµ = dNr dNY dNψΦdNψR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) The integrating region (for Grassmann-even variables) is the N-dimensional torus (S1)N for Y -variables and Euclidean space RN for r-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The differential operator representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) of the HTQM data (V, Q, G±, µ2, g) allows for an easy check of HTQM definitions from sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular the 7-term relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) for G− is a property of the second order differential operator in repre- sentation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The divisor states |1,⃗b⟩ become exponential functions of Y Ψ⃗b(Y ) = ei⟨⃗b,⃗Y ⟩ = eibkYk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) 44 For the tropical form γ = γi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.ikj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.jl(r) dφi1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. ∧ dφik ∧ drj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. ∧ drjl (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) the corresponding evaluation state is Ψγ = γi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.ikj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.jl(r) ψi1 Φ · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. · ψik Φ · ψj1 R · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='. · ψjl R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) Example: The evaluation state for U(1)-invariant Poincare dual of the point on P1 is γ = 1 2πδ(r − r0)dφdr �→ Ψγ = |γ, 0⟩ = 1 2πδ(r − r0)ψΦψR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 B-model In our work [8] we showed that the deformation of A-model, the HTQM (VA, Q, G±, µ2, g), by divisor states Ψ⃗b for toric X is also an HTQM (VB, QX, G±, µ2, g), which we will denote as the B-type HTQM or B-model for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The two-dimensional version of the deformation by compactifying divisors of toric space was discussed in A-I-B mirror paper [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' However, the deformation of observ- ables was not discussed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Divisor states obey G−µ2(Ψ⃗b1, Ψ⃗b2) = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) hence the deformation of the differential Q, which we discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 holds beyond the linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The deformation of the A-model differential by all divisors BX of X is QX = Q − 2π � ⃗b∈BX [G−, µ2(q⃗bΨ⃗b, ·)] = ψj R ∂ ∂rj + 2πi � ⃗b∈BX q⃗b ∂Ψ⃗b ∂Yj ∂ ∂ψj Φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) The same differential can be written as QX = Q + 2πi∂WX(Y ) ∂Yj ∂ ∂ψj Φ = QWX, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) 45 using the mirror superpotential WX(Y ) = � ⃗b∈BX q⃗b Ψ⃗b = � ⃗b∈BX q⃗b ei⟨⃗b,⃗Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) We can absorb some q⃗b by the redefinition of Yj to obtain more familiar (at least for the PN case) form of the superpotential with fewer parameters q⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: The exponential mirror superpotentials for toric spaces were derived by Given- tal [5] and by Hori and Vafa [17] using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Definition: For toric space X the deformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) of an A-model evaluation state Ψγ by the divisor states Ψ⃗b is a mirror state ΨX γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can write the mirror state using the A-model notations in the form ΨX γ = Ψγ + 2πKG−µ2(WX, Ψγ) + (2π)2KG−µ2(WX, KG−µ2(WX, Ψγ)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) For an A-model state Ψγ corresponding to the tropical form γ ∈ Ωk,l(X) the sum (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) terminates after min(k, l) + 1 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, the action of KG− lowers the degree of the form by (1, 1), hence it can be applied to (k, l)-form Ψγ at most min(k, l) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' By preservation of closeness proposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='60), the mirror state ΨX γ is QX- and G−-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' QXΨX γ = G−ΨX γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) Example: For X = P1 the space of boundary normal vectors is BP1 = {+1, −1}, hence the mirror superpotential equals WP1(Y ) = � b∈BP1 qb eibY = q+eiY + q−e−iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) We can absorb the q+ by constant shift of Y to arrive into more familiar form of the super- potential WP1 = eiY + qe−iY (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) with q = q+q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model differential is QP1 = ψR ∂ ∂r + 2πi∂WP1 ∂Y ∂ ∂ψΦ = ψR ∂ ∂r − 2π(q+eiY − q−e−iY ) ∂ ∂ψΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) 46 The mirror state ΨP1 γ for the A-model evaluation observable γ ∈ H∗ dR(P1) contains three terms ΨP1 γ = Ψγ + 2πKG−µ2(WP1, Ψγ) = Ψγ + 2πq+ � ∞ 0 dt e−tH G+G−(eiY Ψγ) + 2πq− � ∞ 0 dt e−tH G+G−(e−iY Ψγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) Indeed, the top form on P1 has degree (1, 1), hence we can apply the deformation only once for each of two boundary divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The mirror state for the constant form γ = 1 ∈ H0 dR(P1) has a trivial deformation ΨP1 1 = Ψ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) Consider the Poincare dual of the point φ = φ0 and r = r0 in tropical coordinates on P1 P0 = δ(φ − φ0)δ(r − r0) dφdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) The U(1)-averaging over φ0 of the form P0 is P = 1 2πδ(r − r0) dφdr ∈ H1,1 dR(P) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) The corresponding A-model state equals ΨP = 1 2πδ(r − r0)ψΦψR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='38) The corresponding mirror state is ΨP1 P = 1 2πδ(r − r0)ψΦψR + q+eiY ∞ � 0 dt δ(r − r0 − t) + q−e−iY ∞ � 0 dt δ(r − r0 + t) = 1 2πδ(r − r0)ψΦψR + q+eiY Θ(r − r0) + q−e−iY Θ(r0 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='39) In our work [8] we proved the mirror symmetry for the A-type HTQM (VA, Q, G±, µ2, g) and the B-type HTQM (VB, QX, G±, µ2, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The proof was essentially a summation over the divisor states Ψ⃗b for all divisors of the toric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The HTQM mirror implies the following statement for the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 47 Theorem (tropical mirror for HTQMs): On toric space X, with boundary divisors ⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=',⃗bB the sum over divisor states in the A-model’s correlation function of evaluation states Ψγa equals to the correlation function of the corresponding mirror states ΨX γa in B-model with mirror superpotential WX i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ∞ � k=0 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψγn, WX, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., WX � �� � k ⟩Q = ⟨ΨX γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨX γn⟩QWX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) Note that the q-dependence in the A-model is due to the divisor states in WX, while in B-model both evaluation states and differential QX have nontrivial q-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 7 Localization of mirror states In the A-model formulation evaluation states and divisor states look very different and does not have any simple relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In case X = P1 the evaluation state for the point observable at r = r0 and divisor states are ΨP = 1 2πδ(r − r0)ψΦψR, Ψ+ = eiY , Ψ− = e−iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 we constructed the P1-mirror state for the evaluation state of a point observable ΨP1 P = 1 2πδ(r − r0)ψΦψR + q+eiY Θ(r − r0) + q−e−iY Θ(r0 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) The expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) for mirror state turns into a pure divisor state Ψ± in the limit r0 → ±∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' lim r0→−∞ΨP1 P = q+eiY = q+Ψ+ lim r0→+∞ΨP1 P = q−e−iY = q−Ψ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) There is a natural geometric interpretation of this relation: The point at finite position r = r0 becomes a compactifying divisor (a point at infinity in case of P1) when we move r0 to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The relation between mirror states and divisor states plays a key role for the localization of the B-model correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Note, that the limit of the state, often referred to as the point-wise limit, in general, may not commute with the amplitude/correlation function evaluation for the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We will discuss this potential problem carefully in the next 48 section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 Mirror states vs divisor states The pair of A-model states, corresponding to the Poincare duals of points r0 and r1 are related by a Q-exact term Ψ1 − Ψ0 = 1 2πδ(r − r1)ψΦψR − 1 2πδ(r − r0)ψΦψR = Q � − 1 2πΘ(r − r1)ψΦ + 1 2πΘ(r − r0)ψΦ � = Qχ01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) The tropical form χ01 = 1 2π [Θ(r − r0) − Θ(r − r1)] ψΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) is a tropical form on P1 since the difference of two Θ-functions has finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can take a limit r1 → −∞ for χ01 to define χ+ = lim r1→−∞ χ01 = 1 2π(Θ(r − r0) − 1)ψΦ = − 1 2πΘ(r0 − r)ψΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) The tropical form χ+ can be used to turn the mirror state into the boundary divisor state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' q+Ψ+ = ΨP1 P + QP1χ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) The relation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) is equivalent to the statement that the mirror state ΨP1 P and the holomor- phic function Ψ+ = eiY represent the same cohomology as a forms in VB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ΨP1 P = q+Ψ+ ∈ H∗(QP1, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) Note that the tropical form χ+ is G−-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' G−χ+ = G− � − 1 2πΘ(r0 − r)ψΦ � = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) hence we can formulate a stronger equality in cohomology ΨP1 P = q+Ψ+ ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) 49 We can take r1 → +∞ limit for (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) to define a different tropical form χ− = 1 2πΘ(r − r0)ψΦ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) such that q−Ψ− = ΨP1 P + QP1χ− (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) Similarly to the previous case G−χ− = G− � 1 2πΘ(r − r0)ψΦ � = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) hence ΨP1 P = q−Ψ− ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) We can choose more general QP1-exact term, so that the mirror state will become a holo- morphic function, which is not one of the compactifying divisor states for P1 ˆΨP = ΨP1 P + QP1 ˆχ = ΨP1 P + QP1 � 1 2πΘ(r − r0)ψΦ + 1 2πqe−2iY ψΦ � = q2e−3iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) The key difference from the χ± is that the tropical form ˆχ is not G−-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' G− ˆχ = � 1 2πΘ(r − r0)ψΦ + q 2π e−2iY ψΦ � = − q π e−2iY ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) hence ˆΨP and ΨP belong to different classes in (QP1 + zG−)-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, a simple computation shows that q2e−3iY = eiY + qz π e−2iY ∈ H∗(QP1 + zG−, VB ⊗ R[[z]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) In the remaining part of this section we will show that all mirror states admit holo- morphic function representatives in H∗(QW + zG−) and discuss some properties of these representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 Spectral sequence for QW-cohomology A mirror state describes a class in H∗(QW, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The QW-differential QW = ψj R ∂ ∂rj + 2πi∂W ∂Yj ∂ ∂ψj Φ = dR + 2πi QW (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) is a sum of two (graded-) commuting differentials, the radial de Rham differential dR = ψj R ∂ ∂rj , d2 R = 0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) and the LGS differential (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) QW = ∂W ∂Yk ∂ ∂ψk Φ , Q2 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) Hence we can use a spectral sequence to evaluate the cohomology of the QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The spectral sequence converges at the second step H∗(QW, VB) = H∗(QW, H∗(dR, VB)), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) since the only cohomology of the radial de Rham operator on RN are constant (r-independent) forms of degree 0 in ψR and is isomorphic to the LSG vector space (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The QW is dif- ferential in LGS theory, hence we can express the cohomology via the Jacobi ring (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) for superpotential W H∗(QW, VB) = JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) An isomorphism (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) means that for every QW-closed B-model state we can find a holo- morphic function from the same QW-cohomology class on VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 Pairing and localization of states Definition: The holomorphic germ for a B-model state Ψ is an evaluation of the state Ψ at r = ψ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Φ(Y ) = Ψ(r, Y, ψΦ, ψR) ��� ψ=r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) 51 Definition: For states Ψ1, Ψ2 in B-model with superpotential W we introduce a pairing gΛ W(Ψ1, Ψ2) = g � Ψ1, eΛQW (L)Ψ2 � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) where g is the B-model pairing, Λ is a real parameter and L is a localization function L = N � k=1 rkψk Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) Note: The exponent in the pairing evaluates into QW(L) = N � k=1 ψk Φψk R + 2πi N � k=1 rk ∂W ∂Yk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) Hence the QW(L) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) is an oscillating function in parity-even variables r and Y what makes the radial integral converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing gΛ W matches with the B-model pairing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) for Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing gΛ W is QW-invariant, indeed gΛ W(QWΨ1, Ψ2) = g � QWΨ1, eΛQW (L)Ψ2 � = −(−1)|Ψ1|g � Ψ1, QWeΛQW (L)Ψ2 � = −(−1)|Ψ1|gΛ W(Ψ1, QWΨ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) We used the QW-invariance of the B-model pairing and an identity QWeQW (L) = QWeQW L+LQW = eQW (L)QW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) The QW-invariance of the pairing implies that it is well-defined on H∗(QW, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Lemma: On QW-closed states the pairing gΛ W is independent of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proof: The derivative of the pairing evaluates into d dΛgΛ W(Ψ1, Ψ2) = d dΛg � Ψ1, eΛQW (L)Ψ2 � = g � Ψ1, (QWL + LQW)eΛQW (L)Ψ2 � = g � Ψ1, LeΛQW (L)QWΨ2 � − (−1)|Ψ1|g � QWΨ1, eΛQW (L)Ψ2 � = 0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) 52 We used the QW invariance of the B-model pairing and QW-closeness of both Ψ1 and Ψ2 and an identity (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ The Λ-independence means that the pairing gΛ W matches with the B-model pairing g on H∗(QW, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model pairing g is non-degenerate on VB and QW-invariant, hence g is non-degenerate on QW-cohomology, so is the gΛ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition: The QW-closed state Ψ and its holomorphic germ Φ are in the same QW- cohomology class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ψ = Φ ∈ H∗(QW, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) Proof: Since QW(L) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) is an oscillating function in parity-even variables r and Y , then the integral will localize near critical points of QW(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Moreover, the localization exponent is scaled by Λ, so we can choose large Λ to eliminate the subleading corrections to the saddle point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The critical points of QW(L) are determined from ∂ ∂rk QW(L) = 2πi∂W ∂Yk = 0, ∂ ∂Ym QW(L) = 2πi N � k=1 rk ∂2W ∂Yk∂Ym = 0, ∂ ∂ψk Φ QW(L) = ψk R = 0, ∂ ∂ψk R QW(L) = −ψk Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) Under the assumption that the critical points of W are isolated, the rank of ∂2W ∂Yl∂Ym is maximal, hence the integral in the limit Λ → ∞ localizes to the critical points of W and origin in all other variables rk = ψk R = ψk Φ = 0, Y = Y0, dW(Y0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) The ratio of determinants for the integration around the critical point is det �∂2QW(L) ∂ψk Φ∂ψl R � (2π)N det � ∂2QW (L) ∂Yk∂rl � = (2π)NΛN (2πΛ)N det � ∂2W ∂Yk∂Yl �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) The leading order saddle point formula for the pairing integral localizes the gΛ W-pairing to gΛ W(Ψ1, Ψ2) = � dW =0 1 det � ∂2W ∂Yk∂Yl � Ψ1 · Ψ2 ��� r=ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) Note that the Λ-dependence drops out from the leading saddle point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We observe that the pairing gΛ W for B-model states localizes to the corresponding holomorphic germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In 53 particular, we have an equality 0 = gΛ W(Ψ, Ψ′) − gΛ W(Φ, Ψ′) = gΛ W(Ψ − Φ, Ψ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) The equality holds for all QW-closed states Ψ′ and pairing gΛ W is non-degenerate on H∗(QW, VB), hence Ψ and Φ represent the same class in QW-cohomology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ψ = Φ ∈ H∗(QW, VB), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) what completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 Higher pairings in B-model Definition: For a B-model with superpotential W we can introduce a C[[z]]-valued pairing on B-model states KW(Ψ1, Ψ2) = � dµ Ψ1 · eΛ{QW +zG−,L}Ψ2 = g � Ψ1, eΛ{QW +zG−,L}Ψ2 � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) with the integration measure (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24), real parameter Λ and localization function L = N � k=1 rkψk Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='38) The pairing KW matches with the B-model pairing (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) for Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing KW is a pairing between H∗(QW − zG−) and H∗(QW + zG−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Indeed, the conjugation formula KW((QW − zG−)Ψ1, Ψ2) = g � (QW − zG−)Ψ1, eΛ{QW +zG−,L}Ψ2 � = −(−1)|Ψ1|g � Ψ1, (QW + zG−)eΛ{QW +zG−,L}Ψ2 � = −(−1)|Ψ1|g � Ψ1, eΛ{QW +zG−,L}(QW + zG−)Ψ2 � = −(−1)|Ψ1|KW(Ψ1, (QW + zG−)Ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='39) Lemma: On H∗(QW − zG−) ⊗ H∗(QW + zG−) the pairing (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) is independent on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 54 Proof: The derivative of the pairing evaluates into d dΛKW(Ψ1, Ψ2) = d dΛg(Ψ1, eΛ{QW +zG−,L}Ψ2) = g(Ψ1, {QW + zG−, L}eΛ{QW +zG−,L}Ψ2) = g(Ψ1, (QW + zG−)LeΛ{QW +zG−,L}Ψ2) + g(Ψ1, L(QW + zG−)eΛ{QW +zG−,L}Ψ2) = −(−1)|Ψ1|g((QW − zG−)Ψ1, LeΛ{QW +zG−,L}Ψ2) + g(Ψ1, LeΛ{QW +zG−,L}(QW + zG−)Ψ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In the relation we used QW-invariance of the B-model pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11), G−-invariance of the B-model pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) an equality (QW + zG−)eΛ{QW +zG−,L} = eΛ{QW +zG−,L}(QW + zG−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ The Λ-independence means that the pairing KW matches with the B-model pairing g on H∗(QW, ker G−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model pairing g is non-degenerate on VB and QW-, G−-invariant, hence g is non-degenerate on H∗(QW, ker G−), so is the KW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Remark: We can introduce an expansion for the pairing in z to define higher pairings KW(Ψ1, Ψ2) = ∞ � k=0 zk K(k) W (Ψ1, Ψ2), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) such that the K(0) W is identical to the gΛ W from the section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The argument of exponent in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) is a sum of two terms: a function {QW, L} = QW(L) = N � k=1 ψk Φψk R + 2πi N � k=1 rk ∂W ∂Yk (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='42) and first-order differential operator {G−, L} = −i N � k=1 rk ∂ ∂Yk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='43) 55 We can use the Zassenhaus formula eA+B = eA eB e− 1 2 [A,B] e 1 6(2[B,[A,B]]+[A,[A,B]]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='44) to express the localization exponent as product of the QW(L)-localizaton exponent and differential operator with coefficients in z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' eΛ{QW +zG−,L} = eΛQW (L) · D(z, Λ, r, Y, ∂Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='45) The representation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='45) allows us to conclude that the higher pairings also localize to ψ = r = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' the value of the pairing KW is the same for QW-, G−-closed states Ψ, Ψ′ and their holomorphic germs Φ, Φ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' KW(Ψ, Ψ′) = KW(Φ, Φ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) The localization exponent to first order in z eΛ{QW +zG−,L} = eΛQW (L) � 1 + πzΛ2 N � k,l=1 rkrl ∂2W ∂Yk∂Yl − izΛ N � k=1 rk ∂ ∂Yk + O(z2) � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='47) allows us to evaluate the K(1) W -pairing K(1) W (Ψ1, Ψ2) = � dµ Ψ1 · eΛQW (L) � πΛ2 rkrl ∂2W ∂Yk∂Yl − iΛrk ∂ ∂Yk � Ψ2 = −π � dW =0 (∂Yk∂YlW)−1(Ψ1∂Yk∂YlΨ2 − Ψ2∂Yk∂YlΨ1) det ∂Yk∂YlW ��� r=ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='48) Remark: In case of single Y -variable the pairing simplifies into K(1) W (Ψ1, Ψ2) = −2π � W ′=0 1 2 Ψ1Ψ′′ 2 − Ψ′′ 1Ψ2 (W ′′)2 ��� r=ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='49) Remark: The pairings K(0) W and K(1) W on holomorphic functions match with the correspond- ing higher pairing components (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) for the LGS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In proposition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) we showed that QW-closed B-model state Ψ and its holomorphic germ represent the same class in H∗(QW, VB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The pairing KW allows us to make a stronger statement: 56 Proposition: The QW-, G−-closed state Ψ and its holomorphic germ Φ are in the same class of H∗(QW + zG−, VB) and there exists a tropical form (z-independent) χ such that Ψ = Φ + QWχ, G−χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='50) Proof: The pairing of both representatives with arbitrary Ψ′ ∈ H∗(QW −zG−) are identical i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' KW(Ψ′, Ψ) = KW(Ψ′, Φ) =⇒ KW(Ψ′, Ψ − Φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='51) The non-degeneracy of the pairing leads to Ψ − Φ = 0 ∈ H∗(QW + zG−), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='52) what implies the the existence of the tropical form χ such that Ψ − Φ = (QW + zG−)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='53) By construction Ψ and Φ are independent of z and we can choose χ, which is z-independent, hence G−χ = 0, what concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ Example: The holomorphic germ for the P1-mirror state (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='39) is Φγ = ΨP1 γ ��� r=ψ=0 = q+eiY Θ(−r0) + q−e−iY Θ(r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='54) For r0 < 0 it coincides with the Ψ+-divisor, while for r0 > 0 it coincides with Ψ−-divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For both cases we showed that the corresponding tropical forms χ± are G−-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 B-model deformation by a holomorphic function In this section we will describe a one-parameter family of deformations for the B-type HTQM (VB, QW, G±, µ2, g) by a holomorphic function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The holomorphic function belongs to the B-model space of states, hence we can use construction from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 for the leading order 57 deformation Qǫ W = QW − 2π[G−, µ2(ǫΦ, ·)] = QW + ǫ 2πi ∂Φ ∂Yk ∂ ∂ψk Φ = Q + 2πi∂W ∂Yk ∂ ∂ψk Φ + 2πiǫ ∂Φ ∂Yk ∂ ∂ψk Φ = QW ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='55) The last equality allows us to describe the deformed differential Qǫ W in the form of differential in B-model with deformed superpotential W ǫ = W + ǫ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='56) The deformation of a B-model state is defined as (a finite) expansion in W and A-model propagators K Ψǫ = Ψ + 2πKWG−µ2(ǫΦ, Ψ) + O(ǫ2) = Ψ + 2πKG−µ2(ǫΦ, Ψ) + 2πKG−µ2(W, 2πKG−µ2(ǫΦ, Ψ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' + O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='57) The deformation of a mirror state Ψ = ΨW γ simplifies into [ΨW γ ]ǫ = ΨW γ + 2πKWG−µ2(ǫΦ, ΨW γ ) + O(ǫ2) = Ψγ + 2πKG−µ2(W, Ψγ) + 2πKG−µ2(ǫΦ, Ψγ) + 2πKG−µ2(W, 2πKG−µ2(W, Ψγ)) + 2πKG−µ2(ǫΦ, 2πKG−µ2(W, Ψγ)) + 2πKG−µ2(W, 2πKG−µ2(ǫΦ, Ψγ)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' + O(ǫ2) = Ψγ + 2πKG−µ2(W + ǫΦ, ΨW γ ) + 2πKG−(W + ǫΦ, 2πKG−µ2(W + ǫΦ, ΨW γ )) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' + O(ǫ2) = ΨWǫ γ Hence we conclude that the deformation [ΨW γ ]ǫ of the mirror state ΨW γ by a holomorphic function in B-model with superpotential W is a mirror state ΨWǫ γ for the same γ but in B-model with deformed superpotential W ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6 Higher pairing for mirror states The holomorphic germs for mirror states in B-model can be used to construct a good section in the corresponding LGS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 58 Definition: The tropical good section in mirror LGS theory for toric space X with su- perpotential W is a linear span of holomorphic germs for mirror states Im Strop W = C⟨ΦW γ | γ ∈ H∗ dR(X)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58) Remark: A similar construction of the good section from holomorphic germs of harmonic states was described in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) of a good section in LGS requires vanishing of higher pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition (higher pairing for tropical good section): Higher LGS pairings (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) for tropical good section vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Moreover KW(ΦW γ1, ΦW γ2) = � X γ1 ∧ γ2, ∀γ1, γ2 ∈ H∗ dR(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) Proof: By construction in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) the mirror states ΨW γ are G−- and QW-closed, hence ΨW γ ∈ H∗(QW ± zG−) and the higher B-model pairing (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) for such states is independent of Λ so we can relate it at different values of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In particular, we use Λ → ∞ limit gives us the LG pairing for holomorphic functions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' KW(ΨW γ1, ΨW γ2) = KW(ΦW γ1, ΦW γ2) = KW(ΦW γ1, ΦW γ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='60) Λ = 0 gives us the B-model pairing g i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' KW(ΨW γ1, ΨW γ2) = g(ΨW γ1, ΨW γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='61) The mirror states ΨW γk can be written in the following schematic form ΨW γ = Ψγ + G−χW γ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='62) for the A-model state χW γ = −2πKµ2(W, Ψγ) − 2πKµ2(W, 2πKG−µ2(W, Ψγ)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='63) 59 The B-model pairing g on representation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='62) simplifies into g(ΨW γ1, ΨW γ2) = g(Ψγ1, Ψγ2) + g(Ψγ1, G−χW γ2) + g(G−χW γ1, Ψγ2) + g(G−χW γ1, G−χW γ2) = g(Ψγ1, Ψγ2) + g(G−Ψγ1, χW γ2) − g(χW γ1, G−Ψγ2) − g(χW γ1, G2 −χW γ2) = g(Ψγ1, Ψγ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='64) We used the G±-invariance of the pairing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) to simplify the expression The A-model pairing on evaluation states Ψγ is the intersection of the cohomology classes on X i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' g(Ψγ1, Ψγ2) = � X γ1 ∧ γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='65) The proof of the proposition is the following: We use (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='60) to turn LGS higher pairing into the B-model higher pairing on mirror states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We use (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='61) to turn the B-model higher pairing into ordinary pairing which according to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='64) simplifies to the intersection of cohomology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The higher parings are z-expansion of KW hence K(k) W (ΦW γ1, ΦW γ2) = 0, ∀ k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='66) what completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ Example: The toric description of P2 consists of three compactifying divisors with primitive normal vectors: (1, 0), (0, 1) and (−1, −1) so the mirror superpotential WP2 = eiY1 + eiY2 + qe−iY1−iY2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='67) The de Rham cohomology of P2 are one-dimensional in degrees 0, 2, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The image of tropical good section is Im Strop P2 = C⟨1, qe−iY1−iY2, eiY1+iY2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='68) 8 Correlation functions for mirror states The correlation function invariance theorem from section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 tells us that ⟨Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn, QXχ⟩QX = 0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1) 60 for QX- G−-closed states Ψa and a G−-closed state χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In previous section we showed that there exists a tropical form χ such that it can turn a mirror state into a holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Moreover, in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 we saw that the deformation of B-model by a holomorphic func- tion leaves it in the same class but with different superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence, our strategy for evaluating correlation functions of mirror states in B-model will be the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Replace one mirror state by the corresponding holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Deform the superpotential and mirror states by the holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Apply the recursion formula for correlation functions to reduce the number of argu- ments by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Repeat steps 1-3 till the number of arguments reaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The simplification of the theory deformation while using the holomorphic germ rather than the mirror state itself suggest an alternative strategy for the correlation function evalu- ation: We replace all mirror states by the corresponding holomorphic germs and evaluate the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The application of this strategy to the 4pt functions immediately gives zero answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The reason for that is our definition of the B-model amplitudes via an expan- sion in A-model amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' In A-model the propagator acts trivially on the holomorphic functions what is the source of zero answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' To resolve the puzzle we recall that the A-model amplitudes for tropical GW invariants have a particular number of divisor states, determined by the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The mirror states can provide additional divisor states, but if we already have too many (from holomorphic germs) the amplitude vanishes due to the degree selection for the tropical GW invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Since, all non-trivial GW invariants have degree bigger than zero, the corresponding A-model amplitudes have at least one divisor state, hence we can always replace a single mirror state by the holomorphic germ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Later we will see that a single replacement is just enough to prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='1 4-point function invariance and holomorphic representatives Using the symmetries of the amplitudes we can rewrite the 4pt function in the form 2⟨Ψ1, Ψ2, Ψ3, Ψ4⟩QW = g(Ψ4, � σ∈S3 µ2(Ψσ(1), KWG−µ2(Ψσ(2), Ψσ(3)))) = g(Ψ4, Ψ123) = g(Φ4 + QWχ+, Ψ123) = g(Φ4, Ψ123) = 2⟨Ψ1, Ψ2, Ψ3, Φ4⟩QW (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2) 61 The equality on the second line requires g(QWχ+, Ψ123) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3) In (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) we showed that χ for P1-model does not belong to tropical forms on P1, hence the difference g(QWχ+, Ψ123) − g(χ+, QWΨ123) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4) might not be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The difference is controlled by the boundary term of the r-integration g(QWχ+, Ψ123) − g(χ+, QWΨ123) = � dµ Q(χ+Ψ123) = � dµ dR(χ+Ψ123) = lim r→−∞ � S1 dY Θ(r4 − r) Ψ123 ��� ψ=0 = lim r→−∞ � S1 dY Ψ123 ��� ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5) We can repeat the analysis for the case different holomorphic representative Φ′ 4 = qe−iY and χ− g(QWχ−, Ψ123) − g(χ−, QWΨ123) = lim r→+∞ � S1 dY Ψ123 ��� ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6) Both boundary terms vanish if Y -independent part of Ψ123 ��� ψ=0 has finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can evaluate the Ψ123 for P1-mirror states � S1dY Ψ123 ��� ψ=0 = 1 2 � σ∈S3 � S1 dY µ2(Ψσ(3), 2πKWG−µ2(Ψσ(1), Ψσ(2))) ��� ψ=0 = πq � σ∈S3 � Θ(min(rσ(1), rσ(2)) − r))Θ(r − rσ(3)) + Θ(r − max(rσ(1), rσ(2)))Θ(rσ(3) − r) � and observe that the products of Θ-functions indeed have the finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can perform the same analysis for ⟨Ψ1, Ψ2, Φ3, Ψ4⟩QW , where we replaced the mirror state Ψ3 by the holomorphic function Φ3 = eiY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The boundary term is controlled by the function � S1dY ˜Ψ123 ��� ψ=0 = � S1 dY µ2(Φ3, 2πKWG−µ2(Ψ1, Ψ2)) ��� ψ=0 + � S1 dY µ2(Ψ2, 2πKWG−µ2(Ψ1, Φ3)) ��� ψ=0 + � S1 dY µ2(Ψ1, 2πKWG−µ2(Ψ2, Φ3)) ��� ψ=0 = 2πqΘ(min(r1, r2) − r) + 2πqΘ(r2 − r)Θ(r − r1) + 2πqΘ(r1 − r)Θ(r − r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7) 62 which does not have the finite support and the limit is finite and non-zero lim r→−∞ � S1 dY ˜Ψ123 ��� ψ=0 = 2πq ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8) Hence we conclude that the simultaneous replacement of two P1-mirror states Ψ3 and Ψ4 by the same holomorphic germs qe−iY does not preserve the invariance of the correlation function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ1, Ψ2, Φ3, Ψ4⟩QW − ⟨Ψ1, Ψ2, Φ3, Φ4⟩QW ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='9) Remark: If we replace Ψ4 by the the different holomorphic function Φ4 = qe−iY then the boundary term is controlled by the different limit lim r→+∞ � S1 dY ˜Ψ123 ��� ψ=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='10) Hence we conclude that for the choice of holomorphic functions Φ3 = eiY and Φ4 = qe−iY preserves the 4-point function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ1, Ψ2, Ψ3, Ψ4⟩QW = ⟨Ψ1, Ψ2, Φ3, Φ4⟩QW = ⟨Ψ1, Ψ2, eiY , qe−iY ⟩QW (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) Note that the equality (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='11) agrees with our heuristic analysis for the possible P1-mirror states replacement, based on the A-model divisor counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The GW invariants for P1 are non-trivial only for degree-1 maps, hence the A-model correlation functions have exactly two divisor states Ψ+ = eiY and Ψ− = e−iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 n-point function invariance and holomorphic representative Conjecture: For the mirror states Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., Ψn for general toric X the state Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n, evaluated on the sum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) over rooted trees with leaves Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn and χ, such that QWχ = Ψn+1 − Ψn+1 ��� ψ=r=0, G−χ = 0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='12) satisfy g(χ, dRΨ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n) − g(dRχ, Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n) = 0, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='13) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' the boundary term vanishes for all boundary divisors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Proposition: The conjecture holds for X = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 63 Proof: For the (n + 1)-point correlation function the boundary term is g(χ∓, dRΨ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n) − g(dRχ∓, Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n) = lim r→±∞ � S1 dY Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n ��� ψ=0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='14) In particular, if Y -independent part of Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n ��� ψ=0 has finite support for any P1-mirror states Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', Ψn then the boundary term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For any pair of P1 B-model states Ψ1 and Ψ2 we can simplify KP1G−µ2(Ψ1, Ψ2) = KG−µ2(Ψ1, Ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='15) This equality follows from the degree counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The B-model states are at most (1, 1)-forms in ψ, so is the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The KG− lowers degree of the state by (1, 1), so there are no higher order terms in expansion of the B-model propagator KP1G− in A-model propagators KG−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can simplify the KG− for product of two P1-mirror states into Ψ′ ab = 2πKG−µ2(ΨP1 a , ΨP1 b ) = eiY Θ(r − max(ra, rb)) + qe−iY Θ(min(ra, rb) − r) and observe that it is degree-0 tropical form, hence the product of two such expressions is also a zero form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The KG−-action on any zero form is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ψ′ abcd = KG−µ2(Ψ′ ab, Ψ′ cd) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='16) This equality allows us to simplify the sum over rooted trees to Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n = 1 4 n−1 � k=1 µ2(Ψ′ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.k, Ψ′ k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n)), Ψ′ 1 = ΨP1 1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='17) using the Ψ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n-expressions defined recursively n = 2 Ψ′ 12 = 2πKG−µ2(ΨP1 1 , ΨP1 2 ) = eiY Θ(r − max(r1, r2)) + qe−iY Θ(min(r1, r2) − r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' n > 2 Ψ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n = 2πKG−µ2(ΨP1 (n, Ψ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) 64 We can prove by induction that Ψ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n = eiY Θ(r − max(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rn)) + qe−iY Θ(min(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rn) − r) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='19) and evaluate Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='.n ��� ψ=0 = 1 4 · n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' � σ∈Sn n−1 � k=1 (eiY Θ(r − max(rσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rσ(k))) + qe−iY Θ(min(rσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rσ(k)) − r)) × (eiY Θ(r − max(rσ(k+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rσ(n))) + qe−iY Θ(min(rσ(k+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='., rσ(n)) − r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='20) Each Y -independent term in the sum has finite support so is the whole sum, what completes the proof of a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3 3-point functions Proposition (3-point localization formula): For QW-closed B-model states Ψ1, Ψ2, Ψ3 the 3-point function equals to the LGS 3-point function for corresponding holomorphic germs and superpotential W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨Ψ1, Ψ2, Ψ3⟩QW = ⟨Φ1, Φ2, Φ3⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='21) Proof: The 3pt function in B-model is ⟨Ψ1, Ψ2, Ψ3⟩QW = g(Ψ1, µ2(Ψ2, Ψ3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='22) The product of two QW-closed states is QW-closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' QWµ2(Ψ2, Ψ3) = µ2(QWΨ2, Ψ3) + µ2(Ψ2, QWΨ3) = 0, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='23) hence we can replace B-model pairing g by the pairing gΛ W ⟨Ψ1, Ψ2, Ψ3⟩QW = gΛ W(Ψ1, µ2(Ψ2, Ψ3)) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='24) and use the localization ⟨Ψ1, Ψ2, Ψ3⟩QW = � dW =0 Ψ1Ψ2Ψ3 det ∂i∂jW ��� r=ψΦ=ψR=0 = � dW =0 Φ1Φ2Φ3 det ∂i∂jW = ⟨Φ1, Φ2, Φ3⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='25) 65 Hence we completed the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ Example: For X = P1 the mirror states for cohomology representatives 1, γP ∈ H∗(QP1) have the holomorphic germs 1, eiY : three 0-forms: The B-model 3-point function vanishes, due to the insufficient degree of the form ⟨ΨP1 1 , ΨP1 1 , ΨP1 1 ⟩QP1 = � dµ 1 = 0 = ⟨1, 1, 1⟩WP1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='26) single 1-form: The B-model 3-point function is the integral of the γP over P1 ⟨ΨP1 1 , ΨP1 1 , ΨP1 P ⟩QP1 = � dµ ΨP1 P = � P1 γP = 1 = ⟨1, 1, eiY ⟩WP1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='27) two 1-forms: The B-model 3-point function vanishes, due to degree selection ⟨ΨP1 1 , ΨP1 P , ΨP1 P ⟩QP1 = 0 = ⟨1, eiY , eiY ⟩WP1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='28) three 1-forms: For three 1-forms, Poincare dual to the three distinct points r1, r2, r3 the B-model 3-point function is ⟨ΨP1 P , ΨP1 P , ΨP1 P ⟩QP1 = � dµ ΨP1 P ΨP1 P ΨP1 P = q � σ∈S3 Θ(rσ(1), rσ(2), rσ(3)) = q = ⟨eiY , eiY , eiY ⟩WP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='29) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 4-point functions Proposition (4-point localization formula): If conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 holds then the 4-point correlation function of mirror states ΨW a equals to the 4-point correlation function of the corresponding holomorphic germs in LGS theory with superpotential W and tropical good section (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 ⟩QW = ⟨Φ1, Φ2, Φ3, Φ4⟩Strop W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='30) Proof: The invariance theorem from section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4, extended by a conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 allows us to replace the mirror state ΨW 4 by the holomorphic germ Φ4 in the 4-point function of mirror 66 states ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 ⟩QW = ⟨ΨW 1 , ΨW 2 , ΨW 3 , Φ4⟩QW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) The holomorphic germ Φ4 is QW- and G−-closed, hence we can use the recursion relation theorem from section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 ⟨ΨW 1 , ΨW 2 , ΨW 3 , Φ4⟩QW = d dǫ ��� ǫ=0⟨Ψǫ 1, Ψǫ 2, Ψǫ 3⟩QW ǫ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='32) for deformed superpotential W ǫ = W + ǫΦ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='33) The deformed states Ψǫ k = ΨW k + ǫ 2πKWG−µ2(ΨW k , Φ4) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='34) are QW ǫ-closed hence the 3-point function can be evaluated in terms of LGS theory ⟨Ψǫ 1, Ψǫ 2, Ψǫ 3⟩QW ǫ = � dW ǫ=0 Φǫ 1 · Φǫ 2 · Φǫ 2 det ∂j∂kW ǫ ��� Y =Y0 = ⟨Φǫ 1, Φǫ 2, Φǫ 3⟩W ǫ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='35) for holomorphic germs of deformed states Φǫ k = Ψǫ k ��� ψ=r=0 = Φk + 2πǫ KWG−µ2(ΨW k , Φ4) ��� ψ=r=0 = Φk + ǫ Ctrop W (Φk, Φ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) The equality (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36) defines tropical contact term Ctrop W (Φk, Φ4) and later we will show that these contact terms match with LGS contact term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='31) for tropical good section (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We can substitute the Φǫ k in terms of Φk and contact term into the 3-point function derivative ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 ⟩QW = d dǫ ��� ǫ=0⟨Φǫ 1, Φǫ 2, Φǫ 3⟩W ǫ = d dǫ ��� ǫ=0⟨Φ1, Φ2, Φ3⟩W ǫ + ⟨Ctrop W (Φ1, Φ4), Φ2, Φ3⟩W + ⟨Φ1, Ctrop W (Φ2, Φ4), Φ3⟩W + ⟨Φ1, Φ2, Ctrop W (Φ3, Φ4)⟩W = ⟨Φ1, Φ2, Φ3, Φ4⟩Strop W (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='37) The last equality is the the LGS recursion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) for the 4pt functions with contact terms defined by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 67 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 Contact terms Definition: For a superpotential W, holomorphic function Φ2 and mirror state ΨW 1 with holomorphic germ Φ1 we define tropical contact term Ctrop W (Φ1, Φ2) = d dǫ ��� ǫ=0 Ψǫ 1 ��� ψ=r=0 = 2πKWG−µ2(ΨW 1 , Φ2) ��� ψ=r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='38) Proposition (contact terms for tropical good section): The tropical contact term equals (as classes in H∗(QW + zG−)) to the LGS contact term for the tropical good section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Ctrop W (Φ1, Φ2) = CStrop W (Φ1, Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='39) Proof: The product µ2(ΨW 1 , Φ2) is a QW-closed state due to QWµ2(ΨW 1 , Φ2) = µ2(QWΨW 1 , Φ2) + µ2(ΨW 1 , QWΦ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='40) Hence it represents some class in H∗(QW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Moreover, the same class can be expressed via µ2(ΨW 1 , Φ2) = µ2(Φ1, Φ2) = Φ1Φ2 ∈ H∗(QW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='41) Using the map πW : RC∗N → JW we can write the class for the product of two holomorphic functions Φ1Φ2 in the form πW(Φ1Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Applying an isomorphism J−1 : JW = H∗(QW) → H∗ dR(X) : Ψ �→ J−1(Ψ) from section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 to the class πW(Φ1Φ2) we can construct class J−1πW(Φ1Φ2) in H∗ dR(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Let us choose representatives γ for each class of cohomology H∗ dR(X), so the class J−1πW(Φ1Φ2) is represented by a tropical form γJ−1πW(Φ1Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' This tropical form defines a mirror state ΨW γJ−1πW (Φ1Φ2), which is G−-closed and represents the same class to µ2(ΨW 1 , Φ2) in H∗(QW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence there exists a tropical form χ such that µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) = QWχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='42) We can use χ to evaluate KWG−µ2(ΨW 1 , Φ2) = KWG−ΨW γJ−1πW (Φ1Φ2) + KWG−QWχ = −G−χ + (QW + zG−)KWG−χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='43) 68 The tropical contact term is the holomorphic germ of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='43) Ctrop W (Φ1, Φ2) = 2π � KWG−µ2(ΨW 1 , Φ2) � ��� ψ=r=0 = −2πG−χ ��� ψ=r=0 + (QW + zG−)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='44) We can represent the QW as a sum of two graded-commuting differentials QW = 2πiQW + dR (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='45) to write a solution to QWχ = (2πiQW + dR)χ = Ψ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='46) using the homotopy ΣW for QW χ = 1 2πiΣWΨ − 1 (2πi)2ΣWdRΣWΨ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='47) The radial de Rham dR adds powers in ψR, hence the holomorphic germs for higher terms in the sum vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' G−ΣWdRΣW � µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψR=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='48) The tropical contact term simplifies to Ctrop W (Φ1, Φ2) = −2πG−χ ��� ψ=r=0 = iG−ΣW � µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψ=r=0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='49) Let us recall the relation iG− = G− between the B-model G− and LGS G−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Both G− and QW act trivially on the radial variables r, ψR, hence we can simplify iG−ΣW � µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψ=r=0 = G−ΣW �� µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψR=r=0 � ��� ψΦ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='50) The mirror states are Hodge type tropical forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' contain the same number of ψΦ and 69 ψR for each degree, hence we can further simplify � µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψR=r=0 = � µ2(ΨW 1 , Φ2) − ΨW γJ−1πW (Φ1Φ2) � ��� ψR=r=ψΦ=0 = µ2(Φ1, Φ2) − ΦW γJ−1πW (Φ1Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='51) For any function Φ we construct another function, defined via SWπW(Φ) = ΦW γJ−1πW (Φ) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='52) to express the tropical contact term in the form Ctrop W (Φ1, Φ2) = G−ΣW � µ2(Φ1, Φ2) − ΦW γJ−1πW (Φ1Φ2) � = G−ΣW (µ2(Φ1, Φ2) − SWπW(Φ1Φ2)) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='53) Earlier in a section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 we saw that, in particular, SW defines a section H∗(QW) → H∗(QW + zG−) hence we can drop exact terms in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ Example: Let X = P1, ΨW 1 is a mirror state for the point observable at r = r1 ΨW 1 = ΨP1 P = δ(r − r1)ψΦψR + eiY Θ(r − r1) + qe−iY Θ(r1 − r) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='54) and holomorphic function Φ2 = qe−iY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='55) The tropical contact term is the holomorphic germ of KWP1G−µ2(ΨW 1 , Φ2) = KG−µ2(ΨW 1 , Φ2) + KG−µ2(WP1, KG−µ2(ΨW 1 , Φ2)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' = KG−µ2(Ψ1, Φ2) = KG−(qe−iY δ(r − r1)ψΦψR) = � ∞ 0 dt e−tHG+G−(qe−iY δ(r − r1)ψΦψR) = qe−iY � ∞ 0 dt δ(r − r1 + t) = qe−iY Θ(r1 − r), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='56) while the germ evaluation gives us Ctrop WP1(Φ1, Φ2) = KWP1G−µ2(ΨW 1 , Φ2) ��� ψ=r=0 = qe−iY Θ(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='57) 70 The holomorphic representative of the mirror state Φ1 = ΨW 1 ��� ψ=r=0 = eiY Θ(−r1) + qe−iY Θ(r1) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58) gives us contact terms of the form Ctrop WP1(qe−iY , qe−iY ) = qe−iY , Ctrop WP1(eiY , qe−iY ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='59) The tropical contact terms matches with the P1-mirror LGS contact terms for tropical good section for P1 Im Strop = C⟨1, eiY ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='60) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='6 5-point function Proposition (5-point localization formula): If conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 holds then the 5-point correlation function of mirror states ΨW a equals to the 5-point correlation function of the corresponding holomorphic germs in LGS theory with superpotential W and tropical good section (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='58), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 , ΨW 5 ⟩QW = ⟨Φ1, Φ2, Φ3, Φ4, Φ5⟩Strop W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='61) Proof: We can rewrite the 5-point function in B-model using the recursion formula for the B-model deformation by a holomorphic germ of the mirror state ΨW 5 ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 , ΨW 5 ⟩QW = ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 , Φ5⟩QW = d dǫ ��� ǫ=0⟨Ψǫ 1, Ψǫ 2, Ψǫ 3, Ψǫ 4⟩QW ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The deformed superpotential is W ǫ = W + ǫΦ5 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='62) and deformed states are Ψǫ k = ΨW k + ǫKWG−µ2(ΨW k , Φ5) = ΨW ǫ k , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='63) while the holomorphic germs are Φǫ k = Ψǫ k ��� ψ=r=0 = Φk + 2πǫ KWG−µ2(ΨW k , Φ5) ��� ψ=r=0 = Φk + ǫ Ctrop W (Φk, Φ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='64) 71 The deformed states are mirror states for W ǫ hence we can repeat the 4-point function analysis from previous section ⟨ΨWǫ 1 , ΨWǫ 2 , ΨWǫ 3 , ΨWǫ 4 ⟩QW ǫ = ⟨Ψǫ 1, Ψǫ 2, Ψǫ 3, Φǫ 4⟩QW ǫ = d dλ ��� λ=0⟨Ψǫ,λ 1 , Ψǫ,λ 2 , Ψǫ,λ 3 ⟩QW ǫ,λ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='65) with deformed superpotential W ǫ,λ = W ǫ + λΦǫ 4 = W + ǫΦ5 + λΦ4 + λǫ Ctrop W (Φ4, Φ5) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='66) and deformed states Ψǫ,λ k = Ψǫ k + λ KW ǫG−µ2(Ψǫ k, Φǫ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='67) Remark: The appearence of the λǫ Ctrop W (Φ4, Φ5) is a signature of the non-linear rela- tion between the linear times ǫ, λ on the image of good section and coordinates T in the superpotential deformation, introduced by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Saito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The states Ψǫ,λ k are mirror states hence we can express the 3-point function in terms of the LGS theory ⟨Ψǫ,λ 1 , Ψǫ,λ 2 , Ψǫ,λ 3 ⟩QW ǫ,λ = � dW ǫ,λ=0 Φǫ,λ 1 Φǫ,λ 2 Φǫ,λ 2 det ∂j∂kW ǫ,λ = ⟨Φǫ,λ 1 , Φǫ,λ 2 , Φǫ,λ 3 ⟩W ǫ,λ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='68) for holomorphic germs Φǫ,λ k = Ψǫ,λ k ��� ψ=r=0 = Φǫ k + λ KW ǫG−µ2(Ψǫ k, Φǫ 4) ��� ψ=r=0 = Φǫ k + λ Ctrop W ǫ (Φǫ k, Φǫ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='69) The Ctrop W ǫ (Φǫ k, Φǫ 4) is a contact term for the good section for deformed superpotential Wǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Hence, we can apply the LSG recursion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) for the 4-point function ⟨Ψǫ 1, Ψǫ 2, Ψǫ 3, Ψǫ 4⟩QW ǫ = d dλ ��� λ=0⟨Ψǫ,λ 1 , Ψǫ,λ 2 , Ψǫ,λ 3 ⟩QW ǫ,λ = d dλ ��� λ=0⟨Φǫ 1, Φǫ 2, Φǫ 3⟩W ǫ,λ + ⟨Ctrop W ǫ (Φǫ 1, Φǫ 4), Φǫ 2, Φǫ 3⟩W ǫ + ⟨Φǫ 1, Ctrop W ǫ (Φǫ 2, Φǫ 4), Φǫ 3⟩W ǫ + ⟨Φǫ 1, Φǫ 2, Ctrop W ǫ (Φǫ 3, Φǫ 4)⟩W ǫ = ⟨Φǫ 1, Φǫ 2, Φǫ 3, Φǫ 4⟩Strop W ǫ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='70) 72 Similarly we can apply the LSG recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) for the 5-point function ⟨ΨW 1 , ΨW 2 , ΨW 3 , ΨW 4 , ΨW 5 ⟩QW = d dǫ ��� ǫ=0⟨Ψǫ 1, Ψǫ 2, Ψǫ 3, Ψǫ 4⟩QW ǫ = d dǫ ��� ǫ=0⟨Φǫ 1, Φǫ 2, Φǫ 3, Φǫ 4⟩Strop W ǫ = d dǫ ��� ǫ=0⟨Φ1, Φ2, Φ3, Φ4⟩Strop W ǫ + ⟨Ctrop W (Φ1, Φ5), Φ2, Φ3, Φ4⟩Strop W + ⟨Φ1, Ctrop W (Φ2, Φ5), Φ3, Φ4⟩Strop W + ⟨Φ1, Φ2, Ctrop W (Φ3, Φ5), Φ4⟩Strop W + ⟨Φ1, Φ2, Φ3, Ctrop W (Φ4, Φ5)⟩Strop W = ⟨Φ1, Φ2, Φ3, Φ4, Φ5⟩Strop W what completes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='7 Parallel transport of a good section In LGS theory, given a good section SW for superpotential W we can extend it to a good section SWt for superpotential Wt via the parallel transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' For a given superpotential W and γ ∈ H∗ dR(X) we construct a mirror state ΨW γ = Ψγ + KG−µ2(W, Ψγ) + KG−µ2(W, KG−µ2(W, Ψγ)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='71) which define an image of good section Im SW = C⟨ΦW γ |γ ∈ H∗ dR(X)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='72) Let W → W + δW, then the change of the mirror state to the leading order is given by ΨW +δW γ − ΨW γ = KWG−µ2(δW, ΨW γ ) + O(δW)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='73) The corresponding change of holomorphic germ ΦW +δW γ − ΦW γ = KWG−µ2(δW, ΨW γ ) ��� r=ψ=0 = Ctrop W (δW, ΦW γ ) = CStrop W (δW, ΦW γ ) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='74) Hence we demonstrated that the tropical good section is parallel with respect to connection determined by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='8 Localization of correlation functions Theorem (localization of correlation functions): The B-model correlation function for the mirror states constructed for observables γk ∈ H∗ dR(X) on toric space X equal to the 73 LGS correlation function ⟨ΨX γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨX γn⟩QX = ⟨ΦX γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΦX γn⟩WX (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='75) The mirror LGS theory has the following data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The LGS theory is on C∗N, where N is the complex dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The holomorphic top form on C∗N written in terms of cylindrical coordinates (r, Y ) is Ω = dY1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ∧ dYN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='76) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The LGS superpotential is a mirror superpotential for X, written in terms of the compactifying divisors for X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' WX = � ⃗b∈BX q⃗b ei⟨⃗b,Y ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='77) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The image of a good section Im Strop W = C⟨ΦW γ | γ ∈ H∗ dR(X)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='78) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The LSG observables are holomorphic germs for mirror states ΦX γ = ΨX γ ��� ψ=r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='79) Proof: We are going to embed the statement of the theorem into more general equality ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn⟩QW = ⟨ΦW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΦW γn⟩W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='80) The theorem is the case when W = WX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' Such rewriting allows us to use an induction in the number of states n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The proof for n = 3 follows from proposition on 3-point localization from section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' We use the invariance theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='4 under assumption of conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='2 to replace the mirror state by its holomorphic germ in the (n + 1)-point correlation function ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn, ΨW γn+1⟩QW = ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn, ΦW γn+1⟩QW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='81) 74 We apply the recursion formula theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 for deformation of HTQM with differential QW by a holomorphic germ state ΦW γn+1 and express the n + 1-point function as a derivative of n-point function in deformed theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn, ΦW γn+1⟩QW = d dǫ ��� ǫ=0⟨ΨW ǫ γ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW ǫ γn ⟩Qǫ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='82) Using our expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='55) for the B-model deformation by a holomorphic function we replace Qǫ W by QW ǫ for superpotential, deformed by the holomorphich germ of ΨW n+1 W ǫ = W + ǫΦW γn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='83) Using an assumption of induction that the equality holds for n-point correlation functions ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn⟩QW = ⟨ΦW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΦW γn⟩W (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='84) we can express the (n+1)-point correlation function in B-model in terms of n-point functions in LSG theory ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn, ΨW γn+1⟩QW = d dǫ ��� ǫ=0⟨ΦW ǫ γ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΦW ǫ γn ⟩W ǫ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='85) for holomorphic functions given by the holomorphic germs of deformed states ΦW ǫ γk = ΨW ǫ γk ��� ψ=r=0 = ΦW γk + 2πKWG−µ2(ΦW γn+1, ΨW γk) ��� ψ=r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='86) According to the tropical good section proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='5 we can identify the second term in the expression above with tropical contact term 2πKWG−µ2(ΦW γn+1, ΨW γk) ��� ψ=r=0 = CStrop W (ΦW γk, ΦW γn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='87) The recursion formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='18) for the LGS correlation functions allows us to rewrite the derivative of n-point function as (n + 1)-point correlation function in LSG theory with superpotential W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ⟨ΨW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΨW γn, ΨW γn+1⟩QW = d dǫ ��� ǫ=0⟨ΦW ǫ γ1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=', ΦW ǫ γn ⟩QW ǫ = ⟨ΦW γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='ΦW γn, ΦW γn+1⟩W (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='88) and complete the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' ■ 75 Acknowledgments We are grateful to Yasha Neiman for many discussions on the topics presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' The work A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' is supported by Wu Wen-Tsun 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 165 (1994) 311–428, arXiv:hep-th/9309140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} +page_content=' 77' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfuP1K/content/2301.01687v1.pdf'} diff --git a/2NE4T4oBgHgl3EQfzw00/content/tmp_files/2301.05276v1.pdf.txt b/2NE4T4oBgHgl3EQfzw00/content/tmp_files/2301.05276v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a855a745023baa43097b94bd42986845e7e5ab22 --- /dev/null +++ b/2NE4T4oBgHgl3EQfzw00/content/tmp_files/2301.05276v1.pdf.txt @@ -0,0 +1,1786 @@ +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA +DEGENERATIONS +ALEX CASAROTTI AND ELISA POSTINGHEL +Abstract. We discuss an approach to the secant non-defectivity of the vari- +eties parametrizing k−th powers of forms of degree d. It employs a Terracini +type argument along with certain degeneration arguments, some of which are +based on toric geometry. This implies a result on the identifiability of the War- +ing decompositions of general forms of degree kd as a sum of kth powers of +degree−d forms, for which an upper bound on the Waring rank was proposed +by Fr¨oberg, Ottaviani and Shapiro. +1. Introduction +Identifiability problems arise naturally in many fields of both applied and classi- +cal algebraic geometry. A variety X ⊂ PN is said to be h−identifiable if the general +point of its h−secant variety has a unique decomposition as a sum of h points of +X. A classical application of identifiability concerns particular polynomial decom- +positions. +The Waring problem for forms asks for a unique decomposition of a +homogeneous polynomial Fd ∈ C[x0, . . . , xn]d as a sum of d−th powers of linear +forms, i.e. +(1.1) +Fd = Ld +1 + · · · + Ld +h, +with Li ∈ C[x0, . . . , xn]1. A necessary condition for identifiability is secant defectiv- +ity: a variety X ⊂ PN of dimension dim(X) = n is said to be not h−(secant) defec- +tive if the h−secant variety Sech(X), defined as the Zariski closure of points in PN +lying in the span of h points of X, has the expected dimension min{N, h(n+1)−1}. +In [COVC17] the authors proved that, for all subgeneric ranks h (i.e. such that +Sech(X) ⊆ PN does not fill up the space), a general form F of rank h is identifi- +able, with a few well known exceptions. In the case of generic rank, the situation is +almost the opposite: in [GM19] it is proved that all forms of generic rank are not +identifiable with the following exceptions: (n, d, h) = (1, 2k − 1, k), (3, 3, 5), (2, 5, 7). +In [FOS12] the authors initiated the investigation of a generalization of the clas- +sical Waring problems for forms. +In particular they show that a general form +Fkd ∈ C[x0, . . . , xn]dk can be written as a sum of at most kn k−th powers of forms +Gi’s of degree d +(1.2) +Fkd = Gk +1 + · · · + Lk +h, +2020 Mathematics Subject Classification. Primary: 14N07. Secondary: 14C20, 14D06, 14M25. +Key words and phrases. Identifiability, Waring problems, secant varieties, linear systems, +degenerations. +Both authors are members of INdAM-GNSAGA. +1 +arXiv:2301.05276v1 [math.AG] 12 Jan 2023 + +2 +ALEX CASAROTTI AND ELISA POSTINGHEL +and that this bound is sharp, i.e. when d is sufficiently large, kn computes the +generic rank. However the secant defectivity of the varieties parametrizing k−th +powers of forms of degree d remains an open problem in general. +In this paper we address both the secant defectivity and identifiability problems +for such Waring decompositions. Denote with V k +n,d the variety parametrizing k−th +powers of homogeneous degree d forms in n + 1 variables: +V k +n,d := {[F k]|F ∈ C[x0, . . . , xn]d}. +Our first main result is about secant non-defectivity. +Theorem 1.1. The variety V k +n,d is non-defective if k ≥ 3 and h ≤ +1 +N+1 +�N+k−3 +N +� +, +where N = +�n+d +d +� +− 1. +Our second result is about identifiability. A bridge from non-defectivity to iden- +tifiability was proposed in [CM22] first and then generalised in the recent [MM22]: +whenever X is a sufficiently regular variety (with non-degenerate Gauss map), then +if X is not h−defective, then X is (h − 1)−identifiable. Using this and Theorem +1.1 we obtain what follows. +Corollary 1.2. A general form F ∈ C[x0, . . . , xn]dk of rank h with k ≥ 3 is +identifiable whenever h ≤ +1 +N+1 +�N+k−3 +N +� +− 1. +We remark that in [Nen17], the author showed that the secant defectivity of V k +n,d +can be bounded asymptotically, using a direct algebraic computational argument, +to kn − dn. When d ≫ k our bound of Theorem 1.1 extends the latter. +In order to prove Theorem 1.1, we brought together a Terracini type argument +and several different degeneration techniques. By a classical application of Ter- +racini’s Lemma, non-defectivity problems for secant varieties translate into the +study of particular linear systems of hypersurfaces with prescribed singularities. +The first systematic study was used in the proof of the celebrated Alexander and +Hirschowitz Theorem for the case of classical Waring problems (1.1), where secant +varieties of Veronese embeddings of Pn correspond with linear systems of hyper- +surfaces of Pn with prescribed double points. In the generalized Waring problem +setting, as in (1.2) for k ≥ 2, a direct translation to linear systems of hypersurfaces +with only double point singularities is not possible. In order to prove secant non- +defectivity in this case it is necessary to impose a bigger base locus to our linear +systems. In particular, we will be interested in studying the dimensions of linear +systems L := LN,k(V, 2h) of hyperurfaces of PN of degree k that are singular at h +general points and that contain the d-thVeronese embedding of Pn, V ⊂ PN. The +study of such linear systems is carried out by combining two types of degenerations +introduced in [Pos12] and in [CDM09] and [Pos13] respectively. On one hand, we +degenerate the ambient space PN to a scheme with two components and, in turn, +the linear system L to a fibered product of two linear systems, one on each compo- +nent, which are somewhat easier to deal with than the original one. In fact one of +them consists of hypersurfaces containing a linear subspace and a collection of dou- +ble points, the other one consists of hypsersurfaces containing V and one fat point +of relatively large multiplicity with support on V . In order to study the latter, we +perform a toric degeneration of the Veronese V to a union of n-dimensional linear +spaces, which will have the effect of reducing further the study of the limit linear +system. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +3 +The study of Waring type problems and identifiability of symmetric tensors has +been implemented also in the applied fields, from chemistry, biology to algebraic +statistics. Recently in [BCMO23], the problem of identifiability for k−th powers +of forms was linked to the identifiability of centered Gaussian mixture models in +applied statistics. +1.1. Organization of the paper. Section 2 contains all definitions and our Ter- +racini type result that translates non-defectivity of V k +n,d to the study of L, Propo- +sition 2.16. +In Section 3 we explain in detail the degenerations techniques, both in the clas- +sical and in the toric setting. +In Section 4 we analyse two auxiliary linear systems arising from the degeneration +of L, Proposition 4.3 and Corollary 4.8. +Section 5 is devoted to the proof of the main technical result, i.e. Theorem 5.2. +Finally in Section 6 we show at what extent our bounds are asymptotically better +then the ones known before in the literature. +1.2. Acknowledgments. The authors would like to thank Giorgio Ottaviani and +Alessandro Oneto for several useful discussions during the preparation of this arti- +cle. +2. Powers of forms +In order to give a coherent and self-contained treatment of the subject, let us +recall some preliminary definitions and results. +We will work over the field of +complex numbers C. +2.1. Veronese embeddings. Let W := Cn+1 and W ∗ the dual vector space. +With Pn = P(W) we denote the projective space over C of dimension n. We set +the following integers +Nd := +�n + d +n +� +− 1, +N k +d := +�Nd + k +Nd +� +− 1. +When d is clear from the context we will indicate Nd simply by N. +Notice that the following identities hold: +h0(Pn, OPn(d)) = Nd + 1 +h0(PNd, OPNd (k)) = +�Nd + k +Nd +� ++ 1 +where h0(Pn, OPn(d)) denotes the number of global sections of the twisting sheaf +OPn(d) on Pn. In other terms, Nd is the dimension of the linear systems of hyper- +surfaces of degree d of PN which, in turn, is the projectivization of the complex +vector spaces of forms of degree d in n + 1 variables. The number N k +d has a similar +interpretation. With this in mind, we can make the following identifications: +PNd = P(Symd(W ∗)), +PN k +d = P(Symk(Symd(W ∗))). +Now we consider the following Veronese embeddings: +νd : Pn −→ V d +n ⊂ P(Symd(W ∗)) +[L] �−→ [Ld] + +4 +ALEX CASAROTTI AND ELISA POSTINGHEL +and +νk : PNd −→ V k +Nd ⊂ P(Symk(Symd(W ∗))) +[F] �−→ [F k] +where L ∈ C[x0, . . . , xn]1 is a linear form and F ∈ C[x0, . . . , xn]d is a form of degree +d. The image of the embeddings are called Veronese varieties. +Remark 2.1. Note that both νd and νk are the maps corresponding to the complete +linear systems associated with the line bundles OPn(d) and OPNd (k). As elements of +P(Symd(W ∗)) (respectively P(Symk(Symd(W ∗)))) the image of νd(p) (respectively +νk(p)), with p a point, corresponds to the hyperplane parametrizing hypersurfaces +of degree d in Pn (respectively of degree k in PNd) passing through p. +We want to parametrize forms in Pn of degree dk, i.e. elements in C[x0, . . . , xn]dk, +that can be written as k−th powers of forms of degree d. +Remark 2.2. Note that the Veronese varieties νdk(Pn) are always contained in the +set of all k−th powers of forms of degree d because, trivially, Ldk = (Ld)k. +Now, we let +φdk : PNd −→ PNdk = P(Symdk(W ∗)) +[F] �−→ [F k] +be the map that assigns to each form F ∈ C[x0, . . . , xn]d its k−th power. +Definition 2.3. We call the scheme theoretic image +V k +n,d = φdk(PNd) ⊆ PNdk +the (d, k)−Veronese variety. +Under the previous identification the classical Veronese varieties correspond to +the (d, 1)−Veronese varieties. On the other hand, for k > 1, V k +n,d is not a standard +Veronese variety, indeed it is easy to see that the target of φdk has dimension +�n+dk +dk +� +, +which is never equal to +�Nd+a +a +� +for any a. A priori we don’t know if the map φdk +is an isomorphism, as it happens for classical Veronese varieties, see Lemma 2.11 +below. +2.2. Secant varieties and identifiability. In this subsection we recall the defi- +nition of secant variety and the notion of identifiability, following [CM22]. +Let X ⊂ PN be a non degenerate reduced variety. Let X(h) be the h-th symmet- +ric product of X, that is the variety parameterizing unordered sets of h points of +X. Let U X +h ⊂ X(h) be the smooth locus, given by sets of h distinct smooth points. +Definition 2.4. A point z ∈ U X +h represents a set of h distinct points, say {z1, . . . , zh}. +We say that a point p ∈ PN is in the span of z, p ∈ ⟨z⟩, if it is a linear combination +of the zi’s. +With this in mind we define the following object. +Definition 2.5. The +abstract h-secant variety is the (hn + h − 1)-dimensional +variety +sech(X) := {(z, p) ∈ U X +h × PN|p ∈ ⟨z⟩} ⊂ X(h) × PN. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +5 +Let π : X(h) × PN → PN be the projection onto the second factor. The h-secant +variety is +Sech(X) := π(sech(X)) ⊂ PN, +and πX +h := π|sech(X) : sech(X) → PN is the h-secant map of X. +If the variety X is irreducible and reduced we say that X is h-defective if +dim Sech(X) < min{dim sech(X), N}. +The following is a classical result. +Theorem 2.6 (Terracini Lemma). Let X ⊂ PN be an irreducible variety. Then +the follwing holds. +• For any p1, . . . , pk ∈ X and z ∈ ⟨p1, . . . , pk⟩, we have +⟨Tp1X, . . . , TpkX⟩ ⊆ TzSeck(X). +• There is a dense open set U ⊂ X(k) such that +⟨Tp1X, . . . , TpkX⟩ = TzSeck(X), +for any general point z ∈ ⟨p1, . . . , pk⟩ with (p1, . . . , pk) ∈ U. +Notions related to that of secant variety are those of rank and of identifiability. +Definition 2.7. Let X ⊂ PN be a non degenerate subvariety. We say that a point +z ∈ PN has rank h with respect to X if z ∈ ⟨p⟩, for some p ∈ U X +h and z ̸∈ ⟨p′⟩ for +any p′ ∈ U X +h′ , with h′ < h. +Definition 2.8. A point z ∈ PN is h-identifiable with respect to X ⊂ PN if z is of +rank h and (πX +h )−1(z) is a single point. The variety X is said to be h-identifiable +if the h-secant map πX +h is birational, that is the general point of Sech(X) is h- +identifiable. +It is clear by Theorem 2.6 that when X is h-defective, or more generally when +πX +h is of fiber type, then X is not h-identifiable. +We now recall the recent result in [MM22], in which the authors generalize the +approach in [CM22] relating identifiability with the non defectivity of the secant +variety. +Theorem 2.9. Let X ⊂ PN be an irreducible and non-degenerate variety of di- +mension n, h ≥ 1 an integer, and assume that: +• (h + 1)n + h ≤ N, +• X has non degenerate Gauss map, +• X is not (h + 1)−defective. +Then X is h−identifiable. +In the next sections we will see how to translate this theorem in the setting of +powers of forms in order to give identifiability results for k−th powers of forms of +degree d. + +6 +ALEX CASAROTTI AND ELISA POSTINGHEL +2.3. Geometric construction of (d, k)−Veronese varieties. Let us recall some +facts from apolarity theory; the main reference is [Ger96]. +Notation 2.10 (Apolarity). We consider two polynomial rings in n + 1 variables, +both endowed with the standard grading: +R = C[x0, . . . , xn] = +� +i∈N +Ri := C[x0, . . . , xn]i +S = C[y0, . . . , yn] = +� +i∈N +Si := C[y0, . . . , yn]i. +Treating elements of S as partial derivatives in the xi’s, the pairing Sk × Rl → C, +sends (Fk, Gl) to the derivative Fk ◦ Gl ∈ R of Gl. If k = l and if I ⊂ R is a +homogeneous ideal, the orthogonal I⊥ +k ⊂ Sk is the following space of polynomials +I⊥ +k = {F ∈ Sk|F ◦ G = 0, ∀G ∈ Rk}. +It is a standard fact of representation theory for the linear group GL(W), see +for instance [Lan12], that the space Symk(Symd(W ∗)) can be decomposed as direct +sum of GL(W)−modules in the following way: +Symk(Symd(W ∗)) = Symdk(W ∗) ⊕ E +where +E = H0(IV d +n (k)) +is the k−th homogeneous part of the ideal of forms that vanish on the Veronese +variety V d +n = νd(Pn), cf. notation of Section 2.1. We have the following exact +sequence: +0 → IV d +n (k) → OPNd (k) → OV d +n (k) → 0 +After passing to cohomology and using the fact that every Veronese variety is +projectively normal, we have: +0 → H0(IV d +n (k)) → H0(OPNd (k)) → H0(OV d +n (k)) → 0 +This shows that +dim(E) = +�Nd + k +k +� +− +�n + dk +dk +� +. +It is easy to show that GL(V )−modules Symdk(W ∗) and E are apolar, i.e. for +every pair of forms F, G ∈ C[x0, . . . , xNd] with F ∈ Symdk(W ∗) and G ∈ E it holds +that F ◦ G = 0. +The constructions above fit into the following commutative diagram: +P(Symk(Symd(W ∗))) +πE +� +Pn +� P(Symd(W ∗)) +νk +� +φdk +� +V k +n,d ⊂ P(Symdk(W ∗)) +where the map πE is the linear projection from the linear space E. We now prove +that the map φdk is in fact an isomorphism. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +7 +Lemma 2.11. With the above notations it holds Sec2(V k +Nd) ∩ E = ∅. In particular +the map φdk is an embedding. +Proof. Note that an element F ∈ Sec2(V k +Nd) is either of the form F = Lk or F = +M k+N k, where L, M, N ∈ C[x0, . . . , xNd]1 are linear forms in P(Symk(Symd(W ∗))), +or equivalently degree d hypersurfaces in Pn. In particular if there exist such a F +with F ∈ E = H0(IVn,d(k)), then Vn,d would be contained in the vanishing locus +of F. Since the zero set of F is either an hyperplane or a union of an hyperplane +with a subscheme of degree k − 1 and Vn,d is non-degenerate and irreducible, the +claim follows. +□ +2.4. Identifiablity for (d, k)−Veronese varieties. From now on we will work +with the projective notation, in particular E has to be intended as the projec- +tivization of the affine E in the previous section. Let us start by characterizing +hyperplanes in E as particular linear subsystems of hypersurfaces in PNd. Denote +with πdk the linear projection from the linear space P(Symdk(W ∗)) to E, i.e. +πdk : P(Symk(Symd(W ∗))) ��� E +Lemma 2.12. Let H0(OPNk +d (1)⊗IP(Symdk(W ∗))) be the complete linear system of hy- +perplane sections of P(Symk(Symd(W ∗))) containing the linear space P(Symdk(W ∗)). +Then +ν∗ +k(H0(OPNk +d (1) ⊗ IP(Symdk(W ∗))) ∼= H0(OPNd (k) ⊗ IVn,d) +Proof. Let H ∈ H0(OPNd (k)⊗IVn,d) be a degree k hypersurface in PNd that contains +the Veronese Vn,d. Then the linear span H of (νk)∗(H) is an hyperplane in PN k +d +that contains Vn,dk := (νk)∗(Vn,d). Since ⟨Vn,dk⟩ = P(Symdk(W ∗)), we have that +H0(OPNd (k) ⊗ IVn,d) ⊆ ν∗ +k(H0(OPNk +d (1) ⊗ IP(Symdk(W ∗))) +To conclude just observe that +h0(OPNd (k) ⊗ IVn,d) = dim(E) = codim(P(Symdk(W ∗))), +where codim here indicates the codimension in P(Symk(Symd(W ∗))), and ν∗ +k in- +duces an isomorphism of global sections. +□ +We need the following easy technical lemma: +Lemma 2.13. The linear projection +πdk : P(Symk(Symd(W ∗))) ��� E = H0(IVn,d(k)) +is generically finite when restricted to VNd,k = νk(P(Symd(W ∗))). +Proof. The map +πdk|νk(P(Symd(W ∗))) : VNd,k ��� E +is induced by a linear subsystem F of the line bundle L = OPNk +d (1) with +ν∗ +k(F) = |OSymd(W ∗)(1) ⊗ IVn,d(k)|. +Now the claim follows easily from the fact that H0(IVn,d(k)) defines the Veronese +variety Vn,d set-theoretically. +□ + +8 +ALEX CASAROTTI AND ELISA POSTINGHEL +Let p1, . . . , ph ∈ X ⊂ PN be general points. +By Lemma 2.6, we have that +Sech(X) has the expected dimension if and only if H0(OX(1) ⊗ Ip2 +1,...,p2 +h) has the +expected dimension, i.e. +dim H0(OX(1) ⊗ Ip2 +1,...,p2 +h) = max{0, N − (n + 1)h} +Notation 2.14. If p1, . . . , ph ∈ X are general points, then we denote Lh,X := +|OX(1) ⊗ Ip2 +1,...,p2 +h|. +Before moving on to the explicit description of the identifiability for V k +n,d, let us +first prove a general proposition about linear systems of projected varieties: +Proposition 2.15. Let X ⊂ PN be a smooth non-degenerate projective variety. +Moreover let PN = ⟨F, E⟩ with F, E skew linear subspaces. +Let πE : PN → F +and πF : PN → E be the natural projections. If the projections restricted to X +are generically finite and Lh,X has the expected dimension, then Lh,πF (X) has the +expected dimension if and only if Lh,πE(X) has the expected dimension. +Proof. Note that by symmetry of E and F it suffices to prove only one of the +implications. Let q1, . . . , qh be general points on πE(X) and call pi = π−1 +E (qi) and +zi = πF (pi). Since πE|X : X �→ πE(X) is an isomorphism we have that x1, . . . , xh +are general and so also z1, . . . , zh. +Since πFX is generically finite the space of +hyperplanes Lh,πF (X) = |OE(1) ⊗ Iz2 +1,...,z2 +h| correspond to Lh,X ⊗ IF . +Now the +splitting PN = ⟨F, E⟩ induces the linear projection +π : Lh,X �→ Lh,X ⊗ IE +such that Ker(π) = Lh,πF (X). Since by assumption both the source and the kernel +have the expected dimension by the rank-nullity theorem the assertion follows. +□ +We are finally able to characterize the identifiability properties for the case of +powers of forms. Consider the following linear system of all hypersurfaces of PNd +of degree k containing the Veronese variety Vn,d ⊂ PNd and double at the points +p1, . . . , ph that lie in general position in PNd: +LNd(Vn,d, 2h) := OPNd (k) ⊗ IVn,d ⊗ Ip2 +1,...,p2 +h. +Proposition 2.16. In the above notation, let p1, . . . , ph be general points in PNd = +P(Symd(W ∗)). The linear system LNd(Vn,d, 2h) has the expected dimension if and +only if Sech(V k +n,d) has the expected dimension. +Proof. With the notations of Proposition 2.15 we have E = H0(IVn,d(k)) and F = +Symdk(W ∗). +We have that πE restricted to X = VNd,k is an isomorphism by +Lemma 2.11, in particular it is generically finite. The same holds for πF = πdk by +Lemma 2.13. Now Lh,VNd,k has the expected dimension by Alexander-Hirschowitz +(see Theorem 4.1 below) and +Lh,πdk(VNd,k) = |OPNd (k) ⊗ IVn,d ⊗ Ip2 +1,...,p2 +h| +by Lemma 2.12. +□ +3. Degeneration techniques +In this section we will discuss two types of degenerations that will provide main +tools for the proofs of the results of this article. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +9 +3.1. The FP−degeneration. In this section we recall a degeneration procedure +introduced in [Pos12], which consists in degenerating the projective space PN to +a reducible variety with two components, and then studying degenerations of line +bundles on the general fibre. +3.1.1. Degenerating the ambient space. Let ∆ be a complex disc centred at the +origin and consider the product Y = PN ×∆ with the natural projections πY +1 : Y → +PN and πY +2 : Y → ∆. The second projection is a flat morphism and we denote by +Yt := PN ×{t} the fibre over t ∈ ∆. We will refer to Y0 and to Yt, with t ̸= 0, as the +central fibre and the general fibre respectively. Let f : X → Y denote the blow-up +of Y at a point (p, 0) ∈ Y0 in the central fibre. Consider the following diagram, +where πX +i := πY +i ◦ f, for i = 1, 2: +X +f +� +πX +2 +� +πX +1 +� +Y +πY +1 +� +πY +2 +� +PN +∆ +The morphism πX +2 : X → ∆ is flat with fibres denoted by Xt, t ∈ ∆. For the general +fibre we have Xt ∼= Yt = PN, while the central fibre X0 is the reduced union of +the strict transform of Y0, that we shall denote with F, and the exceptional divisor +P ∼= PN of f. The two components P and F meet transversally and we will denote +by R the intersection: R := F ∩ P ∼= PN−1. We will say that PN degenerates to +X0 = P ∪ F. +We will now endow the general fibre Xt with a line bundle and we will describe +its limits on X0 via this degeneration. In order to do so, we will give bases for the +Picard groups of the components of X0. +Notation 3.1. We denote by HP the hyperplane class of P, so that the Picard +group of the exceptional component is generated by HP. Moreover we denote with +HF the hyperplane class of F, pull-back of a general hyperplane of Y0 ∼= PN, and +with E := P|F the exceptional class in F: HF and E generate the Picard group of F. +In these bases, R has class HP in N1(P) and E in N1(F). A line bundle on X0 +will correspond to a line bundle on P and a line bundle on F, which match on the +intersection R. In other terms, we can describe the Picard group of X0 as a fibre +product +Pic(X0) = Pic(P) ×Pic(R) Pic(F). +Consider the line bundle OX (k) = (πX +1 )∗(OPN (k)) and the following twist by a +negative multiple of the exceptional divisor: +MX (k, a) := OX (k) ⊗ OX (−aP). +The line bundle MX (k, a) will induce a line bundle on each fibre Xt by restriction: +Mt(k, a) := MX (k, a)|Xt, t ∈ ∆. +For t ̸= 0, since P ∩ Xt = ∅, we have +Mt(k, a) = OXt(k) + +10 +ALEX CASAROTTI AND ELISA POSTINGHEL +while on the components of the central fibre we have +MP(k, a) := MX (k, a)|P = OP(aHP), +MF(k, a) := MX (k, a)|F = OF(kHF − aE). +The resulting line bundle on X0 is a flat limit of the bundle OXt(k) ∼= OPn(k), for +t → 0. +3.1.2. Degenerating the Veronese variety. We will use the same notation as in Sec- +tion 3.1.1. Let us set +N := Nd = +�n + d +n +� +− 1, +and let V := Vn,d = vd(Pn) ⊂ PN denote the d-th Veronese embedding of Pn in PN, +and consider the 1-parameter family V = V × ∆ ⊂ Y with the natural projections +πY +1 |V : V → PN and πY +2 |V : V → ∆. The second projection is a flat morphism +and we denote by Vt := V × {t} the fibre over t ∈ ∆. We pick a general point +(p, 0) ∈ V0 ⊂ Y0 in the central fibre of πY +2 : Y → ∆ supported on the Veronese +variety and we consider the blow-up f : X → Y at (p, 0). This induces the blow-up +f|V : �V → V of V at (p, 0) and the fibres of (πY +2 ◦ f)|�V are as follows: the general +fibre is a Veronese variety +�Vt ∼= V, +while the central fibre is the reduced union of two components, +�V0 = �VF ∪ Λ, +where �VF is the strict transform of V0 under the blow-up at p, while Λ ∼= Pn is the +exceptional divisor on �V . Moreover we can write +ΛR := �VF ∩ Λ ⊂ Λ, +and observe that ΛR ∼= Pn−1. +Consider the line bundle MX (k, a) and twist it by the ideal sheaf of �V: +MX (k, a; �V) := MX (k, a) ⊗ I�V = OX (k) ⊗ OX (−aP) ⊗ I�V. +This restricts to the following line bundles on the fibres Xt: +Mt(k, a; V ) = OXt(k) ⊗ IVt, t ∈ ∆ \ {0}, +MP(k, a; Λ) = OP(aHP) ⊗ IΛ, +MF(k, a; �V ) = OF(kHF − aE) ⊗ I�V . +The resulting line bundle on X0 is a flat limit of the bundle OPn(k) ⊗ IV on the +general fibre. +3.1.3. Degenerating a collection of points in general position. We continue to use +the notation of Sections 3.1.1-3.1.2. On the general fibre of Y → ∆, i.e. for t ̸= 0, +we consider a collection of points {x1,t . . . , xh,t} ⊂ Yt in general position and that, +in particular, lie off the Veronese Variety Vt ⊂ Yt. After choosing a point that +lies generically on the Veronese in the central fibre, p ∈ V0 ⊂ Y0, we degenerate +each point x1,t ∈ Yt to an infinitely near point to p ∈ V0 as follows. For every +i ∈ {1, . . . , h}, consider the curve (πY +1 )−1(xi) ⊂ Y and its pull-back Ci on X. The + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +11 +union � +i Ci intersects each fibre Xt transversally in h distinct points. For t ̸= 0, the +points � +i Ci ∩ Xt are in general position. Moreover, by the generality assumption, +the curves (πY +1 )−1(pi) ⊂ Y are not tangent to V0 ∈ Y0, therefore the intersection +points Ci ∩ X0 lie on P but not inside Λ. +Consider the blow-up of X along the union of curves �h +i=1 Ct, gh : � +X → X, with +exceptional divisors ECi. Since these curves are disjoint, the result does not depend +of the order of blow-up. The general fibre, that we continue to call Xt by abuse +of notation, is isomorphic to a PN blown-up at h points in general position and its +Picard group will be generated by the hyperplane class and by the classes of the +exceptional divisors: +Pic(Xt) = Z⟨H, E1,t, . . . , Eh,t⟩. +The central fibre has two components, the pull-back of F and the strict transform +of P ∼= PN, which is isomorphic to a PN blown-up at h points in general position: +abusing notation, we call F and P the two components, so that X0 = P ∪ F. The +Picard group of �P is +Pic(P) = Z⟨HP, E1,t, . . . , Eh,t⟩. +For a vector m = (m1, . . . , mh) ∈ Nn, consider the following sheaf on � +X: +M � +X (k, a; ˜V, m) := O(k) ⊗ O(−aP) ⊗ O(−(m1EC1 + · · · + mhECh)) ⊗ I�V . +It restricts to +Mt(k, a; V, m) = O(k) ⊗ O(−(m1E1,t + · · · + mhEh,t)) ⊗ IVt, t ̸= 0, +on the general fibre, and to +MP(k, a; Λ, m) = OP(aHP − (m1E1 + · · · + mhEh)) ⊗ IΛ, +MF(k, a; �V , m) = OF(kHF − aE) ⊗ I�V . +on the components of the central fibre. The resulting line bundle on X0 is a flat +limit of the bundle on the general fibre. +3.1.4. Matching conditions. We will abbreviate the notation of the previous sections +by setting +M � +X := M � +X (k, a; ˜V, m) +and +Mt := Mt(k, a; V, m) +MP := MP(k, a; Λ, m) +MF := MF(k, a; ˜V , m). +We are interested in computing the dimension of the space of global sections of the +line bundle on the central fibre, which by upper-semicontinuity is an upper bound +for the dimension of the space of global sections of the line bundle on the general +fibre: +(3.1) +h0(X0, M0) ≥ h0(Xt, Mt). + +12 +ALEX CASAROTTI AND ELISA POSTINGHEL +In order to do so, we consider the natural restrictions of to the intersection R = P∩F +of the central fibre: +0 → ˆ +MP → MP → MP|R → 0, +0 → ˆ +MF → MF → MF|R → 0, +where +ˆ +MP = +ˆ +MP(k, a; Λ, m) and +ˆ +MF = +ˆ +MF(k, a; ˜V , m) denote the kernels of the +restriction maps. Since R = HP on P and R = E on F, we have +ˆ +MP = OP((a − 1)HP − (m1E1 + · · · + mhEh) ⊗ IΛ +ˆ +MF = OF(kHF − (a + 1)E) ⊗ I ˜V . +Consider the restriction maps of global sections: +rP : H0(P, MP) → H0(R, MP|R), +rF : H0(P, MF) → H0(R, MF|R). +We notice that the spaces of global sections of the restricted systems are both +subspaces of the space of global sections of the degree-a line bundle on R ∼= PN−1: +H0(R, MP|R), H0(R, MF|R) ⊆ H0(R, OR(a)). +A global sections of M0 consists of an element of H0(P, MP) and an element of +H0(P, MP) which match in H0(R, OR(a)), i.e. the space space of global sections +H0(X0, M0) is described as a fibre product via the following commutative diagram: +H0(X0, M0) +H0(P, MP) +H0(F, MF) +H0(R, MP|R ∩ MF|R) +rP +rF +This yields the formula for the dimension of the spaces of global sections +h0(X0, M0) = h0(P, ˆ +MP) + h0(F, ˆ +MF) + h0(R, MP|R ∩ MF|R), +which, in terms of dimensions of line bundles, it reads +(3.2) +dim M0 = dim ˆ +MP + dim ˆ +MF + dim MP|R ∩ MF|R + 2. +Remark 3.2. There is an obvious isomorpshism between Mt(k, a; V, m), line bun- +dle on Xt, and the line bundle LN,k(V, m) := OPN (k) ⊗ IV ⊗ IZ on PN, where +Z is a union of fat points in general position in PN with multiplicities respectively +m1, . . . , mh. Such isomorphism is given by taking strict transforms of elements of +LN,k(V, m). +Similarly, since P is the blow-up of PN at h points in general position and Λ ⊂ +P is a general linear subspace, then there is an isomorphisms MP(k, a; Λ, m) ∼= +LN,a(Λ, m) := OPN (a) ⊗ IΛ ⊗ IZ. +Finally, since F is a PN blown-up at a point p on the Veronese variety V ⊂ +PN and since ˜V is the strict transform of V via this blow-up, then there is an +isomorphisms MF(k, a; ˜V , m) ∼= LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +13 +3.2. Toric degeneration of the Veronese variety. We refer to [Ful93] for details +on projective toric varieties associated with convex lattice polytopes and to [GKZ94] +for details on coherent triangulations. Let ∆n ⊂ Rn be the n-dimensional simplex +obtained as the convex hull of the points (0, . . . , 0), (1, 0, . . . , 0), . . . , (0, . . . , 0, 1). +Consider d∆n = ∆N + · · · + ∆N, where + here denotes the Minkowski sum of +polytopes in Rn. The polytope d∆n defines the d-th Veronese embedding of Pn +in PN, with N = Nd = +�N+d +N +� +− 1, that we shall call V = Vn,d as in the previous +section. Consider the lattice Zn ⊂ Rn and the set of lattice points A := d∆n ∩ Zn. +We have ♯A = N + 1 and each such point corresponds to a coordinate point of the +ambient space PN. +3.2.1. Degenerating the Veronese to a union of linear spaces. Take a regular trian- +gulation of d∆n, that is a decomposition of d∆n into a finite union of simplices +dn +� +i=1 +Si, +where +• each Si is obtained as the convex hull of n + 1 non-aligned points of A, +• ♯Si ∩ Zn = n + 1, +• for i ̸= j, Si ∩ Sj is a common faces of Si and Sj (possibly empty), +• there is a strictly convex piecewise linear function λ : Rn → R whose +domains of linearity are precisely the Si’s. +We can always assume that S1 is the convex hull of the lattice points (0, . . . , 0), +(1, 0, . . . , 0), . . . , (0, . . . , 0, 1), so that it lies at a corner of d∆n. Consider for ex- +ample, for n = 2 and d = 3, the triangulation into 9 and the piecewise linear +function inducing shown in Figure 1. In this figure, S1 is the triangle with vertices +@ +@ +@ +@ +@ +@@ +@ +@ +@ +@@ +@ +@ +@ +Figure 1. A regular triangulation of 3∆2. +(0, 0), (1, 0), (0, 1). Each Si defines a Pn as a toric variety, which we will call Πi, +for i = 1, . . . , dn. Since a regular triangulation of d∆n induces a 1-parameter em- +bedded degeneration of Vn,d ⊂ PN to the union of toric varieties described by the +Si’s (see [CDM09] and [Pos13] for details on the 2-dimensional case), we have a +degeneration of the Veronese variety to a union of dn n-planes +Π := +dn +� +i=1 +Πi. +The intersection table of these planes is encoded in the combinatorial data described +by the triangulation, that is: if Si ∩ Sj is r-dimensional, then Πi ∩ Πj ∼= Pr, for +0 ≤ r ≤ n − 1. + +14 +ALEX CASAROTTI AND ELISA POSTINGHEL +Remark 3.3. Because of the choice of S1 made, we will say that Π1 is a sink. In +practice this means that it is possible to choose a hyperplane of PN that contains +every Si, i > 1, but that does not contain S1. +Moreover, the union of planes Π ⊂ PN is a torus invariant subscheme. In fact, +consider the simplex ∆N, which defines PN as a toric variety, with an action of +the algebraic torus (C∗)N. Each r-dimensional face of ∆N corresponds to a torus +invariant linear subspace of dimension r of PN. In particular vertices of ∆N are in +one-to-one correspondence with N + 1 linearly independent points, which we may +assume to be the coordinate points of PN, up to a change of coordinates. Each +r-dimensional face of ∆N corresponds to a Pr spanned by r + 1 coordinate points. +Since each Πi is the linear span of n + 1 coordinate points of PN, then the union +Π is embedded in a copy of PN and it is invariant under the action of the torus +(C∗)N. In particular, each Πi will correspond to a marked n-dimensional face of +∆N and we have dn such marked faces. +3.2.2. Degenerating a linear system intepolating the Veronese. We now consider the +linear systems on PN of degree−k hypersurfaces containing the Veronese variety on +the one hand, and the union of n−planes Π on the other hand: +LN,k(Vn,d) := OPN (k) ⊗ IVn,d, +LN,k(Π) := OPN (k) ⊗ IΠ +Lemma 3.4. In the above notation, we have +dim LN,k(Vn,d) ≤ dim LN,k(Π). +Proof. Since Π is a flat degeneration of Vn,d, then the statement follows by semi- +continuity of the function dim. +□ +4. Some auxiliary linear systems +4.1. Hypersurfaces containing a linear subspace and h double points in +general position. It is a well celebrated result of Alexander and Hirschowitz that +if we impose h double points in general position to the hypersurfaces of degree d +of PN, there is only a finite list of cases where the dimension is larger than that +obtained via a parameter count, i.e. +edim LN,d(2h) = max +� +−1, +�N + d +N +� +− h(N + 1) − 1 +� +. +Theorem 4.1 (Alexander-Hirschowitz Theorem). The linear system LN,d(2h) is +non-special except in the following cases: +• d = 2 and N ≥ 2, 2 ≤ h ≤ N; +• d = 3 and (N, h) = (4, 7); +• d = 4 and (N, h) = (2, 5), (3, 9), (4, 14). +The interested reader may see [Ale88],[AH92b],[AH92],[AH95],[AH97]] for the +original proof based on specialisation of points (Horace method), and [BO08] and +[Cha05] for a simplified proof. An alternative proof via a different degeneration +construction can be found here [Pos10,Pos12]. This inspired the degeneration ap- +proach developed in Section 3.1 that will be used to prove the main result, Theorem +1.1. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +15 +In this section we want to present an analogous result about linear systems with +h imposed double points in general position and a linear subspace. Let Λ ⊂ PN be +a general linear subspace of dimension n and let Z ⊂ Pn be a double point scheme +with support a set of points in general position. Let IΛ be the ideal sheaf of Λ ⊂ PN +and let IZ be the ideal sheaf of Z ⊂ PN. Consider the sheaf +LN,k(Λ, 2h) := OPN (k) ⊗ IΛ ⊗ IZ. +Since the Hilbert polynomial of Λ ⊂ PN at degree k is +�n + k +n +� +or, in other terms, +(4.1) +h0(OPN (k) ⊗ IΛ) = +�N + k +N +� +− +�n + k +n +� +, +and since h double points in general position of PN impose h(N + 1) conditions to +the hypersurfaces of PN of degree k, we can give the following definitions. +Definition 4.2. The virtual dimension of the linear system LN,k,Λ(2h) of hyper- +surfaces of Pn that vanish along a linear subspace of dimension n, Λ ⊂ PN, and +double at h points in general position is the following integer: +vdim LN,k(Λ, 2h) = +�N + k +N +� +− +�n + k +n +� +− h(N + 1) − 1. +The expected dimension of LN,k(Λ, 2h) is +edim LN,k(Λ, 2h) = max +� +−1, vdim LN,k(Λ, 2h) +� +. +Since Λ and the scheme of double points are disjoint the virtual dimension, +which is obtained by a simple parameter count, provides a lower bound to the +actual dimension: +(4.2) +dim LN,k(Λ, 2h) ≥ edim LN,k(Λ, 2h). +Proposition 4.3. Let Λ ⊂ PN be linear subspace of dimension n and let ZΛ ⊂ Pn +be a double point scheme supported on a collection of points in general position in +PN. Then if +(4.3) +h ≤ +1 +N + 1 +�N + k − 1 +N +� +, +and (N, k − 1, h) is not in the list of exceptions of Theorem 4.1, and k ≥ 2, then +(4.4) +dim LN,k(Λ, 2h) = edim LN,k(Λ, 2h). +Proof. If k = 2 and h = 0, the conclusion follows from (4.1). If k = 2 and h = 1, it +is easy to see that all elements of LN,2(Λ, 2) are pointed quadric cones containing +Λ and hence we have the isomorphism LN,2(Λ, 2) ∼= LN−1,2(Λ). By (4.1), we have +that dim LN−1,2(Λ) = +�N+1 +2 +� +− +�n+2 +2 +� +. We conclude noticing that the latter equals +the expected dimension of LN,2(Λ, 2). +Now, assume k ≥ 4 and consider the following exact sequence obtained by re- +stricting LN,k(Λ, 2h) to a general hyperplane H ⊂ PN such that Λ ⊆ H: +(4.5) +0 → LN,k−1(2h) → LN,k(Λ, 2h) → LN,k(Λ, 2h)|H ⊆ LN−1,k(Λ). + +16 +ALEX CASAROTTI AND ELISA POSTINGHEL +Under the assumption (4.3) and using Theorem 4.1, the kernel system LN,k−1(2h) +has dimension equal to its virtual dimension: +dim LN,k−1(2h) = +�N + k − 1 +N +� +− h(N + 1) − 1, +and in particular H1(PN, dim LN,k−1(2h)) = 0, so that we have the following exact +sequence in cohomology: +0 → H0(PN, LN,k−1(2h)) → H0(PN, LN,k(Λ, 2h)) → H0(H, LN,k(Λ, 2h)|H) → 0. +Moreover by (4.1) +h0(PN−1, LN−1,k(Λ)) = +�N − 1 + k +N − 1 +� +− +�n + k +n +� +, +and so +h0(H, LN,k(Λ, 2h)|H) ≤ +�N − 1 + k +N − 1 +� +− +�n + k +n +� +. +From the exact sequence of global sections we obtain: +h0(PN, LN,k(Λ, 2h)) = h0(PN, LN,k−1(2h)) + h0(H, LN,k(Λ, 2h)|H) +≤ +�N + k +N +� +− +�n + k +n +� +− h(N + 1). +We conclude the proof of this case using (4.2). +Finally, assume that k = 3. In this case, the bound on the number of points is +h ≤ N +2 + 1. We consider the restriction to a general hyperplane containing Λ as in +(4.5). The kernel system is special by Theorem 4.1, and one can easily check that +it has dimension +dim LN,2(2h) = +�N + 2 +2 +� +− h(N + 1) + +�h +2 +� +− 1, +see for instance [Pos10, Section 1.2.1]. +Moreover, as a simple consequence of +B´ezout’s Theorem, the linear system LN,3(Λ, 2h) contains in its base locus the lines +spanned by pairs of points, each of which intersects H in a point. We claim that +the base locus of LN,3(Λ, 2h) is supported on the union of Λ and these lines. This +implies that the restricted system is the complete linear system of cubics containing +Λ and passing simply through the +�h +2 +� +trace points: +LN,3(Λ, 2h)|H = LN−1,3(Λ, 1( +h +2)). +We claim that the linear system on the right hand side of the above expression +is non-special, namely that the scheme given by Λ and the simple points impose +independent conditions to the cubics of PN−1. This shows that +dim LN,3(Λ, 2h) = dim LN,2(2h) + dim LN−1,3(Λ, 1( +h +2)) + 1 += +��N + 2 +2 +� +− h(N + 1) + +�h +2 +� +− 1 +� ++ +��N + 2 +3 +� +− +�n + 3 +3 +� +− +�h +2 +� +− 1 +� ++ 1, +which implies that LN,3(Λ, 2h) is non-special. +We are left to proving the two claims. For the second claim, first of all notice +that there is a hyperplane inside H, containing all +�h +2 +� +points. This can be taken to + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +17 +be the intersection with H of a hyperplane of PN containing the h original points. +Call H1 ⊂ H the intersection and restrict the linear system LN−1,3(Λ, 1( +h +2)) to it, +giving rise to the following exact sequence: +0 → LN−1,2(Λ) → LN−1,3(Λ, 1( +h +2)) → LN−2,3(1( +h +2)). +Since the two external linear systems are non-special, the so is the middle one, +concluding the proof of the claim. +As for the first claim: we show that LN,3(Λ, 2h) has no additional base locus +other than Λ and the lines spanned by pairs of points. Let’s call p1, . . . , ph the h +assigned pints in general position. Assume that q is a point in PN in linearly general +position with respect to p1, . . . , ph. Since h ≤ N +2 + 1 < N, there is a hyperplane +A containing p1, . . . , ph but not containing q. Since n < N, there is a hyperplane +B containing Λ but not containing q. The cubic 2A + B belongs to LN,3(Λ, 2h) +proving that q cannot be a base point. Assume now that q is a point in PN not +in linearly general position with respect to p1, . . . , ph, which means that there is +a linear space spanned by some of the pi’s containing q, but such that q does not +belong to any of the lines ⟨pi, pj⟩. Let ⟨pi : i ∈ Iq⟩ be the minimum such linear +span and choose two distinct indices j1, j2 ∈ Iq. Let A1 be a hyperplane containing +all pi’s with i ̸= j1 and let A2 be a hyperplane containing all pi’s with i ̸= j2. +Let B be a hyperplabe containing Λ, pi1 and pj2 and not containing q. The cubic +A1 + A2 + B belongs to LN,3(Λ, 2h) proving that q cannot be a base point. Finally, +since the multiplicity of the general element of the linear system of cubics along +the line q ∈ ⟨pi : i ∈ Iq⟩ is exactly 1, then the above cases are exhaustive and this +conclude the proof of the claim. +□ +4.2. Hypersurfaces containing the Veronese variety and a fat point. Let +V = Vn,d ⊂ Pn be the d-th Veronese embedding of Pn and let {pa} ⊂ V ⊂ Pn be a +fat point scheme with support on V . Let IV be the ideal sheaf of V ⊂ PN and let +Ipa be the ideal sheaf of Z ⊂ PN. Consider the sheaf +LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa. +We are interested in computing the dimension of the space of global sections. The +Hilbert polynomial of V ⊂ PN at degree k is +�n + kd +n +� +or, equivalently, we have +dim OPN (k) ⊗ IV = +�N + k +N +� +− +�n + kd +n +� +− 1. +The scheme given by a point of multiplicity a of PN imposes +(4.6) +�N + a − 1 +N +� +conditions to the hypersurfaces of PN of degree k. Therefore the virtual dimension +of LN,k(V, a), obtained by a parameter count, is +�N + k +N +� +− +�n + kd +n +� +− +�N + a − 1 +N +� +− 1. + +18 +ALEX CASAROTTI AND ELISA POSTINGHEL +It does not yield a useful notion of expected dimension for the linear system +LN,k(V, a) due to the fact that the two subschemes V and {pa} of PN have nonempty +intersection so that some of the conditions imposed by them individually to the hy- +persurfaces of degree k of PN will overlap. For instance, if we first impose V and +then {p}, clearly the latter will not give any independent condition, because, by +the containment relation p ∈ V , p is a base point of the linear system LN,k(V ) = +OPN (k) ⊗ IV . When the support of a fat point subscheme Z = {pa} ⊂ PN, whose +length is given in (4.6), lies on the n-dimensional subvariety V , the restriction +Z|V ⊂ V is a subscheme of length +�n + a − 1 +n +� +. +Therefore we may define the following notion of expected dimension. +4.2.1. A notion of expected dimension. We introduce the following refined param- +eter count. +Definition 4.4. Let V ⊂ PN be the d-th Veronese embedding of PN and let {pa} ⊂ +PN be a fat point scheme supported on V . The expected dimension of LN,k(V, a), +denoted by edim LN,k(V, a), is the following integer: +max +� +−1, +�N + k +N +� +− +�n + kd +n +� +− +��N + a − 1 +N +� +− +�n + a − 1 +n +�� +− 1 +� +. +That the integer of Definition 4.4 is a lower bound to the actual dimension of +LN,k(V, a) is not an obvious statement. We will show that it does when a ≤ k. +Proposition 4.5. Let νd : Pn → PN be the d-the Veronese embedding. Let V = +Vn,d := νd(Pn) ⊂ PN and let ZV = {pa} ⊂ Pn be a fat point of multiplicity a ≤ k +supported on V . Then +(4.7) +dim LN,k(V, a) ≥ edim LN,k(V, a). +Proof. We consider the linear system LN,k(a) = OPN ⊗ IZV of the degree-k hyper- +surfaces of PN with a point of multiplicity a with support on V . Restriction to V +gives the following Castelnuovo sequence: +0 → LN,k(V, a) → LN,k(a) → LN,k(a)|V . +It is an easy observation that a fat point of multiplicity a imposes independent +conditions to the hypersurfaces of fixed degree of PN, as long as the multiplicity +does not exceed the degree. Therefore we can obtain the dimension of the linear +system LN,k(a) by a parameter count: +dim LN,k,V (a) = +�N + k +N +� +− +�N + a − 1 +N +� +− 1 +In particular h1(PN, LN,k(a)) = 0, so that we have the following sequence in coho- +mology: +0 → H0(LN,k(V, a)) → H0(LN,k(a)) → H0(LN,k(a)|V ) +→ H1(LN,k(V, a)) → 0. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +19 +Since the Veronese morphism νd : Pn → PN gives an isomorphism of Pn to its image +V , then the pull-back of LN,k(a)|V is a linear system of degree-kd hypersurfaces of +Pn: +ν∗ +d(LN,k(a)|V ) ⊆ OPn(kd) ⊗ IZ′ =: Ln,kd(a) +where Z′ ⊂ Pn is a fat point of multiplicity a with support a general point of V . +Since nk ≥ a by the assumption, the linear system Ln,kd(a) has dimension +dim Ln,kd(a) = +�n + kd +n +� +− +�n + a − 1 +n +� +− 1. +From this we obtain +dim LN,k(a)|V ≤ +�n + kd +n +� +− +�n + a − 1 +n +� +− 1. +Putting everything together: +h0(LN,k(V, a)) = h0(LN,k(a)) − h0(LN,k(a)|V ) + h1(LN,k(V, a)) +≥ h0(LN,k(a)) − h0(LN,k(a)|V ) +≥ +��N + k +N +� +− +�N + a − 1 +N +�� +− +��n + kd +n +� +− +�n + a − 1 +n +�� +, +which concludes the proof. +□ +4.2.2. Dimensionality via apolarity and toric geometry. Let V = Vn,d ⊂ PN be the +Veronese variety and let Π ⊂ PN be a union of n-planes, degeneration of V , as in +Section 3.2. Let p ∈ V and p0 ∈ Π1 ⊂ Π be a general points. Consider the linear +systems on PN +LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa, +LN,k(Π, a) := OPN (k) ⊗ Iv ⊗ Ipa +0. +Building from Lemma 3.4, we obtain the following result. +Proposition 4.6. In the above notation, we have +dim LN,k(V, a) ≤ dim LN,k(Π, a). +Proof. Since p is a general point on V , we may assume that it degenerates to a +general point p0 ∈ S1. Since Π∪{pa +0} is a flat degeneration of the scheme V ∪{pa}, +then the Hilbert functions of the former is at most that of latter, by semi-continuity. +This concludes the proof. +□ +Proposition 4.7. In the above notation and for any 1 ≤ a ≤ k, then +dim LN,k(Π, a) = +�N + k +N +� +− +�n + kd +n +� +− +�N + a − 1 +N +� ++ +�n + a − 1 +n +� +− 1. +Proof. Given the union Π := �dn +i=1 Πi of torus invariant n-planes of Pn, with Π1 a +sink and p0 supported generically on Π1, there is a torus invariant hyperplane H +such that Π1 ∩ H is an (n − 1)-plane and Πi ⊂ H for 2 ≤ i ≤ dn (cf. Remark 3.3). +We can always assume that p0 is a coordinare point of PN and we can call p1, . . . , pN +the other coordinate (torus invariant) points of PN. Hence we can choose, without +loss of generality, that Π1 = ⟨p0, . . . , pn⟩ and H = ⟨p1, . . . , pN⟩, so that p0 ∈ Π1 +and pi /∈ Πi for i ≥ 2. + +20 +ALEX CASAROTTI AND ELISA POSTINGHEL +Let R = C[x0, . . . , xN] be the homogeneous polynomial ring of PN and consider +the ideals Ip0 ⊂ C[x0, . . . , xN] and IΠi ⊂ C[x0, . . . , xN] +Ip0 = ⟨x1, . . . , nN⟩, +IΠ1 = ⟨xn+1, . . . , nN⟩, +IΠi = ⟨xin+1, . . . , niN ⟩, i ≥ 2, +IH = ⟨x0⟩. +By construction, for i ≥ 2 , we have 0 ∈ {in+1, . . . , iN}. Using Notation 2.10, we +compute: +� +I−1 +pa +0 +� +k = {yk−l +0 +Fl(y1, . . . , yN) : Fl ∈ Sl, 0 ≤ l ≤ a − 1}, +� +I−1 +Π1 +� +k = {Fk(y0, . . . , yn) : Fk ∈ Sk}, +� +I−1 +Πi +� +k = {Fk(yi0, . . . , yin) : Fk ∈ Sk}, i ≥ 2, +where for i ≥ 2, the index set {i0, . . . , in} is the complement of {in+1, . . . , iN} ⊂ +{0, . . . , N}. We have the following intersections +� +I−1 +pa +0 +� +k ∩ +� +I−1 +Πi +� +k = ∅, i ≥ 2, +� +I−1 +pa +0 +� +k ∩ +� +I−1 +Π1 +� +k = {yk−l +0 +Fl(y1, . . . , yn) : Fl ∈ Sl}. +We compute the dimension of the latter intersection: +dim +� +I−1 +pa +0 +� +k ∩ +� +I−1 +Π1 +� +k = dim{yk−l +0 +Fl(y1, . . . , yn) : Fl ∈ Sl, 0 ≤ l ≤ a − 1} += +a−1 +� +l=0 +dim{Fl(y1, . . . , yn) : Fl ∈ Sl} += +a−1 +� +l=0 +�n − 1 + l +n − 1 +� += +�n + a − 1 +n +� +, +where the last equality follows a standard relation of Newton coefficients, commonly +known as the hockey stick identity. +The number of conditions imposed to the linear system of degree-k hypersurfaces +of PN by the scheme {pa +0} ∪ Π is the dimension of the linear span of +� +I−1 +pa +0 +� +k and +� +I−1 +Πi +� +k, for i = 1 . . . , dn, which is the following integer: +dim +� +I−1 +pa +0 +� +k + dim +�� +I−1 +Πi +� +k , i = 1 . . . , dn� +− dim +� +I−1 +qa +0 +� +k ∩ +� +I−1 +Π1 +� +k += +�N + a − 1 +N +� ++ +�n + kd +n +� +− +�n + a − 1 +n +� +. +□ +Corollary 4.8. The linear system dim LN,k(V, a) has the expected dimension ac- +cording to Definition 4.4. +Proof. It follows from Propositions 4.5 and 4.7. +□ + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +21 +5. Proof of the main theorem +We are ready to prove our main theorem, Theorem 1.1. Thanks to Proposition +2.16, computing the dimension of the h-secant varieties of the (d, k)-Veronese variety +V k +n,d ⊂ PNdk is equivalent to computing the dimension of the linear system in PN +of all hypersurfaces containing the standard Veronese variety V = Vn,d ⊂ PN and +double at h general points. +Let Vn,d ⊂ PN be the d-th Veronese embedding of Pn and let Z ⊂ Pn be a double +point scheme with support a set of points in general position. Let IV be the ideal +sheaf of V ⊂ PN and let IZ be the ideal sheaf of Z ⊂ PN. Consider the sheaf +LN,k(V, 2h) := OPN (k) ⊗ IV ⊗ IZ. +Since the Hilbert polynomial of V ⊂ PN in degree k is +�n+kd +n +� +or, in other terms, +(5.1) +dim OPN (k) ⊗ IV = +�N + k +N +� +− +�n + kd +n +� +− 1, +and since h double points in general position of PN impose h(N + 1) conditions to +the hypersurfaces of PN of degree k, we can give the following definitions. +Definition 5.1. The virtual dimension of the linear system LN,k,V (2h) of hyper- +surfaces of Pn that vanish along the Veronese variety V = Vn,d ⊂ PN and double +at h points in general position is the following integer: +vdim LN,k(V, 2h) = +�N + k +N +� +− +�n + kd +n +� +− h(N + 1) − 1. +The expected dimension is +edim LN,k(V, 2h) = max +� +−1, vdim LN,k(V, 2h) +� +. +Since V and the scheme of double points are disjoint, the virtual dimension, +which is obtained by a simple parameter count, provides a lower bound to the +actual dimension: +(5.2) +dim LN,k(V, 2h) ≥ edim LN,k(V, 2h). +Using a degeneration argument, we shall show that if the number of points h +is not too large, then the linear system LN,k,V (V, 2h) has dimension equal to the +expected dimension. +Theorem 5.2. Let νd : Pn → PN be the d-the Veronese embedding. +Let V = +Vn,d := νd(Pn) ⊂ PN and let ZV ⊂ Pn be a double point scheme supported on h +points in general position of PN. Then if k ≥ 3 and +(5.3) +h ≤ +1 +N + 1 +�N + k − 3 +N +� +then +(5.4) +dim LN,k(V, 2h) = edim LN,k(V, 2h). +Proof. Using (5.2), it is enough to prove that the inequality dim LN,k(V, 2h) ≤ +edim LN,k(V, 2h) holds. + +22 +ALEX CASAROTTI AND ELISA POSTINGHEL +If k = 3, then h = 0 so the statement follows from (5.1). For k ≥ 4, we will +prove the statement by means of the FP−degeneration introducedin Section 3.1.2, +applied to the line bundle +L � +X :=M � +X (k, k − 1; �V, 2, . . . , 2). +By Remark 3.2, the line bundle on the general fibre is isomorphic to +Lt :=LN,k(V, 2h), +while on the central fibre the linear systems on the two components are the follow- +ing: +LP :=LN,k−1(Λ, 2h), +LF :=LN,k(V, k − 1). +We consider the restriction to R = P ∩ F: the kernels on the two components are, +respectively: +ˆLP :=LN,k−2(Λ, 2h), +ˆLF :=LN,k(V, k). +Since R ∼= PN−1, the two restricted systems satisfy the following: +RP :=LP|R ⊂ LN−1,k−1(ΛR), +RF :=LF|R ⊂ LN−1,k−1(ΛR), +where we recall that ΛR = Λ ∩ R ∼= Pn−1. +We first look at the exceptional component P. By Proposition 4.3, since k−2 ≥ 2 +and +h ≤ +1 +N + 1 +�N + k − 3 +N +� +, +both linear systems LP and ˆLP have the expected dimension, that is +dim LP = +�N + k − 1 +N +� +− +�n + k − 1 +n +� +− h(N + 1) − 1 +dim ˆLP = +�N + k − 2 +N +� +− +�n + k − 2 +n +� +− h(N + 1) − 1. +(5.5) +Moreover, we have a short exact sequence of spaces of global sections: +0 → H0(P, ˆLP) → H0(P, LP) → H0(R, RP) → 0. +In particular, we can compute +dim RP = dim LP − dim ˆLP + 1 += +�N + k − 2 +N − 1 +� +− +�n + k − 2 +n − 1 +� +− 1 += dim LN−1,k−1(ΛR). +We conclude that RP is the complete linear system +RP = LN−1,k−1(ΛR). + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +23 +On the component F, using Corollary 4.8, we have that both LF and ˆLF have +the expected dimension, that is +dim LF = +�N + k +N +� +− +�n + kd +n +� +− +��N + k − 2 +N +� +− +�n + k − 2 +n +�� +− 1 +dim ˆLF = +�N + k +N +� +− +�n + kd +n +� +− +��N + k − 1 +N +� +− +�n + k − 1 +n +�� +− 1. +(5.6) +We claim that +RF = LN−1,k−1(ΛR), +so that, together with the above argument, we have +(5.7) +RP ∩ RF = OPN−1(k − 1) ⊗ IΛR. +In order to prove the claim, we observe that by semicontinuity, and precisely For- +mula (3.1), and by (5.2), we have +(5.8) +dim(L0) ≥ dim(Lt) ≥ edim Lt = +�N + k +N +� +− +�n + kd +n +� +− h(N + 1) − 1. +Using Formula (3.2), i.e., +dim L0 = dim ˆLP + dim ˆLF + RP ∩ RF + 2 +(5.9) +and observing that RP ∩ RF = RF, we obtain +dim RF ≥ edim Lt − dim ˆLP − dim ˆLF − 2 = +�N + k − 2 +N − 1 +� +− +�n + k − 2 +n − 1 +� +− 1; +the proof of the latter equality is easy and left to the reader. Since +dim RF ≤ dim OPN−1(k − 1) ⊗ IΛR = +�N + k − 2 +N − 1 +� +− +�n + k − 2 +n − 1 +� +− 1, +the claim follows and we have +(5.10) +dim RP ∩ RF = +�N + k − 2 +N − 1 +� +− +�n + k − 2 +n − 1 +� +− 1, +Using (5.5), (5.6), (5.9) and (5.10), we obtain dim L0 = edim Lt. We conclude using +(5.8). +□ +Theorem 1.1 is now just a corollary of what we just proved. +Corollary 5.3. For k ≥ 3, if (5.3) holds, then the (d, k)−Veronese variety V k +n,d ⊂ +PNdk is non-defective. +Proof. It follows from Theorem 5.2 and Proposition 2.16. +□ +Theorem 1.2 is an easy consequence of Theorem 1.1 and Theorem 2.9. +Corollary 5.4. For k ≥ 3, if h < +1 +N+1 +�N+k−3 +N +� +, then holds, then the (d, k)−Veronese +variety V k +n,d ⊂ PNdk is h-identifiable. +Proof. It follows from Corollary 5.3 and Theorem 2.9. +□ + +24 +ALEX CASAROTTI AND ELISA POSTINGHEL +6. Asymptotical Bound +In this section we relate our bound in Theorem 1.1 with the one given in [Nen17]. +We first of all state Nenashev’s result for the sake of completeness. +Theorem 6.1. [Nen17, Theorem 1] Let I be a homogeneous ideal generated by +h ∈ N0 generic elements of some nonempty variety D ⊆ Symr(Cn) of r-forms that +is closed under linear transformations. Fix an integer s ≥ 0. If +h ≤ +��r + s + n − 1 +n − 1 +� +/ +�s + n − 1 +n − 1 +�� +− +�s + n − 1 +n − 1 +� +then the dimension of I in degree (r + s) is maximal, i.e. it equal s h +�s+n−1 +n−1 +� +. +Note that when r = d(k − 1) and s = r the degree r + s = dk component of I +gives us exactly the dimension of the h− secant variety Sech(V k +n,d), where D is the +tangential variety of V k +n,d, i.e. +D = {F k−1G|F, G ∈ C[x0, . . . , xn]d}. +As a consequence we have the following. +Corollary 6.2. The dimension of Sech(V k +n,d) is the expected one, i.e. +dim Sech(V k +n,d) = h +�n + d +d +� +− 1 +for h ≤ ( +n+dk +dk ) +( +n+d +d ) − +�n+d +d +� +. +Note that if we fix k, n and let d ≫ 0 we have that +�n+dk +dk +� +�n+d +d +� − +�n + d +d +� +∼ kn − dn +and if d ≫ k the bound is trivial. In Theorem 1.1 and under the same assumptions +we get +1 +N + 1 +�N + k − 3 +N +� +∼ dn(k−4) +which gives non trivial bounds for d ≫ k when k > 4. + +WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS +25 +The figure shows in red the bound given by Nenashev and in blue the bound of +Theorem 1.1 as a function of d. In this case we have set the values k = 5 and n = 2. +For d > 3 our bound is better and continues to give informations also in the range +d > 4. +References +[Ale88] J. Alexander, Singularixt´es imposables en position g´en´erale `a une hypersurface pro- +jective, Compositio Math. 68 (1988), no. 3, 305–354. ↑14 +[AH92a] J. Alexander and A. Hirschowitz, La m´ethode d’Horace ´eclat´ee: application `a +l’interpolation en degr´e quatre, Invent. Math. 107 (1992), no. 3, 585–602, DOI +10.1007/BF01231903. ↑14 +[AH92b] +, Un lemme d’Horace diff´erentiel: application aux singularit´es hyperquartiques +de P5, J. Algebraic Geom. 1 (1992), no. 3, 411–426. ↑14 +[AH95] +, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), +no. 2, 201–222. ↑14 +[AH97] +, Generic hypersurface singularities, Proc. Indian Acad. Sci. Math. Sci. 107 +(1997), no. 2, 139–154. ↑14 +[BCMO23] A. T. Blomenhofer, A. Casarotti, M. Michalek, and A. Oneto, Identifiability for mix- +tures of centered Gaussians and sums of powers of quadratics, 2023 Joint Mathematics +Meetings (JMM 2023), 2023. ↑3 +[BO08] M. C. Brambilla and G. Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure +Appl. Algebra 212 (2008), no. 5, 1229–1251, DOI 10.1016/j.jpaa.2007.09.014. ↑14 +[CM22] A. Casarotti and M. Mella, From non-defectivity to identifiability, J. Eur. Math. Soc., +posted on 2022, DOI https://doi.org/10.4171/jems/1198. ↑2, 4, 5 +[Cha05] K. A. Chandler, The geometric interpretation of Fr¨oberg-Iarrobino conjectures on +infinitesimal neighbourhoods of points in projective space, J. Algebra 286 (2005), no. 2, +421–455, DOI 10.1016/j.jalgebra.2005.01.010. ↑14 +[COVC17] L. Chiantini, G. Ottaviani, N. Vannieuwenhoven, and Luca and Ottaviani Chiantini +Giorgio and Vannieuwenhoven, On generic identifiability of symmetric tensors of sub- +generic rank, Transactions of the American Mathematical Society 369 (2017), no. 6, +4021–4042. ↑1 +[CDM09] C. Ciliberto, O. Dumitrescu, and R. Miranda, Degenerations of the Veronese and +applications, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 5, Linear systems +and subschemes, 771–798. ↑2, 13 + +8 +6 +2 +1 +1.5 +2 +2.5 +3 +3.5 +426 +ALEX CASAROTTI AND ELISA POSTINGHEL +[FOS12] R. Fr¨oberg, G. Ottaviani, and B. Shapiro, On the Waring problem for polynomial +rings, Proceedings of the National Academy of Sciences 109 (2012), no. 15, 5600– +5602, DOI 10.1073/pnas.1120984109. ↑1 +[Ful93] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, +Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in +Geometry. ↑13 +[GM19] F. Galuppi and M. Mella, Identifiability of homogeneous polynomials and Cremona +transformations, Journal f¨ur die reine und angewandte Mathematik (Crelles Journal) +2019 (2019), no. 757, 279–308. ↑1 +[GKZ94] I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, +and multidimensional determinants, Mathematics: Theory & Applications, Birkh¨auser +Boston, Inc., Boston, MA, 1994. ↑13 +[Ger96] A. V. Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of +Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar +at Queen’s, Vol. X (Kingston, ON, 1995), 1996, pp. 2–114. ↑5 +[Lan12] Joseph M Landsberg, Tensors: geometry and applications, Vol. 381, 2012. ↑6 +[MM22] A. Massarenti and M. Mella, Bronowski’s conjecture and the identifiability of projec- +tive varieties, https://arxiv.org/abs/2210.13524 (2022). ↑2, 5 +[Nen17] G. Nenashev, A note on Fr¨oberg’s conjecture for forms of equal degrees, Comptes +Rendus Mathematique 355 (2017), no. 3, 272–276. ↑2, 24 +[Pos10] E. Postinghel, Degenerations and applications: polynomial interpolation and secant +degree, Universit`a Roma Tre, Italy, 2010. ↑14, 16 +[Pos12] +, A new proof of the Alexander-Hirschowitz interpolation theorem, Ann. Mat. +Pura Appl. (4) 191 (2012), no. 1, 77–94, DOI 10.1007/s10231-010-0175-9. ↑2, 9, 14 +[Pos13] +, Secant degree of toric surfaces and delightful planar toric degenerations, Adv. +Geom. 13 (2013), no. 2, 211–228, DOI 10.1515/advgeom-2012-0023. ↑2, 13 +Dipartimento di Matematica +Universit`a degli Studi di Trento +via Sommarive 14 I-38123 +Povo di Trento (TN), Italy +Email address: alex.casarotti@unitn.it +Dipartimento di Matematica +Universit`a degli Studi di Trento +via Sommarive 14 I-38123 +Povo di Trento (TN), Italy +Email address: elisa.postinghel@unitn.it + diff --git a/2NE4T4oBgHgl3EQfzw00/content/tmp_files/load_file.txt b/2NE4T4oBgHgl3EQfzw00/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e414fb7ba4c2ca9c476d3828b55f81ef61e82ff1 --- /dev/null +++ b/2NE4T4oBgHgl3EQfzw00/content/tmp_files/load_file.txt @@ -0,0 +1,1036 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf,len=1035 +page_content='WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS ALEX CASAROTTI AND ELISA POSTINGHEL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We discuss an approach to the secant non-defectivity of the vari- eties parametrizing k−th powers of forms of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It employs a Terracini type argument along with certain degeneration arguments, some of which are based on toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This implies a result on the identifiability of the War- ing decompositions of general forms of degree kd as a sum of kth powers of degree−d forms, for which an upper bound on the Waring rank was proposed by Fr¨oberg, Ottaviani and Shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Introduction Identifiability problems arise naturally in many fields of both applied and classi- cal algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A variety X ⊂ PN is said to be h−identifiable if the general point of its h−secant variety has a unique decomposition as a sum of h points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A classical application of identifiability concerns particular polynomial decom- positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The Waring problem for forms asks for a unique decomposition of a homogeneous polynomial Fd ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]d as a sum of d−th powers of linear forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1) Fd = Ld 1 + · · · + Ld h, with Li ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A necessary condition for identifiability is secant defectiv- ity: a variety X ⊂ PN of dimension dim(X) = n is said to be not h−(secant) defec- tive if the h−secant variety Sech(X), defined as the Zariski closure of points in PN lying in the span of h points of X, has the expected dimension min{N, h(n+1)−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In [COVC17] the authors proved that, for all subgeneric ranks h (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' such that Sech(X) ⊆ PN does not fill up the space), a general form F of rank h is identifi- able, with a few well known exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the case of generic rank, the situation is almost the opposite: in [GM19] it is proved that all forms of generic rank are not identifiable with the following exceptions: (n, d, h) = (1, 2k − 1, k), (3, 3, 5), (2, 5, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In [FOS12] the authors initiated the investigation of a generalization of the clas- sical Waring problems for forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular they show that a general form Fkd ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]dk can be written as a sum of at most kn k−th powers of forms Gi’s of degree d (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2) Fkd = Gk 1 + · · · + Lk h, 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Primary: 14N07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Secondary: 14C20, 14D06, 14M25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Identifiability, Waring problems, secant varieties, linear systems, degenerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Both authors are members of INdAM-GNSAGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='05276v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='AG] 12 Jan 2023 2 ALEX CASAROTTI AND ELISA POSTINGHEL and that this bound is sharp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' when d is sufficiently large, kn computes the generic rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' However the secant defectivity of the varieties parametrizing k−th powers of forms of degree d remains an open problem in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this paper we address both the secant defectivity and identifiability problems for such Waring decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Denote with V k n,d the variety parametrizing k−th powers of homogeneous degree d forms in n + 1 variables: V k n,d := {[F k]|F ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Our first main result is about secant non-defectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The variety V k n,d is non-defective if k ≥ 3 and h ≤ 1 N+1 �N+k−3 N � , where N = �n+d d � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Our second result is about identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A bridge from non-defectivity to iden- tifiability was proposed in [CM22] first and then generalised in the recent [MM22]: whenever X is a sufficiently regular variety (with non-degenerate Gauss map), then if X is not h−defective, then X is (h − 1)−identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Using this and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 we obtain what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A general form F ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]dk of rank h with k ≥ 3 is identifiable whenever h ≤ 1 N+1 �N+k−3 N � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We remark that in [Nen17], the author showed that the secant defectivity of V k n,d can be bounded asymptotically, using a direct algebraic computational argument, to kn − dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' When d ≫ k our bound of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 extends the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1, we brought together a Terracini type argument and several different degeneration techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By a classical application of Ter- racini’s Lemma, non-defectivity problems for secant varieties translate into the study of particular linear systems of hypersurfaces with prescribed singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The first systematic study was used in the proof of the celebrated Alexander and Hirschowitz Theorem for the case of classical Waring problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1), where secant varieties of Veronese embeddings of Pn correspond with linear systems of hyper- surfaces of Pn with prescribed double points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the generalized Waring problem setting, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2) for k ≥ 2, a direct translation to linear systems of hypersurfaces with only double point singularities is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In order to prove secant non- defectivity in this case it is necessary to impose a bigger base locus to our linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular, we will be interested in studying the dimensions of linear systems L := LN,k(V, 2h) of hyperurfaces of PN of degree k that are singular at h general points and that contain the d-thVeronese embedding of Pn, V ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The study of such linear systems is carried out by combining two types of degenerations introduced in [Pos12] and in [CDM09] and [Pos13] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' On one hand, we degenerate the ambient space PN to a scheme with two components and, in turn, the linear system L to a fibered product of two linear systems, one on each compo- nent, which are somewhat easier to deal with than the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In fact one of them consists of hypersurfaces containing a linear subspace and a collection of dou- ble points, the other one consists of hypsersurfaces containing V and one fat point of relatively large multiplicity with support on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In order to study the latter, we perform a toric degeneration of the Veronese V to a union of n-dimensional linear spaces, which will have the effect of reducing further the study of the limit linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 3 The study of Waring type problems and identifiability of symmetric tensors has been implemented also in the applied fields, from chemistry, biology to algebraic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Recently in [BCMO23], the problem of identifiability for k−th powers of forms was linked to the identifiability of centered Gaussian mixture models in applied statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Section 2 contains all definitions and our Ter- racini type result that translates non-defectivity of V k n,d to the study of L, Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In Section 3 we explain in detail the degenerations techniques, both in the clas- sical and in the toric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In Section 4 we analyse two auxiliary linear systems arising from the degeneration of L, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Section 5 is devoted to the proof of the main technical result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Finally in Section 6 we show at what extent our bounds are asymptotically better then the ones known before in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The authors would like to thank Giorgio Ottaviani and Alessandro Oneto for several useful discussions during the preparation of this arti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Powers of forms In order to give a coherent and self-contained treatment of the subject, let us recall some preliminary definitions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will work over the field of complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Veronese embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let W := Cn+1 and W ∗ the dual vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' With Pn = P(W) we denote the projective space over C of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We set the following integers Nd := �n + d n � − 1, N k d := �Nd + k Nd � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' When d is clear from the context we will indicate Nd simply by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Notice that the following identities hold: h0(Pn, OPn(d)) = Nd + 1 h0(PNd, OPNd (k)) = �Nd + k Nd � + 1 where h0(Pn, OPn(d)) denotes the number of global sections of the twisting sheaf OPn(d) on Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In other terms, Nd is the dimension of the linear systems of hyper- surfaces of degree d of PN which, in turn, is the projectivization of the complex vector spaces of forms of degree d in n + 1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The number N k d has a similar interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' With this in mind, we can make the following identifications: PNd = P(Symd(W ∗)), PN k d = P(Symk(Symd(W ∗))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now we consider the following Veronese embeddings: νd : Pn −→ V d n ⊂ P(Symd(W ∗)) [L] �−→ [Ld] 4 ALEX CASAROTTI AND ELISA POSTINGHEL and νk : PNd −→ V k Nd ⊂ P(Symk(Symd(W ∗))) [F] �−→ [F k] where L ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]1 is a linear form and F ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]d is a form of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The image of the embeddings are called Veronese varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that both νd and νk are the maps corresponding to the complete linear systems associated with the line bundles OPn(d) and OPNd (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' As elements of P(Symd(W ∗)) (respectively P(Symk(Symd(W ∗)))) the image of νd(p) (respectively νk(p)), with p a point, corresponds to the hyperplane parametrizing hypersurfaces of degree d in Pn (respectively of degree k in PNd) passing through p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We want to parametrize forms in Pn of degree dk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' elements in C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]dk, that can be written as k−th powers of forms of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that the Veronese varieties νdk(Pn) are always contained in the set of all k−th powers of forms of degree d because, trivially, Ldk = (Ld)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now, we let φdk : PNd −→ PNdk = P(Symdk(W ∗)) [F] �−→ [F k] be the map that assigns to each form F ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]d its k−th power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We call the scheme theoretic image V k n,d = φdk(PNd) ⊆ PNdk the (d, k)−Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Under the previous identification the classical Veronese varieties correspond to the (d, 1)−Veronese varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' On the other hand, for k > 1, V k n,d is not a standard Veronese variety, indeed it is easy to see that the target of φdk has dimension �n+dk dk � , which is never equal to �Nd+a a � for any a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A priori we don’t know if the map φdk is an isomorphism, as it happens for classical Veronese varieties, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Secant varieties and identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this subsection we recall the defi- nition of secant variety and the notion of identifiability, following [CM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X ⊂ PN be a non degenerate reduced variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X(h) be the h-th symmet- ric product of X, that is the variety parameterizing unordered sets of h points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let U X h ⊂ X(h) be the smooth locus, given by sets of h distinct smooth points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A point z ∈ U X h represents a set of h distinct points, say {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , zh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We say that a point p ∈ PN is in the span of z, p ∈ ⟨z⟩, if it is a linear combination of the zi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' With this in mind we define the following object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The abstract h-secant variety is the (hn + h − 1)-dimensional variety sech(X) := {(z, p) ∈ U X h × PN|p ∈ ⟨z⟩} ⊂ X(h) × PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 5 Let π : X(h) × PN → PN be the projection onto the second factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The h-secant variety is Sech(X) := π(sech(X)) ⊂ PN, and πX h := π|sech(X) : sech(X) → PN is the h-secant map of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If the variety X is irreducible and reduced we say that X is h-defective if dim Sech(X) < min{dim sech(X), N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The following is a classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6 (Terracini Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X ⊂ PN be an irreducible variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then the follwing holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For any p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pk ∈ X and z ∈ ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pk⟩, we have ⟨Tp1X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , TpkX⟩ ⊆ TzSeck(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' There is a dense open set U ⊂ X(k) such that ⟨Tp1X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , TpkX⟩ = TzSeck(X), for any general point z ∈ ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pk⟩ with (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pk) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Notions related to that of secant variety are those of rank and of identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X ⊂ PN be a non degenerate subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We say that a point z ∈ PN has rank h with respect to X if z ∈ ⟨p⟩, for some p ∈ U X h and z ̸∈ ⟨p′⟩ for any p′ ∈ U X h′ , with h′ < h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A point z ∈ PN is h-identifiable with respect to X ⊂ PN if z is of rank h and (πX h )−1(z) is a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The variety X is said to be h-identifiable if the h-secant map πX h is birational, that is the general point of Sech(X) is h- identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It is clear by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6 that when X is h-defective, or more generally when πX h is of fiber type, then X is not h-identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We now recall the recent result in [MM22], in which the authors generalize the approach in [CM22] relating identifiability with the non defectivity of the secant variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X ⊂ PN be an irreducible and non-degenerate variety of di- mension n, h ≥ 1 an integer, and assume that: (h + 1)n + h ≤ N, X has non degenerate Gauss map, X is not (h + 1)−defective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then X is h−identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the next sections we will see how to translate this theorem in the setting of powers of forms in order to give identifiability results for k−th powers of forms of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 6 ALEX CASAROTTI AND ELISA POSTINGHEL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Geometric construction of (d, k)−Veronese varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let us recall some facts from apolarity theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' the main reference is [Ger96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='10 (Apolarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We consider two polynomial rings in n + 1 variables, both endowed with the standard grading: R = C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn] = � i∈N Ri := C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]i S = C[y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn] = � i∈N Si := C[y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn]i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Treating elements of S as partial derivatives in the xi’s, the pairing Sk × Rl → C, sends (Fk, Gl) to the derivative Fk ◦ Gl ∈ R of Gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If k = l and if I ⊂ R is a homogeneous ideal, the orthogonal I⊥ k ⊂ Sk is the following space of polynomials I⊥ k = {F ∈ Sk|F ◦ G = 0, ∀G ∈ Rk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It is a standard fact of representation theory for the linear group GL(W), see for instance [Lan12], that the space Symk(Symd(W ∗)) can be decomposed as direct sum of GL(W)−modules in the following way: Symk(Symd(W ∗)) = Symdk(W ∗) ⊕ E where E = H0(IV d n (k)) is the k−th homogeneous part of the ideal of forms that vanish on the Veronese variety V d n = νd(Pn), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' notation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We have the following exact sequence: 0 → IV d n (k) → OPNd (k) → OV d n (k) → 0 After passing to cohomology and using the fact that every Veronese variety is projectively normal, we have: 0 → H0(IV d n (k)) → H0(OPNd (k)) → H0(OV d n (k)) → 0 This shows that dim(E) = �Nd + k k � − �n + dk dk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It is easy to show that GL(V )−modules Symdk(W ∗) and E are apolar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' for every pair of forms F, G ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xNd] with F ∈ Symdk(W ∗) and G ∈ E it holds that F ◦ G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The constructions above fit into the following commutative diagram: P(Symk(Symd(W ∗))) πE � Pn � P(Symd(W ∗)) νk � φdk � V k n,d ⊂ P(Symdk(W ∗)) where the map πE is the linear projection from the linear space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We now prove that the map φdk is in fact an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' With the above notations it holds Sec2(V k Nd) ∩ E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular the map φdk is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that an element F ∈ Sec2(V k Nd) is either of the form F = Lk or F = M k+N k, where L, M, N ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xNd]1 are linear forms in P(Symk(Symd(W ∗))), or equivalently degree d hypersurfaces in Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular if there exist such a F with F ∈ E = H0(IVn,d(k)), then Vn,d would be contained in the vanishing locus of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since the zero set of F is either an hyperplane or a union of an hyperplane with a subscheme of degree k − 1 and Vn,d is non-degenerate and irreducible, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Identifiablity for (d, k)−Veronese varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' From now on we will work with the projective notation, in particular E has to be intended as the projec- tivization of the affine E in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let us start by characterizing hyperplanes in E as particular linear subsystems of hypersurfaces in PNd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Denote with πdk the linear projection from the linear space P(Symdk(W ∗)) to E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' πdk : P(Symk(Symd(W ∗))) ��� E Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let H0(OPNk d (1)⊗IP(Symdk(W ∗))) be the complete linear system of hy- perplane sections of P(Symk(Symd(W ∗))) containing the linear space P(Symdk(W ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then ν∗ k(H0(OPNk d (1) ⊗ IP(Symdk(W ∗))) ∼= H0(OPNd (k) ⊗ IVn,d) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let H ∈ H0(OPNd (k)⊗IVn,d) be a degree k hypersurface in PNd that contains the Veronese Vn,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then the linear span H of (νk)∗(H) is an hyperplane in PN k d that contains Vn,dk := (νk)∗(Vn,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since ⟨Vn,dk⟩ = P(Symdk(W ∗)), we have that H0(OPNd (k) ⊗ IVn,d) ⊆ ν∗ k(H0(OPNk d (1) ⊗ IP(Symdk(W ∗))) To conclude just observe that h0(OPNd (k) ⊗ IVn,d) = dim(E) = codim(P(Symdk(W ∗))), where codim here indicates the codimension in P(Symk(Symd(W ∗))), and ν∗ k in- duces an isomorphism of global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ We need the following easy technical lemma: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The linear projection πdk : P(Symk(Symd(W ∗))) ��� E = H0(IVn,d(k)) is generically finite when restricted to VNd,k = νk(P(Symd(W ∗))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The map πdk|νk(P(Symd(W ∗))) : VNd,k ��� E is induced by a linear subsystem F of the line bundle L = OPNk d (1) with ν∗ k(F) = |OSymd(W ∗)(1) ⊗ IVn,d(k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now the claim follows easily from the fact that H0(IVn,d(k)) defines the Veronese variety Vn,d set-theoretically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 8 ALEX CASAROTTI AND ELISA POSTINGHEL Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph ∈ X ⊂ PN be general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6, we have that Sech(X) has the expected dimension if and only if H0(OX(1) ⊗ Ip2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',p2 h) has the expected dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' dim H0(OX(1) ⊗ Ip2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',p2 h) = max{0, N − (n + 1)h} Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph ∈ X are general points, then we denote Lh,X := |OX(1) ⊗ Ip2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',p2 h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Before moving on to the explicit description of the identifiability for V k n,d, let us first prove a general proposition about linear systems of projected varieties: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let X ⊂ PN be a smooth non-degenerate projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover let PN = ⟨F, E⟩ with F, E skew linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let πE : PN → F and πF : PN → E be the natural projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If the projections restricted to X are generically finite and Lh,X has the expected dimension, then Lh,πF (X) has the expected dimension if and only if Lh,πE(X) has the expected dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that by symmetry of E and F it suffices to prove only one of the implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , qh be general points on πE(X) and call pi = π−1 E (qi) and zi = πF (pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since πE|X : X �→ πE(X) is an isomorphism we have that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xh are general and so also z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since πFX is generically finite the space of hyperplanes Lh,πF (X) = |OE(1) ⊗ Iz2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',z2 h| correspond to Lh,X ⊗ IF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now the splitting PN = ⟨F, E⟩ induces the linear projection π : Lh,X �→ Lh,X ⊗ IE such that Ker(π) = Lh,πF (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since by assumption both the source and the kernel have the expected dimension by the rank-nullity theorem the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ We are finally able to characterize the identifiability properties for the case of powers of forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the following linear system of all hypersurfaces of PNd of degree k containing the Veronese variety Vn,d ⊂ PNd and double at the points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph that lie in general position in PNd: LNd(Vn,d, 2h) := OPNd (k) ⊗ IVn,d ⊗ Ip2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',p2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the above notation, let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph be general points in PNd = P(Symd(W ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The linear system LNd(Vn,d, 2h) has the expected dimension if and only if Sech(V k n,d) has the expected dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' With the notations of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='15 we have E = H0(IVn,d(k)) and F = Symdk(W ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We have that πE restricted to X = VNd,k is an isomorphism by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='11, in particular it is generically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The same holds for πF = πdk by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now Lh,VNd,k has the expected dimension by Alexander-Hirschowitz (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 below) and Lh,πdk(VNd,k) = |OPNd (k) ⊗ IVn,d ⊗ Ip2 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=',p2 h| by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degeneration techniques In this section we will discuss two types of degenerations that will provide main tools for the proofs of the results of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The FP−degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this section we recall a degeneration procedure introduced in [Pos12], which consists in degenerating the projective space PN to a reducible variety with two components, and then studying degenerations of line bundles on the general fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degenerating the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let ∆ be a complex disc centred at the origin and consider the product Y = PN ×∆ with the natural projections πY 1 : Y → PN and πY 2 : Y → ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The second projection is a flat morphism and we denote by Yt := PN ×{t} the fibre over t ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will refer to Y0 and to Yt, with t ̸= 0, as the central fibre and the general fibre respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let f : X → Y denote the blow-up of Y at a point (p, 0) ∈ Y0 in the central fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the following diagram, where πX i := πY i ◦ f, for i = 1, 2: X f � πX 2 � πX 1 � Y πY 1 � πY 2 � PN ∆ The morphism πX 2 : X → ∆ is flat with fibres denoted by Xt, t ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For the general fibre we have Xt ∼= Yt = PN, while the central fibre X0 is the reduced union of the strict transform of Y0, that we shall denote with F, and the exceptional divisor P ∼= PN of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The two components P and F meet transversally and we will denote by R the intersection: R := F ∩ P ∼= PN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will say that PN degenerates to X0 = P ∪ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will now endow the general fibre Xt with a line bundle and we will describe its limits on X0 via this degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In order to do so, we will give bases for the Picard groups of the components of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We denote by HP the hyperplane class of P, so that the Picard group of the exceptional component is generated by HP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover we denote with HF the hyperplane class of F, pull-back of a general hyperplane of Y0 ∼= PN, and with E := P|F the exceptional class in F: HF and E generate the Picard group of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In these bases, R has class HP in N1(P) and E in N1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A line bundle on X0 will correspond to a line bundle on P and a line bundle on F, which match on the intersection R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In other terms, we can describe the Picard group of X0 as a fibre product Pic(X0) = Pic(P) ×Pic(R) Pic(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the line bundle OX (k) = (πX 1 )∗(OPN (k)) and the following twist by a negative multiple of the exceptional divisor: MX (k, a) := OX (k) ⊗ OX (−aP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The line bundle MX (k, a) will induce a line bundle on each fibre Xt by restriction: Mt(k, a) := MX (k, a)|Xt, t ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For t ̸= 0, since P ∩ Xt = ∅, we have Mt(k, a) = OXt(k) 10 ALEX CASAROTTI AND ELISA POSTINGHEL while on the components of the central fibre we have MP(k, a) := MX (k, a)|P = OP(aHP), MF(k, a) := MX (k, a)|F = OF(kHF − aE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The resulting line bundle on X0 is a flat limit of the bundle OXt(k) ∼= OPn(k), for t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degenerating the Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will use the same notation as in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let us set N := Nd = �n + d n � − 1, and let V := Vn,d = vd(Pn) ⊂ PN denote the d-th Veronese embedding of Pn in PN, and consider the 1-parameter family V = V × ∆ ⊂ Y with the natural projections πY 1 |V : V → PN and πY 2 |V : V → ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The second projection is a flat morphism and we denote by Vt := V × {t} the fibre over t ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We pick a general point (p, 0) ∈ V0 ⊂ Y0 in the central fibre of πY 2 : Y → ∆ supported on the Veronese variety and we consider the blow-up f : X → Y at (p, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This induces the blow-up f|V : �V → V of V at (p, 0) and the fibres of (πY 2 ◦ f)|�V are as follows: the general fibre is a Veronese variety �Vt ∼= V, while the central fibre is the reduced union of two components, �V0 = �VF ∪ Λ, where �VF is the strict transform of V0 under the blow-up at p, while Λ ∼= Pn is the exceptional divisor on �V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover we can write ΛR := �VF ∩ Λ ⊂ Λ, and observe that ΛR ∼= Pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the line bundle MX (k, a) and twist it by the ideal sheaf of �V: MX (k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' �V) := MX (k, a) ⊗ I�V = OX (k) ⊗ OX (−aP) ⊗ I�V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This restricts to the following line bundles on the fibres Xt: Mt(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V ) = OXt(k) ⊗ IVt, t ∈ ∆ \\ {0}, MP(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Λ) = OP(aHP) ⊗ IΛ, MF(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' �V ) = OF(kHF − aE) ⊗ I�V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The resulting line bundle on X0 is a flat limit of the bundle OPn(k) ⊗ IV on the general fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degenerating a collection of points in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We continue to use the notation of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' On the general fibre of Y → ∆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' for t ̸= 0, we consider a collection of points {x1,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xh,t} ⊂ Yt in general position and that, in particular, lie off the Veronese Variety Vt ⊂ Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' After choosing a point that lies generically on the Veronese in the central fibre, p ∈ V0 ⊂ Y0, we degenerate each point x1,t ∈ Yt to an infinitely near point to p ∈ V0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , h}, consider the curve (πY 1 )−1(xi) ⊂ Y and its pull-back Ci on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 11 union � i Ci intersects each fibre Xt transversally in h distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For t ̸= 0, the points � i Ci ∩ Xt are in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover, by the generality assumption, the curves (πY 1 )−1(pi) ⊂ Y are not tangent to V0 ∈ Y0, therefore the intersection points Ci ∩ X0 lie on P but not inside Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the blow-up of X along the union of curves �h i=1 Ct, gh : � X → X, with exceptional divisors ECi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since these curves are disjoint, the result does not depend of the order of blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The general fibre, that we continue to call Xt by abuse of notation, is isomorphic to a PN blown-up at h points in general position and its Picard group will be generated by the hyperplane class and by the classes of the exceptional divisors: Pic(Xt) = Z⟨H, E1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , Eh,t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The central fibre has two components, the pull-back of F and the strict transform of P ∼= PN, which is isomorphic to a PN blown-up at h points in general position: abusing notation, we call F and P the two components, so that X0 = P ∪ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The Picard group of �P is Pic(P) = Z⟨HP, E1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , Eh,t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For a vector m = (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , mh) ∈ Nn, consider the following sheaf on � X: M � X (k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ˜V, m) := O(k) ⊗ O(−aP) ⊗ O(−(m1EC1 + · · · + mhECh)) ⊗ I�V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It restricts to Mt(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V, m) = O(k) ⊗ O(−(m1E1,t + · · · + mhEh,t)) ⊗ IVt, t ̸= 0, on the general fibre, and to MP(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Λ, m) = OP(aHP − (m1E1 + · · · + mhEh)) ⊗ IΛ, MF(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' �V , m) = OF(kHF − aE) ⊗ I�V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' on the components of the central fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The resulting line bundle on X0 is a flat limit of the bundle on the general fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will abbreviate the notation of the previous sections by setting M � X := M � X (k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ˜V, m) and Mt := Mt(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V, m) MP := MP(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Λ, m) MF := MF(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ˜V , m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We are interested in computing the dimension of the space of global sections of the line bundle on the central fibre, which by upper-semicontinuity is an upper bound for the dimension of the space of global sections of the line bundle on the general fibre: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1) h0(X0, M0) ≥ h0(Xt, Mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 12 ALEX CASAROTTI AND ELISA POSTINGHEL In order to do so, we consider the natural restrictions of to the intersection R = P∩F of the central fibre: 0 → ˆ MP → MP → MP|R → 0, 0 → ˆ MF → MF → MF|R → 0, where ˆ MP = ˆ MP(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Λ, m) and ˆ MF = ˆ MF(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ˜V , m) denote the kernels of the restriction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since R = HP on P and R = E on F, we have ˆ MP = OP((a − 1)HP − (m1E1 + · · · + mhEh) ⊗ IΛ ˆ MF = OF(kHF − (a + 1)E) ⊗ I ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the restriction maps of global sections: rP : H0(P, MP) → H0(R, MP|R), rF : H0(P, MF) → H0(R, MF|R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We notice that the spaces of global sections of the restricted systems are both subspaces of the space of global sections of the degree-a line bundle on R ∼= PN−1: H0(R, MP|R), H0(R, MF|R) ⊆ H0(R, OR(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A global sections of M0 consists of an element of H0(P, MP) and an element of H0(P, MP) which match in H0(R, OR(a)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' the space space of global sections H0(X0, M0) is described as a fibre product via the following commutative diagram: H0(X0, M0) H0(P, MP) H0(F, MF) H0(R, MP|R ∩ MF|R) rP rF This yields the formula for the dimension of the spaces of global sections h0(X0, M0) = h0(P, ˆ MP) + h0(F, ˆ MF) + h0(R, MP|R ∩ MF|R), which, in terms of dimensions of line bundles, it reads (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2) dim M0 = dim ˆ MP + dim ˆ MF + dim MP|R ∩ MF|R + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' There is an obvious isomorpshism between Mt(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V, m), line bun- dle on Xt, and the line bundle LN,k(V, m) := OPN (k) ⊗ IV ⊗ IZ on PN, where Z is a union of fat points in general position in PN with multiplicities respectively m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Such isomorphism is given by taking strict transforms of elements of LN,k(V, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Similarly, since P is the blow-up of PN at h points in general position and Λ ⊂ P is a general linear subspace, then there is an isomorphisms MP(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Λ, m) ∼= LN,a(Λ, m) := OPN (a) ⊗ IΛ ⊗ IZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Finally, since F is a PN blown-up at a point p on the Veronese variety V ⊂ PN and since ˜V is the strict transform of V via this blow-up, then there is an isomorphisms MF(k, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ˜V , m) ∼= LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Toric degeneration of the Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We refer to [Ful93] for details on projective toric varieties associated with convex lattice polytopes and to [GKZ94] for details on coherent triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let ∆n ⊂ Rn be the n-dimensional simplex obtained as the convex hull of the points (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0), (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider d∆n = ∆N + · · · + ∆N, where + here denotes the Minkowski sum of polytopes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The polytope d∆n defines the d-th Veronese embedding of Pn in PN, with N = Nd = �N+d N � − 1, that we shall call V = Vn,d as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the lattice Zn ⊂ Rn and the set of lattice points A := d∆n ∩ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We have ♯A = N + 1 and each such point corresponds to a coordinate point of the ambient space PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degenerating the Veronese to a union of linear spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Take a regular trian- gulation of d∆n, that is a decomposition of d∆n into a finite union of simplices dn � i=1 Si, where each Si is obtained as the convex hull of n + 1 non-aligned points of A, ♯Si ∩ Zn = n + 1, for i ̸= j, Si ∩ Sj is a common faces of Si and Sj (possibly empty), there is a strictly convex piecewise linear function λ : Rn → R whose domains of linearity are precisely the Si’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We can always assume that S1 is the convex hull of the lattice points (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0), (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 0, 1), so that it lies at a corner of d∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider for ex- ample, for n = 2 and d = 3, the triangulation into 9 and the piecewise linear function inducing shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this figure, S1 is the triangle with vertices @ @ @ @ @ @@ @ @ @ @@ @ @ @ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A regular triangulation of 3∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' (0, 0), (1, 0), (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Each Si defines a Pn as a toric variety, which we will call Πi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since a regular triangulation of d∆n induces a 1-parameter em- bedded degeneration of Vn,d ⊂ PN to the union of toric varieties described by the Si’s (see [CDM09] and [Pos13] for details on the 2-dimensional case), we have a degeneration of the Veronese variety to a union of dn n-planes Π := dn � i=1 Πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The intersection table of these planes is encoded in the combinatorial data described by the triangulation, that is: if Si ∩ Sj is r-dimensional, then Πi ∩ Πj ∼= Pr, for 0 ≤ r ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 14 ALEX CASAROTTI AND ELISA POSTINGHEL Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Because of the choice of S1 made, we will say that Π1 is a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In practice this means that it is possible to choose a hyperplane of PN that contains every Si, i > 1, but that does not contain S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover, the union of planes Π ⊂ PN is a torus invariant subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In fact, consider the simplex ∆N, which defines PN as a toric variety, with an action of the algebraic torus (C∗)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Each r-dimensional face of ∆N corresponds to a torus invariant linear subspace of dimension r of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular vertices of ∆N are in one-to-one correspondence with N + 1 linearly independent points, which we may assume to be the coordinate points of PN, up to a change of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Each r-dimensional face of ∆N corresponds to a Pr spanned by r + 1 coordinate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since each Πi is the linear span of n + 1 coordinate points of PN, then the union Π is embedded in a copy of PN and it is invariant under the action of the torus (C∗)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular, each Πi will correspond to a marked n-dimensional face of ∆N and we have dn such marked faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Degenerating a linear system intepolating the Veronese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We now consider the linear systems on PN of degree−k hypersurfaces containing the Veronese variety on the one hand, and the union of n−planes Π on the other hand: LN,k(Vn,d) := OPN (k) ⊗ IVn,d, LN,k(Π) := OPN (k) ⊗ IΠ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the above notation, we have dim LN,k(Vn,d) ≤ dim LN,k(Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since Π is a flat degeneration of Vn,d, then the statement follows by semi- continuity of the function dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Some auxiliary linear systems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Hypersurfaces containing a linear subspace and h double points in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It is a well celebrated result of Alexander and Hirschowitz that if we impose h double points in general position to the hypersurfaces of degree d of PN, there is only a finite list of cases where the dimension is larger than that obtained via a parameter count, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' edim LN,d(2h) = max � −1, �N + d N � − h(N + 1) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 (Alexander-Hirschowitz Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The linear system LN,d(2h) is non-special except in the following cases: d = 2 and N ≥ 2, 2 ≤ h ≤ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' d = 3 and (N, h) = (4, 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' d = 4 and (N, h) = (2, 5), (3, 9), (4, 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The interested reader may see [Ale88],[AH92b],[AH92],[AH95],[AH97]] for the original proof based on specialisation of points (Horace method), and [BO08] and [Cha05] for a simplified proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' An alternative proof via a different degeneration construction can be found here [Pos10,Pos12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This inspired the degeneration ap- proach developed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 that will be used to prove the main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 15 In this section we want to present an analogous result about linear systems with h imposed double points in general position and a linear subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let Λ ⊂ PN be a general linear subspace of dimension n and let Z ⊂ Pn be a double point scheme with support a set of points in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let IΛ be the ideal sheaf of Λ ⊂ PN and let IZ be the ideal sheaf of Z ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the sheaf LN,k(Λ, 2h) := OPN (k) ⊗ IΛ ⊗ IZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since the Hilbert polynomial of Λ ⊂ PN at degree k is �n + k n � or, in other terms, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1) h0(OPN (k) ⊗ IΛ) = �N + k N � − �n + k n � , and since h double points in general position of PN impose h(N + 1) conditions to the hypersurfaces of PN of degree k, we can give the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The virtual dimension of the linear system LN,k,Λ(2h) of hyper- surfaces of Pn that vanish along a linear subspace of dimension n, Λ ⊂ PN, and double at h points in general position is the following integer: vdim LN,k(Λ, 2h) = �N + k N � − �n + k n � − h(N + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The expected dimension of LN,k(Λ, 2h) is edim LN,k(Λ, 2h) = max � −1, vdim LN,k(Λ, 2h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since Λ and the scheme of double points are disjoint the virtual dimension, which is obtained by a simple parameter count, provides a lower bound to the actual dimension: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2) dim LN,k(Λ, 2h) ≥ edim LN,k(Λ, 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let Λ ⊂ PN be linear subspace of dimension n and let ZΛ ⊂ Pn be a double point scheme supported on a collection of points in general position in PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3) h ≤ 1 N + 1 �N + k − 1 N � , and (N, k − 1, h) is not in the list of exceptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1, and k ≥ 2, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4) dim LN,k(Λ, 2h) = edim LN,k(Λ, 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If k = 2 and h = 0, the conclusion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If k = 2 and h = 1, it is easy to see that all elements of LN,2(Λ, 2) are pointed quadric cones containing Λ and hence we have the isomorphism LN,2(Λ, 2) ∼= LN−1,2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1), we have that dim LN−1,2(Λ) = �N+1 2 � − �n+2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We conclude noticing that the latter equals the expected dimension of LN,2(Λ, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Now, assume k ≥ 4 and consider the following exact sequence obtained by re- stricting LN,k(Λ, 2h) to a general hyperplane H ⊂ PN such that Λ ⊆ H: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5) 0 → LN,k−1(2h) → LN,k(Λ, 2h) → LN,k(Λ, 2h)|H ⊆ LN−1,k(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 16 ALEX CASAROTTI AND ELISA POSTINGHEL Under the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3) and using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1, the kernel system LN,k−1(2h) has dimension equal to its virtual dimension: dim LN,k−1(2h) = �N + k − 1 N � − h(N + 1) − 1, and in particular H1(PN, dim LN,k−1(2h)) = 0, so that we have the following exact sequence in cohomology: 0 → H0(PN, LN,k−1(2h)) → H0(PN, LN,k(Λ, 2h)) → H0(H, LN,k(Λ, 2h)|H) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1) h0(PN−1, LN−1,k(Λ)) = �N − 1 + k N − 1 � − �n + k n � , and so h0(H, LN,k(Λ, 2h)|H) ≤ �N − 1 + k N − 1 � − �n + k n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' From the exact sequence of global sections we obtain: h0(PN, LN,k(Λ, 2h)) = h0(PN, LN,k−1(2h)) + h0(H, LN,k(Λ, 2h)|H) ≤ �N + k N � − �n + k n � − h(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We conclude the proof of this case using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Finally, assume that k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this case, the bound on the number of points is h ≤ N 2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We consider the restriction to a general hyperplane containing Λ as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The kernel system is special by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1, and one can easily check that it has dimension dim LN,2(2h) = �N + 2 2 � − h(N + 1) + �h 2 � − 1, see for instance [Pos10, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Moreover, as a simple consequence of B´ezout’s Theorem, the linear system LN,3(Λ, 2h) contains in its base locus the lines spanned by pairs of points, each of which intersects H in a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We claim that the base locus of LN,3(Λ, 2h) is supported on the union of Λ and these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This implies that the restricted system is the complete linear system of cubics containing Λ and passing simply through the �h 2 � trace points: LN,3(Λ, 2h)|H = LN−1,3(Λ, 1( h 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We claim that the linear system on the right hand side of the above expression is non-special, namely that the scheme given by Λ and the simple points impose independent conditions to the cubics of PN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This shows that dim LN,3(Λ, 2h) = dim LN,2(2h) + dim LN−1,3(Λ, 1( h 2)) + 1 = ��N + 2 2 � − h(N + 1) + �h 2 � − 1 � + ��N + 2 3 � − �n + 3 3 � − �h 2 � − 1 � + 1, which implies that LN,3(Λ, 2h) is non-special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We are left to proving the two claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For the second claim, first of all notice that there is a hyperplane inside H, containing all �h 2 � points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This can be taken to WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 17 be the intersection with H of a hyperplane of PN containing the h original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Call H1 ⊂ H the intersection and restrict the linear system LN−1,3(Λ, 1( h 2)) to it, giving rise to the following exact sequence: 0 → LN−1,2(Λ) → LN−1,3(Λ, 1( h 2)) → LN−2,3(1( h 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since the two external linear systems are non-special, the so is the middle one, concluding the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' As for the first claim: we show that LN,3(Λ, 2h) has no additional base locus other than Λ and the lines spanned by pairs of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let’s call p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph the h assigned pints in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Assume that q is a point in PN in linearly general position with respect to p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since h ≤ N 2 + 1 < N, there is a hyperplane A containing p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph but not containing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since n < N, there is a hyperplane B containing Λ but not containing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The cubic 2A + B belongs to LN,3(Λ, 2h) proving that q cannot be a base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Assume now that q is a point in PN not in linearly general position with respect to p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , ph, which means that there is a linear space spanned by some of the pi’s containing q, but such that q does not belong to any of the lines ⟨pi, pj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let ⟨pi : i ∈ Iq⟩ be the minimum such linear span and choose two distinct indices j1, j2 ∈ Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let A1 be a hyperplane containing all pi’s with i ̸= j1 and let A2 be a hyperplane containing all pi’s with i ̸= j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let B be a hyperplabe containing Λ, pi1 and pj2 and not containing q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The cubic A1 + A2 + B belongs to LN,3(Λ, 2h) proving that q cannot be a base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Finally, since the multiplicity of the general element of the linear system of cubics along the line q ∈ ⟨pi : i ∈ Iq⟩ is exactly 1, then the above cases are exhaustive and this conclude the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Hypersurfaces containing the Veronese variety and a fat point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let V = Vn,d ⊂ Pn be the d-th Veronese embedding of Pn and let {pa} ⊂ V ⊂ Pn be a fat point scheme with support on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let IV be the ideal sheaf of V ⊂ PN and let Ipa be the ideal sheaf of Z ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the sheaf LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We are interested in computing the dimension of the space of global sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The Hilbert polynomial of V ⊂ PN at degree k is �n + kd n � or, equivalently, we have dim OPN (k) ⊗ IV = �N + k N � − �n + kd n � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The scheme given by a point of multiplicity a of PN imposes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6) �N + a − 1 N � conditions to the hypersurfaces of PN of degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Therefore the virtual dimension of LN,k(V, a), obtained by a parameter count, is �N + k N � − �n + kd n � − �N + a − 1 N � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 18 ALEX CASAROTTI AND ELISA POSTINGHEL It does not yield a useful notion of expected dimension for the linear system LN,k(V, a) due to the fact that the two subschemes V and {pa} of PN have nonempty intersection so that some of the conditions imposed by them individually to the hy- persurfaces of degree k of PN will overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For instance, if we first impose V and then {p}, clearly the latter will not give any independent condition, because, by the containment relation p ∈ V , p is a base point of the linear system LN,k(V ) = OPN (k) ⊗ IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' When the support of a fat point subscheme Z = {pa} ⊂ PN, whose length is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6), lies on the n-dimensional subvariety V , the restriction Z|V ⊂ V is a subscheme of length �n + a − 1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Therefore we may define the following notion of expected dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A notion of expected dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We introduce the following refined param- eter count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let V ⊂ PN be the d-th Veronese embedding of PN and let {pa} ⊂ PN be a fat point scheme supported on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The expected dimension of LN,k(V, a), denoted by edim LN,k(V, a), is the following integer: max � −1, �N + k N � − �n + kd n � − ��N + a − 1 N � − �n + a − 1 n �� − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' That the integer of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4 is a lower bound to the actual dimension of LN,k(V, a) is not an obvious statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We will show that it does when a ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let νd : Pn → PN be the d-the Veronese embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let V = Vn,d := νd(Pn) ⊂ PN and let ZV = {pa} ⊂ Pn be a fat point of multiplicity a ≤ k supported on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='7) dim LN,k(V, a) ≥ edim LN,k(V, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We consider the linear system LN,k(a) = OPN ⊗ IZV of the degree-k hyper- surfaces of PN with a point of multiplicity a with support on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Restriction to V gives the following Castelnuovo sequence: 0 → LN,k(V, a) → LN,k(a) → LN,k(a)|V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It is an easy observation that a fat point of multiplicity a imposes independent conditions to the hypersurfaces of fixed degree of PN, as long as the multiplicity does not exceed the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Therefore we can obtain the dimension of the linear system LN,k(a) by a parameter count: dim LN,k,V (a) = �N + k N � − �N + a − 1 N � − 1 In particular h1(PN, LN,k(a)) = 0, so that we have the following sequence in coho- mology: 0 → H0(LN,k(V, a)) → H0(LN,k(a)) → H0(LN,k(a)|V ) → H1(LN,k(V, a)) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 19 Since the Veronese morphism νd : Pn → PN gives an isomorphism of Pn to its image V , then the pull-back of LN,k(a)|V is a linear system of degree-kd hypersurfaces of Pn: ν∗ d(LN,k(a)|V ) ⊆ OPn(kd) ⊗ IZ′ =: Ln,kd(a) where Z′ ⊂ Pn is a fat point of multiplicity a with support a general point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since nk ≥ a by the assumption, the linear system Ln,kd(a) has dimension dim Ln,kd(a) = �n + kd n � − �n + a − 1 n � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' From this we obtain dim LN,k(a)|V ≤ �n + kd n � − �n + a − 1 n � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Putting everything together: h0(LN,k(V, a)) = h0(LN,k(a)) − h0(LN,k(a)|V ) + h1(LN,k(V, a)) ≥ h0(LN,k(a)) − h0(LN,k(a)|V ) ≥ ��N + k N � − �N + a − 1 N �� − ��n + kd n � − �n + a − 1 n �� , which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Dimensionality via apolarity and toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let V = Vn,d ⊂ PN be the Veronese variety and let Π ⊂ PN be a union of n-planes, degeneration of V , as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let p ∈ V and p0 ∈ Π1 ⊂ Π be a general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the linear systems on PN LN,k(V, a) := OPN (k) ⊗ IV ⊗ Ipa, LN,k(Π, a) := OPN (k) ⊗ Iv ⊗ Ipa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Building from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the above notation, we have dim LN,k(V, a) ≤ dim LN,k(Π, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since p is a general point on V , we may assume that it degenerates to a general point p0 ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since Π∪{pa 0} is a flat degeneration of the scheme V ∪{pa}, then the Hilbert functions of the former is at most that of latter, by semi-continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In the above notation and for any 1 ≤ a ≤ k, then dim LN,k(Π, a) = �N + k N � − �n + kd n � − �N + a − 1 N � + �n + a − 1 n � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Given the union Π := �dn i=1 Πi of torus invariant n-planes of Pn, with Π1 a sink and p0 supported generically on Π1, there is a torus invariant hyperplane H such that Π1 ∩ H is an (n − 1)-plane and Πi ⊂ H for 2 ≤ i ≤ dn (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We can always assume that p0 is a coordinare point of PN and we can call p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pN the other coordinate (torus invariant) points of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Hence we can choose, without loss of generality, that Π1 = ⟨p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pn⟩ and H = ⟨p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , pN⟩, so that p0 ∈ Π1 and pi /∈ Πi for i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 20 ALEX CASAROTTI AND ELISA POSTINGHEL Let R = C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xN] be the homogeneous polynomial ring of PN and consider the ideals Ip0 ⊂ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xN] and IΠi ⊂ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xN] Ip0 = ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , nN⟩, IΠ1 = ⟨xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , nN⟩, IΠi = ⟨xin+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , niN ⟩, i ≥ 2, IH = ⟨x0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By construction, for i ≥ 2 , we have 0 ∈ {in+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , iN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Using Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='10, we compute: � I−1 pa 0 � k = {yk−l 0 Fl(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yN) : Fl ∈ Sl, 0 ≤ l ≤ a − 1}, � I−1 Π1 � k = {Fk(y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn) : Fk ∈ Sk}, � I−1 Πi � k = {Fk(yi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yin) : Fk ∈ Sk}, i ≥ 2, where for i ≥ 2, the index set {i0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , in} is the complement of {in+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , iN} ⊂ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We have the following intersections � I−1 pa 0 � k ∩ � I−1 Πi � k = ∅, i ≥ 2, � I−1 pa 0 � k ∩ � I−1 Π1 � k = {yk−l 0 Fl(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn) : Fl ∈ Sl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We compute the dimension of the latter intersection: dim � I−1 pa 0 � k ∩ � I−1 Π1 � k = dim{yk−l 0 Fl(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn) : Fl ∈ Sl, 0 ≤ l ≤ a − 1} = a−1 � l=0 dim{Fl(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , yn) : Fl ∈ Sl} = a−1 � l=0 �n − 1 + l n − 1 � = �n + a − 1 n � , where the last equality follows a standard relation of Newton coefficients, commonly known as the hockey stick identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The number of conditions imposed to the linear system of degree-k hypersurfaces of PN by the scheme {pa 0} ∪ Π is the dimension of the linear span of � I−1 pa 0 � k and � I−1 Πi � k, for i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , dn, which is the following integer: dim � I−1 pa 0 � k + dim �� I−1 Πi � k , i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , dn� − dim � I−1 qa 0 � k ∩ � I−1 Π1 � k = �N + a − 1 N � + �n + kd n � − �n + a − 1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The linear system dim LN,k(V, a) has the expected dimension ac- cording to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It follows from Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof of the main theorem We are ready to prove our main theorem, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Thanks to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='16, computing the dimension of the h-secant varieties of the (d, k)-Veronese variety V k n,d ⊂ PNdk is equivalent to computing the dimension of the linear system in PN of all hypersurfaces containing the standard Veronese variety V = Vn,d ⊂ PN and double at h general points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let Vn,d ⊂ PN be the d-th Veronese embedding of Pn and let Z ⊂ Pn be a double point scheme with support a set of points in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let IV be the ideal sheaf of V ⊂ PN and let IZ be the ideal sheaf of Z ⊂ PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Consider the sheaf LN,k(V, 2h) := OPN (k) ⊗ IV ⊗ IZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since the Hilbert polynomial of V ⊂ PN in degree k is �n+kd n � or, in other terms, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1) dim OPN (k) ⊗ IV = �N + k N � − �n + kd n � − 1, and since h double points in general position of PN impose h(N + 1) conditions to the hypersurfaces of PN of degree k, we can give the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The virtual dimension of the linear system LN,k,V (2h) of hyper- surfaces of Pn that vanish along the Veronese variety V = Vn,d ⊂ PN and double at h points in general position is the following integer: vdim LN,k(V, 2h) = �N + k N � − �n + kd n � − h(N + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The expected dimension is edim LN,k(V, 2h) = max � −1, vdim LN,k(V, 2h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since V and the scheme of double points are disjoint, the virtual dimension, which is obtained by a simple parameter count, provides a lower bound to the actual dimension: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2) dim LN,k(V, 2h) ≥ edim LN,k(V, 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Using a degeneration argument, we shall show that if the number of points h is not too large, then the linear system LN,k,V (V, 2h) has dimension equal to the expected dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let νd : Pn → PN be the d-the Veronese embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Let V = Vn,d := νd(Pn) ⊂ PN and let ZV ⊂ Pn be a double point scheme supported on h points in general position of PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Then if k ≥ 3 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3) h ≤ 1 N + 1 �N + k − 3 N � then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4) dim LN,k(V, 2h) = edim LN,k(V, 2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2), it is enough to prove that the inequality dim LN,k(V, 2h) ≤ edim LN,k(V, 2h) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 22 ALEX CASAROTTI AND ELISA POSTINGHEL If k = 3, then h = 0 so the statement follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For k ≥ 4, we will prove the statement by means of the FP−degeneration introducedin Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2, applied to the line bundle L � X :=M � X (k, k − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' �V, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2, the line bundle on the general fibre is isomorphic to Lt :=LN,k(V, 2h), while on the central fibre the linear systems on the two components are the follow- ing: LP :=LN,k−1(Λ, 2h), LF :=LN,k(V, k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We consider the restriction to R = P ∩ F: the kernels on the two components are, respectively: ˆLP :=LN,k−2(Λ, 2h), ˆLF :=LN,k(V, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since R ∼= PN−1, the two restricted systems satisfy the following: RP :=LP|R ⊂ LN−1,k−1(ΛR), RF :=LF|R ⊂ LN−1,k−1(ΛR), where we recall that ΛR = Λ ∩ R ∼= Pn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We first look at the exceptional component P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3, since k−2 ≥ 2 and h ≤ 1 N + 1 �N + k − 3 N � , both linear systems LP and ˆLP have the expected dimension, that is dim LP = �N + k − 1 N � − �n + k − 1 n � − h(N + 1) − 1 dim ˆLP = �N + k − 2 N � − �n + k − 2 n � − h(N + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5) Moreover, we have a short exact sequence of spaces of global sections: 0 → H0(P, ˆLP) → H0(P, LP) → H0(R, RP) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In particular, we can compute dim RP = dim LP − dim ˆLP + 1 = �N + k − 2 N − 1 � − �n + k − 2 n − 1 � − 1 = dim LN−1,k−1(ΛR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We conclude that RP is the complete linear system RP = LN−1,k−1(ΛR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 23 On the component F, using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8, we have that both LF and ˆLF have the expected dimension, that is dim LF = �N + k N � − �n + kd n � − ��N + k − 2 N � − �n + k − 2 n �� − 1 dim ˆLF = �N + k N � − �n + kd n � − ��N + k − 1 N � − �n + k − 1 n �� − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6) We claim that RF = LN−1,k−1(ΛR), so that, together with the above argument, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='7) RP ∩ RF = OPN−1(k − 1) ⊗ IΛR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In order to prove the claim, we observe that by semicontinuity, and precisely For- mula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1), and by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2), we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8) dim(L0) ≥ dim(Lt) ≥ edim Lt = �N + k N � − �n + kd n � − h(N + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Using Formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=', dim L0 = dim ˆLP + dim ˆLF + RP ∩ RF + 2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='9) and observing that RP ∩ RF = RF, we obtain dim RF ≥ edim Lt − dim ˆLP − dim ˆLF − 2 = �N + k − 2 N − 1 � − �n + k − 2 n − 1 � − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' the proof of the latter equality is easy and left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Since dim RF ≤ dim OPN−1(k − 1) ⊗ IΛR = �N + k − 2 N − 1 � − �n + k − 2 n − 1 � − 1, the claim follows and we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='10) dim RP ∩ RF = �N + k − 2 N − 1 � − �n + k − 2 n − 1 � − 1, Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='10), we obtain dim L0 = edim Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We conclude using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 is now just a corollary of what we just proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For k ≥ 3, if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3) holds, then the (d, k)−Veronese variety V k n,d ⊂ PNdk is non-defective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2 is an easy consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For k ≥ 3, if h < 1 N+1 �N+k−3 N � , then holds, then the (d, k)−Veronese variety V k n,d ⊂ PNdk is h-identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' It follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' □ 24 ALEX CASAROTTI AND ELISA POSTINGHEL 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Asymptotical Bound In this section we relate our bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 with the one given in [Nen17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' We first of all state Nenashev’s result for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' [Nen17, Theorem 1] Let I be a homogeneous ideal generated by h ∈ N0 generic elements of some nonempty variety D ⊆ Symr(Cn) of r-forms that is closed under linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Fix an integer s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' If h ≤ ��r + s + n − 1 n − 1 � / �s + n − 1 n − 1 �� − �s + n − 1 n − 1 � then the dimension of I in degree (r + s) is maximal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' it equal s h �s+n−1 n−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that when r = d(k − 1) and s = r the degree r + s = dk component of I gives us exactly the dimension of the h− secant variety Sech(V k n,d), where D is the tangential variety of V k n,d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' D = {F k−1G|F, G ∈ C[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' , xn]d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' As a consequence we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The dimension of Sech(V k n,d) is the expected one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' dim Sech(V k n,d) = h �n + d d � − 1 for h ≤ ( n+dk dk ) ( n+d d ) − �n+d d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Note that if we fix k, n and let d ≫ 0 we have that �n+dk dk � �n+d d � − �n + d d � ∼ kn − dn and if d ≫ k the bound is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 and under the same assumptions we get 1 N + 1 �N + k − 3 N � ∼ dn(k−4) which gives non trivial bounds for d ≫ k when k > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' WARING IDENTIFIABILITY FOR POWERS OF FORMS VIA DEGENERATIONS 25 The figure shows in red the bound given by Nenashev and in blue the bound of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1 as a function of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' In this case we have set the values k = 5 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' For d > 3 our bound is better and continues to give informations also in the range d > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' References [Ale88] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Alexander, Singularixt´es imposables en position g´en´erale `a une hypersurface pro- jective, Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 68 (1988), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3, 305–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [AH92a] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Alexander and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Hirschowitz, La m´ethode d’Horace ´eclat´ee: application `a l’interpolation en degr´e quatre, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 107 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3, 585–602, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1007/BF01231903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [AH92b] , Un lemme d’Horace diff´erentiel: application aux singularit´es hyperquartiques de P5, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3, 411–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [AH95] , Polynomial interpolation in several variables, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 4 (1995), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2, 201–222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [AH97] , Generic hypersurface singularities, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Indian Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 107 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2, 139–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [BCMO23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Blomenhofer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Casarotti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Michalek, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Oneto, Identifiability for mix- tures of centered Gaussians and sums of powers of quadratics, 2023 Joint Mathematics Meetings (JMM 2023), 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑3 [BO08] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Brambilla and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Ottaviani, On the Alexander-Hirschowitz theorem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Algebra 212 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 5, 1229–1251, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='jpaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [CM22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Casarotti and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Mella, From non-defectivity to identifiability, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=', posted on 2022, DOI https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='4171/jems/1198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 4, 5 [Cha05] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Chandler, The geometric interpretation of Fr¨oberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Algebra 286 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2, 421–455, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='jalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14 [COVC17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Chiantini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Ottaviani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Vannieuwenhoven, and Luca and Ottaviani Chiantini Giorgio and Vannieuwenhoven, On generic identifiability of symmetric tensors of sub- generic rank, Transactions of the American Mathematical Society 369 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 6, 4021–4042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑1 [CDM09] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Ciliberto, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Dumitrescu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Miranda, Degenerations of the Veronese and applications, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Belg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Simon Stevin 16 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 5, Linear systems and subschemes, 771–798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 13 8 6 2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='5 426 ALEX CASAROTTI AND ELISA POSTINGHEL [FOS12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Fr¨oberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Ottaviani, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Shapiro, On the Waring problem for polynomial rings, Proceedings of the National Academy of Sciences 109 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 15, 5600– 5602, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1120984109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑1 [Ful93] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 131, Princeton University Press, Princeton, NJ, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' The William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Roever Lectures in Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑13 [GM19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Galuppi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Mella, Identifiability of homogeneous polynomials and Cremona transformations, Journal f¨ur die reine und angewandte Mathematik (Crelles Journal) 2019 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 757, 279–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑1 [GKZ94] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Gel’fand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Kapranov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkh¨auser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=', Boston, MA, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑13 [Ger96] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Geramita, Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein ideals, The Curves Seminar at Queen’s, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' X (Kingston, ON, 1995), 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑5 [Lan12] Joseph M Landsberg, Tensors: geometry and applications, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 381, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑6 [MM22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Massarenti and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Mella, Bronowski’s conjecture and the identifiability of projec- tive varieties, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='13524 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 5 [Nen17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Nenashev, A note on Fr¨oberg’s conjecture for forms of equal degrees, Comptes Rendus Mathematique 355 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 3, 272–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 24 [Pos10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Postinghel, Degenerations and applications: polynomial interpolation and secant degree, Universit`a Roma Tre, Italy, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑14, 16 [Pos12] , A new proof of the Alexander-Hirschowitz interpolation theorem, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Pura Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' (4) 191 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 1, 77–94, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1007/s10231-010-0175-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 9, 14 [Pos13] , Secant degree of toric surfaces and delightful planar toric degenerations, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 13 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' 2, 211–228, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='1515/advgeom-2012-0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content=' ↑2, 13 Dipartimento di Matematica Universit`a degli Studi di Trento via Sommarive 14 I-38123 Povo di Trento (TN), Italy Email address: alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='casarotti@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='it Dipartimento di Matematica Universit`a degli Studi di Trento via Sommarive 14 I-38123 Povo di Trento (TN), Italy Email address: elisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='postinghel@unitn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE4T4oBgHgl3EQfzw00/content/2301.05276v1.pdf'} diff --git a/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/2301.02838v1.pdf.txt b/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/2301.02838v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..389b66e7bee475e781b82d153ee489c65bd48eff --- /dev/null +++ b/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/2301.02838v1.pdf.txt @@ -0,0 +1,2992 @@ +arXiv:2301.02838v1 [math.NT] 7 Jan 2023 +CONGRUENCES OF THE q-FIBONACCI SEQUENCE RELATED WITH ITS +FINITE TRANSCENDENCE +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Abstract. By using Andrews’s explicit formulae of q-Fibonacci sequence introduced by +Schur, we prove certain congruences of the q-Fibonacci sequence which relate the sequence +with the original Fibonacci sequence. As a corollary, we show that it yields a transcendental +element in the Q-algebra A of integers modulo infinitely large primes under the generalized +Riemann hypothesis. +Contents +1. +Introduction +1 +2. +Congruences of q-Fibonacci sequence +2 +2.1. +Review on q-analogues +2 +2.2. +Preliminaries +3 +2.3. +The case where p ≡ 2, 3, 4 mod 5 +5 +2.4. +The case where p ≡ 1 mod 5 +12 +3. +Finite algebraic numbers and finite transcendence of q-Fibonacci sequence +15 +3.1. +Review on finite algebraic numbers +15 +3.2. +Finite transcendence of q-Fibonacci sequence +16 +References +20 +1. Introduction +In 1917, Schur ([10]) introduced the so-called the q-Fibonacci sequence {Fn(q)} which is the +sequence of Q[q] defined by the initial value (F0(q), F1(q)) = (0, 1) and the recurrence relation +Fn+2(q) − Fn+1(q) − qnFn(q) = 0 +for every n ∈ N. It recovers the ordinary Fibonacci sequence {Fn} when q = 1. Andrews ([1]) +gave an explicit formula (cf. Theorem 2.1) of the q-Fibonacci sequence to prove some kind of +the Rogers-Ramanujan identities. +Let P be the set of prime numbers and let vp(α) denote the p-adic valuation of α for α ∈ Q× +and p ∈ P. For a pair (α, p) ∈ Q× × P with vp(α) = 0, ordp(α) denotes the order of α in +the multiplicative group (Z/pZ)× and Ip(α) := (p − 1)/ ordp(α), i.e. Ip(α) is the index of the +subgroup of (Z/pZ)× generated by α. So when α is a primitive root, we have ordp(α) = p−1 and +Ip(α) = 1. The values ordp(α) and Ip(α) are called the residual order of α and the residual index +of α respectively. Our main theorem is on congruence which relates the q-Fibonacci sequence +with the ordinary one: +Date: January 10, 2023. +1 + +2 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Theorem 1.1. For α ∈ Q× and p ∈ P satisfying vp(α) = vp(α − 1) = 0 and ordp(α) ̸≡ 0 +mod 5, +Fp(α) ≡ FIp(α)−1 or FIp(α)+1 +mod p +holds. +The following quotient Q-algebra +A := + +� +p∈P +Z/pZ + + +� + +� +p∈P +Z/pZ + + +appeared in [4], has been studied in several literatures ([8], [9], [11], etc) in relation with the +study of finite multiple zeta values (FMZVs, in short) introduced by Kaneko and Zagier ([3]). +Rosen ([8]) introduced the notion of finite algebraic numbers in A by using recurrent sequences. +It should be noted that solutions of Q-polynomials in A are not always finite algebraic numbers +in A . In this paper we consider finite transcendental numbers, elements in A which are not +roots of non-zero Q-polynomials in A (Definition 3.6). It is expected that non-zero FMZVs +are finite transcendental, however so far no single example has been otained as far as authors +know. Our interests in this paper is to construct examples of finite transcendental number in +the algebra A +We show that the (Fp(α))p ∈ A is a finite transcendental number by combining Theorem 1.1 +and Moree’s result ([6]) on the density of certain primes related to residual index. +Theorem 1.2. Under the generalized Riemann hypothesis (GRH, in short), (Fp(g))p ∈ A is a +finite transcendental number when g ∈ Z>1 is square-free. +2. Congruences of q-Fibonacci sequence +In this section we prove our main theorem (Theorem 1.1). +2.1. Review on q-analogues. +We recall the following standard notation: +• For n ∈ Z>0, the q-integer [n]q is defined by +[n]q := 1 − qn +1 − q . +• For n ∈ Z>0, the q-factorial [n]q! is defined by +[n]q! := [n]q[n − 1]q · · · [1]q. +• For a pair of integers n, m, the q-binomial coefficient +� n +m +� +q is defined as follows: +� n +m +� +q := +� [n]q···[n−m+1]q +[m]q···[1]q +if 0 ≤ m ≤ n +0 +else. +Andrews gave a general explicit formula of q-Fibonacci sequence. +Theorem 2.1 ([1, Theorem]). For any non-negative integer n, +Fn+1(q) = +∞ +� +j=−∞ +(−1)jqj(5j+1)/2 +� +n +⌊(n − 5j)/2⌋ +� +q +(2.1) +holds, where ⌊x⌋ denotes the greatest integer not exceeding x. + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +3 +2.2. Preliminaries. +In this subsection, we prepare some lemmas of q-integers and q-binomial coefficients. We show +the prime congruence of the q-Fibonacci sequence by the lemmas. Let Z(p) be the localization of +Z with respect to the prime ideal (p) generated by a prime number p. Note that the isomorphism +of field Z/pZ ∼= Z(p)/pZ(p) holds. +Lemma 2.2. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. Let l and k ∈ Z be +integers satisfying 1 ≤ l ≤ p − 1 and k ≡ l mod ordp(α). Then [k]α +[l]α +∈ Z(p) \ {0} and +(2.2) +[k]α +[l]α +≡ +� +k +l +mod p +if k ≡ 0 +mod ordp(α) +1 +mod p +if k ̸≡ 0 +mod ordp(α) +holds. +Remark 2.3. Note that for n ∈ Z, α ∈ Q \ {0, 1} and p be a prime number satisfying vp(α) = +vp(α − 1) = 0, [n]α ≡ 0 mod p holds if and only if n ≡ 0 mod ordp(α). We omit the proof. +Proof of Lemma 2.2. Let i be an integer satisfying 0 ≤ i ≤ ordp(α)−1 and k ≡ i mod ordp(α). +If i ̸= 0, then [l]α ̸≡ 0 mod p. We have +[k]α +[l]α += αk−iαi − 1 +αl−iαi − 1 ≡ αi − 1 +αi − 1 = 1 +mod p. +If i = 0, we have +[k]q +[l]q +���� +q=α += qk − 1 +ql − 1 +���� +q=α += (1 − qordp(α)) �k/ ordp(α)−1 +s=0 +qs ordp(α) +(1 − qordp(α)) �l/ ordp(α)−1 +t=0 +qt ordp(α) +����� +q=α += +�k/ ordp(α)−1 +s=0 +αs ordp(α) +�l/ ordp(α)−1 +t=0 +αt ordp(α) +≡ k/ ordp(α) +l/ ordp(α) +mod p += k +l . +□ +Lemma 2.4. Let α ∈ Q\{0, 1} and p ∈ P satisfying vp(α) = vp(α−1) = 0. Let k be a positive +integer with 0 ≤ k ≤ p − 1 − ordp(α). Put Ck := [p − k − 1]α · · · [p − k − ordp(α)]α +[k + ordp(α)]α · · · [k + 1]α +. Then +Ck ∈ Z(p) \ {0} and +� +p − 1 +k + ordp(α) +� +α +≡ Ck +�p − 1 +k +� +α +mod p +holds. Especially if there exists l ∈ Z such that k = l ordp(α), then we have Cl ordp(α) ≡ Ip(α) − l +l + 1 +mod p. +Proof. By the definition, we have +� +p − 1 +k + ordp(α) +� +α += [p − 1]α · · · [p − k − ordp(α)]α +[k + ordp(α)]α · · · [1]α += [p − k − 1]α · · · [p − k − ordp(α)]α +[k + ordp(α)]α · · · [k + 1]α +× [p − 1]α · · · [p − k]α +[k]α · · · [1]α += Ck +�p − 1 +k +� +α +. + +4 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Note that for any 1 ≤ i ≤ ordp(α), there exists uniquely ji ∈ {1, . . . , ordp(α)} such that +k + i ≡ p − k − ji mod ordp(α). This correspondence is one-to-one. We have +Ck = [p − k − 1]α · · · [p − k − ordp(α)]α +[k + ordp(α)]α · · · [k + 1]α += [p − k − j1]α +[k + 1]α +· [p − k − j2]α +[k + 2]α +· · · [p − k − jordp(α)]α +[k + ordp(α)]α +. +By 1 ≤ k + 1, . . . , k + ordp(α) ≤ p − 1, Lemma 2.2 implies [p − k − ji]α +[k + i]α +∈ Z(p) \ {0} for every +1 ≤ i ≤ ordp(α) and Ck ∈ Z(p) \ {0} holds. +In tha case where k is given by k = l ordp(α), Lemma 2.2 implies +Cl ordp(α) ≡ [p − k − 1]α +[k + ordp(α)]α +× +[p − k − 2]α +[k + ordp(α) − 1]α +× · · · × [p − k − ordp(α)]α +[k + 1]α +≡ p − k − 1 +k + ordp(α) ≡ Ip(α) − l +l + 1 +mod p. +□ +Lemma 2.5. Let α ∈ Q\{0, 1} and p ∈ P satisfying vp(α) = vp(α−1) = 0. Let k be a positive +integer satisfying 0 ≤ k < ordp(α). Then +�p − 1 +k +� +α +≡ +� +1 +mod p +if k = 0 +0 +mod p +if k ̸= 0 +holds. +Proof. We have a conclusion immediately if k = 0. Assume 1 ≤ k < ordp(α). Note that [k]α! ̸≡ 0 +mod p holds since k ̸≡ 0 mod ordp(α) holds. By [p − 1]α ≡ 0 mod p, we have +�p − 1 +k +� +α +≡ 0 +mod p. +□ +Lemma 2.6. For α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0, +�p − 1 +k +� +α +≡ +� � +Ip(α) +k/ ordp(α) +� +mod p +if k ∈ (ordp(α)) +0 +mod p +else +holds. +Proof. The case of k ̸∈ (ordp(α)) is clear by Lemma 2.4 and Lemma 2.5. If we suppose k = +l ordp(α), then we have +�p − 1 +k +� +α +≡ (Ip(α) − (l − 1))(Ip(α) − (l − 2)) · · · (Ip(α) − 0) +(l − 1 + 1)(l − 2 + 1) · · · (0 + 1) +· 1 +≡ +�Ip(α) +l +� +mod p. +□ +The following proposition is prime congruences of the q-Fibonacci sequence. +Proposition 2.7. For α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0, +Fp(α) ≡ +� +k∈Sp,1(α) +α +p−1−2k ordp(α) +10 +�Ip(α) +k +� +− +�α +p +� +� +k∈Sp,2(α) +�Ip(α) +k +� +mod p + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +5 +holds, where +Sp,i(α) = {k ∈ Z | 2k ordp(α) ≡ p − i +mod 5} +for 1 ≤ i ≤ 4. +Proof. For k ∈ Z, we have +⌊(p − 1 − 5j)/2⌋ = k ordp(α) for some j +⇔ k ordp(α) ≤ (p − 1 − 5j)/2 < k ordp(α) + 1 for some j +⇔ 2k ordp(α) = p − 1 − 5j or p − 2 − 5j for some j +⇔ k ∈ Sp,1(α) or k ∈ Sp,2(α). +We note that +j = p − i − 2k ordp(α) +5 +∈ Z +holds for k ∈ Sp,i(α). Since we have +α +j(5j+1) +2 += α +(p−1−2k ordp(α))(p−2k ordp(α)) +10 += +� +αp−2k ordp(α)� p−1−2k ordp(α) +10 += α +p−1−2k ordp(α) +10 +for every k ∈ Sp,1(α) and +α +j(5j+1) +2 += α +(p−2−2k ordp(α))(p−1−2k ordp(α)) +10 += +� +α +p−1−2k ordp(α) +2 +� p−2−2k ordp(α) +5 += +�α +p +� p−2−2k ordp(α) +5 += +�α +p +� +for every k ∈ Sp,2(α), we obtain +Fp(α) ≡ +� +k∈Sp,1(α) +α +p−1−2k ordp(α) +10 +�Ip(α) +k +� +− +�α +p +� +� +k∈Sp,2(α) +�Ip(α) +k +� +mod p +by Theorem 2.1 and Lemma 2.6. +□ +2.3. The case where p ≡ 2, 3, 4 mod 5. +In this subsection, we show the part of our main theorem (Theorem 1.1) which is the case +where p ≡ 2, 3, 4 mod 5. +Proposition 2.8. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. If p ̸≡ 0, 1 mod 5 +holds, then +Fp(α) ≡ +� aIp(α) +mod p +if p ≡ 2 +mod 5 +bIp(α) +mod p +if p ≡ 3, 4 +mod 5 +holds, where we define the rational sequence (am)m and (bm)m by +am = (−1)m � +k∈5Z +�� +m +k + 3m +� +− +�m +k +�� +bm = (−1)m � +k∈5Z +�� +m +k + 3m +� +− +� +m +k + 4m +�� +. +Remark 2.9. It is noted that am and bm are finite sums. + +6 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Proof of Proposition 2.8. Since Ip(α) ordp(α) = p − 1 ̸≡ 0 mod 5, we have ordp(α) ̸≡ 0 mod 5 +and Ip(α) ̸≡ 0 mod 5. Hence we get +Sp,i(α) = {k ∈ Z | 2k(p − 1) ≡ Ip(α)(p − 1) + Ip(α)(1 − i) +mod 5} += {k ∈ Z | (2k − Ip(α))(p − 1) ≡ Ip(α)(1 − i) +mod 5} += {k ∈ Z | k ≡ 3Ip(α){1 + (1 − i)/(p − 1)} +mod 5} +and +(2.3) +3Ip(α) +� +1 + (1 − i) +p − 1 +� +≡ + + + +3Ip(α) +mod 5 +if i = 1 +0 +mod 5 +if i = 2 and p ≡ 2 +mod 5 +3pIp(α) +mod 5 +if i = 2 and p ≡ 3, 4 +mod 5. +By using Proposition 2.7, we have +Fp(α) ≡ +� +k∈Sp,1(α) +α +p−1−2k ordp(α) +10 +�Ip(α) +k +� +− +�α +p +� +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ +� +k∈Sp,1(α) +α +ordp(α)(Ip(α)−2k) +10 +�Ip(α) +k +� +− α +ordp(α)Ip(α) +2 +� +k∈Sp,2(α) +�Ip(α) +k +� +mod p. +i) If the case where 2|Ip(α), we have +Fp(α) ≡ +� +k∈Sp,1(α) +αordp(α) +Ip(α)−2k +10 +�Ip(α) +k +� +− αordp(α) +Ip(α) +2 +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ +� +k∈Sp,1(α) +�Ip(α) +k +� +− +� +k∈Sp,2(α) +�Ip(α) +k +� +mod p. +If p ≡ 2 mod 5, then the equation (2.3) implies +Fp(α) ≡ +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +�Ip(α) +k +�� += aIp(α) +mod p. +If p ≡ 3 mod 5, by 3 · 3Ip(α) ≡ 4Ip(α) mod p, we have +Fp(α) ≡ +� +k∈Sp,1(α) +�Ip(α) +k +� +− +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +� +Ip(α) +k + 4Ip(α) +�� +mod p += bIp(α). + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +7 +If p ≡ 4 mod 5, by 3 · 4Ip(α) ≡ 2Ip(α) mod p, we have +Fp(α) ≡ +� +k∈Sp,1(α) +�Ip(α) +k +� +− +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +� +Ip(α) +k + 2Ip(α) +�� +≡ +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +� +Ip(α) +−k − Ip(α) +�� +≡ +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +� +Ip(α) +k + 4Ip(α) +�� +mod p += bIp(α). +ii) In the case where 2 ̸ |Ip(α), we have +Fp(α) ≡ +� +k∈Sp,1(α) +α +ordp(α) +2 +Ip(α)−2k +5 +�Ip(α) +k +� +− α +ordp(α) +2 +Ip(α) +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ +� +k∈Sp,1(α) +(−1) +Ip(α)−2k +5 +�Ip(α) +k +� +− (−1)Ip(α) +� +k∈Sp,2(α) +�Ip(α) +k +� +≡ − +� +k∈Sp,1(α) +�Ip(α) +k +� ++ +� +k∈Sp,2(α) +�Ip(α) +k +� +mod p. +In a similar way as (i), we have +Fp(α) ≡ + + + + + + + + + +− +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +�Ip(α) +k +�� += aIp(α) +mod p +if p ≡ 2 +mod 5 +− +� +k∈5Z +�� +Ip(α) +k + 3Ip(α) +� +− +� +Ip(α) +k + 4Ip(α) +�� += bIp(α) +mod p +if p ≡ 3, 4 +mod 5. +□ +To prove part of our main theorem, we show that the sequences {an}n and {bn} describe +the ordinary Fibonacci sequence in Proposition 2.12. To prove it, we define a dummy sequence +c5(i+1) as follows: +c5(i+1) = a5i+3 + b5i+4. +Lemma 2.10. For any non-negative integer m, we have the following relations: +i) b1 = F0 and a2 = F1. +ii) If m ≡ 1 mod 5, then bm + am+1 = am+2. +iii) If m ≡ 2 mod 5, then am + am+1 = bm+2. +iv) If m ≡ 4 mod 5, then bm + cm+1 = bm+2. +v) If m ≡ 0 mod 5, then cm + bm+1 = am+2. +Proof. i) They are verified directly by +b1 = −(0 − 0) = 0 = F0 and a2 = 2 − 1 = 1 = F1. +ii) By Pascal’s rule and a property of the following finite summation +� +k∈5Z +� m +k − i +� += +� +k∈5Z +� +m +k + 5 − i +� + +8 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +for i = 0, 1, 2, 3, 4, we have +bm + am+1 − am+2 += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 4 +�� ++ (−1)m+1 � +k∈5Z +��m + 1 +k + 1 +� +− +�m + 1 +k +�� +− (−1)m+2 � +k∈5Z +��m + 2 +k + 4 +� +− +�m + 2 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 4 +� +− +�m + 1 +k + 1 +� ++ +�m + 1 +k +� +− +�m + 2 +k + 4 +� ++ +�m + 2 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 4 +� +− +�m +k +� +− +� m +k + 1 +� ++ +� m +k − 1 +� ++ +�m +k +� +− +�m + 1 +k + 3 +� +− +�m + 1 +k + 4 +� ++ +�m + 1 +k − 1 +� ++ +�m + 1 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 4 +� +− +� m +k + 1 +� ++ +� m +k + 4 +� +− +�m + 1 +k + 3 +� +− +�m + 1 +k + 4 +� ++ +�m + 1 +k + 4 +� ++ +�m + 1 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 1 +� +− +�m + 1 +k + 3 +� ++ +�m + 1 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +� m +k + 1 +� +− +� m +k + 2 +� +− +� m +k + 3 +� ++ +� m +k − 1 +� ++ +�m +k +�� += (−1)m � +k∈5Z +� +− +� m +k + 1 +� +− +� m +k + 2 +� ++ +� m +k + 4 +� ++ +�m +k +�� += (−1)m � +k∈5Z +� +− +� m +k + 1 +� +− +� m +k + 2 +� ++ +� +m +m − k − 4 +� ++ +� +m +m − k +�� += (−1)m � +k∈5Z +� +− +� m +k + 1 +� +− +� m +k + 2 +� ++ +� m +k + 2 +� ++ +� m +k + 1 +�� += 0. +iii) Similarly, we have +a5m+2 + a5m+3 − b5m+4 += (−1)m � +k∈5Z +�� m +k + 1 +� +− +�m +k +� +− +�m + 1 +k + 4 +� ++ +�m + 1 +k +� +− +�m + 2 +k + 2 +� ++ +�m + 2 +k + 1 +�� += (−1)m � +k∈5Z +�� m +k + 1 +� +− +�m +k +� +− +� m +k + 3 +� +− +� m +k + 4 +� ++ +� m +k + 4 +� ++ +�m +k +� +− +�m + 1 +k + 1 +� +− +�m + 1 +k + 2 +� ++ +�m + 1 +k +� ++ +�m + 1 +k + 1 +�� += (−1)m � +k∈5Z +�� m +k + 1 +� +− +� m +k + 3 +� +− +�m +k +� +− +� m +k + 1 +� +− +� m +k + 1 +� +− +� m +k + 2 +� ++ +� m +k + 4 +� ++ +�m +k +� ++ +�m +k +� ++ +� m +k + 1 +�� + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +9 += (−1)m � +k∈5Z +� +− +� m +k + 3 +� +− +� m +k + 2 +� ++ +� m +k + 4 +� ++ +�m +k +�� += (−1)m � +k∈5Z +� +− +� m +k + 4 +� +− +�m +k +� ++ +� m +k + 4 +� ++ +�m +k +�� += 0. +iv) Similarly, we have +bm + cm+1 − bm+2 = am−1 + 2bm − bm+2 += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� ++ 2 +� m +k + 2 +� +− 2 +� m +k + 1 +� +− +�m + 2 +k + 3 +� ++ +�m + 2 +k + 4 +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� ++ 2 +� m +k + 2 +� +− 2 +� m +k + 1 +� +− +� m +k + 1 +� +− 2 +� m +k + 2 +� +− +� m +k + 3 +� ++ +� m +k + 2 +� ++ 2 +� m +k + 3 +� ++ +� m +k + 4 +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� +− 3 +� m +k + 1 +� ++ +� m +k + 2 +� ++ +� m +k + 3 +� ++ +� m +k + 4 +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� +− 3 +� m +k + 1 +� ++ +� m +k + 2 +� ++ +� m +k + 1 +� ++ +�m +k +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� +− 2 +� m +k + 1 +� ++ +� m +k + 2 +� ++ +�m +k +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 4 +� ++ +�m − 1 +k +� +− 2 +�m − 1 +k +� +− 2 +�m − 1 +k + 1 +� ++ +�m − 1 +k + 1 +� ++ +�m − 1 +k + 2 +� ++ +�m − 1 +k + 4 +� ++ +�m − 1 +k +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 1 +� ++ +�m − 1 +k + 2 +�� += (−1)m � +k∈5Z +� +− +�m − 1 +k + 1 +� ++ +�m − 1 +k + 1 +�� += 0. +v) By iv), we have +cm + bm+1 − am+2 = −bm−1 + 2bm+1 − am+2 += (−1)m � +k∈5Z +��m − 1 +k + 2 +� +− +�m − 1 +k + 1 +� +− 2 +�m + 1 +k + 3 +� ++ 2 +�m + 1 +k + 4 +� +− +�m + 2 +k + 1 +� ++ +�m + 2 +k +�� + +10 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA += (−1)m � +k∈5Z +��m − 1 +k + 2 +� +− +�m − 1 +k + 1 +� +− 2 +�m + 1 +k + 3 +� ++ 2 +�m + 1 +k + 4 +� +− +�m + 1 +k +� +− +�m + 1 +k + 1 +� ++ +�m + 1 +k + 4 +� ++ +�m + 1 +k +�� += (−1)m � +k∈5Z +��m − 1 +k + 2 +� +− +�m − 1 +k + 1 +� +− 2 +�m + 1 +k + 3 +� ++ 3 +�m + 1 +k + 4 +� +− +�m + 1 +k + 1 +�� += (−1)m � +k∈5Z +��m − 1 +k + 2 +� +− +�m − 1 +k + 1 +� +− 2 +�m − 1 +k + 1 +� +− 4 +�m − 1 +k + 2 +� +− 2 +�m − 1 +k + 3 +� ++3 +�m − 1 +k + 2 +� ++ 6 +�m − 1 +k + 3 +� ++ 3 +�m − 1 +k + 4 +� +− +�m − 1 +k + 4 +� +− 2 +�m − 1 +k +� +− +�m − 1 +k + 1 +�� += (−1)m � +k∈5Z +� +−4 +�m − 1 +k + 1 +� ++ 4 +�m − 1 +k + 3 +� ++ 2 +�m − 1 +k + 4 +� +− 2 +�m − 1 +k +�� += (−1)m � +k∈5Z +� +−4 +�m − 1 +k + 1 +� ++ 4 +�m − 1 +k + 1 +� ++ 2 +�m − 1 +k +� +− 2 +�m − 1 +k +�� += 0. +□ +Lemma 2.11. For any non-negative integer m ≥ 0, we have following relations: +i) If m ≡ 1 mod 5, then am = am+2. +ii) If m ≡ 2 mod 5, then bm = bm+2. +iii) If m ≡ 3 mod 5, then bm = cm+2. +iv) If m ≡ 4 mod 5, then am = bm+2. +Proof. i) It can be proved by direct calculation: +am − am+2 = (−1)m � +k∈5Z +�� m +k + 3 +� +− +�m +k +� +− +�m + 2 +k + 4 +� ++ +�m + 2 +k +�� += (−1)m � +k∈5Z +�� m +k + 3 +� +− +�m +k +� +− +� m +k + 2 +� +− 2 +� m +k + 3 +� +− +� m +k + 4 +� ++ +� m +k + 3 +� ++ 2 +� m +k + 4 +� ++ +�m +k +�� += (−1)m � +k∈5Z +�� m +k + 4 +� +− +� m +k + 2 +�� += (−1)m � +k∈5Z +�� m +k + 4 +� +− +� m +k + 4 +�� += 0. +ii) Similarly, we obtain +bm − bm+2 = (−1)m � +k∈5Z +�� m +k + 1 +� +− +� m +k + 3 +� +− +�m + 2 +k + 2 +� ++ +�m + 2 +k + 1 +�� += (−1)m � +k∈5Z +�� m +k + 1 +� +− +� m +k + 3 +� +− +�m +k +� +− 2 +� m +k + 1 +� +− +� m +k + 2 +� ++ +� m +k + 4 +� ++ 2 +�m +k +� ++ +� m +k + 1 +�� += (−1)m � +k∈5Z +�� m +k + 4 +� ++ +�m +k +� +− +� m +k + 3 +� +− +� m +k + 2 +�� + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +11 += (−1)m � +k∈5Z +�� m +k + 4 +� ++ +�m +k +� +− +� m +k + 4 +� +− +�m +k +�� += 0. +iii) Similarly, we obtain +bm − cm+2 = bm − am − bm+1 += (−1)m � +k∈5Z +� +− +� m +k + 2 +� ++ +�m +k +� ++ +�m + 1 +k + 2 +� +− +�m + 1 +k + 1 +�� += (−1)m � +k∈5Z +� +− +� m +k + 2 +� ++ +�m +k +� ++ +� m +k + 1 +� ++ +� m +k + 2 +� +− +�m +k +� +− +� m +k + 1 +�� += 0. +iv) Similarly, we obtain +a5i+4 − b5(i+1)+2 = (−1)m � +k∈5Z +�� m +k + 2 +� +− +�m +k +� +− +�m + 2 +k + 3 +� ++ +�m + 2 +k + 4 +�� += (−1)m � +k∈5Z +�� m +k + 2 +� +− +�m +k +� +− +� m +k + 1 +� +− 2 +� m +k + 2 +� +− +� m +k + 3 +� ++ +� m +k + 2 +� ++ 2 +� m +k + 3 +� ++ +� m +k + 4 +�� += (−1)m � +k∈5Z +�� m +k + 3 +� ++ +� m +k + 4 +� +− +�m +k +� +− +� m +k + 1 +�� += (−1)m � +k∈5Z +�� m +k + 3 +� ++ +� m +k + 4 +� +− +� m +k + 4 +� +− +� m +k + 3 +�� += 0. +□ +By the above lemmas, we have a part of conclusion: +Proposition 2.12. For any positive integer m ≥ 1, the following holds: +(2.4) +Fm−1 = + + + + + + + + + + + + + + + +cm = bm−2 +if m ≡ 0 +mod 5 +bm = am−2 +if m ≡ 1 +mod 5 +am +if m ≡ 2 +mod 5 +am = am−2 +if m ≡ 3 +mod 5 +bm = bm−2 +if m ≡ 4 +mod 5, +where a−1 = b−1 = 0. +Proof. By Lemma 2.10 and Lemma 2.11, we have the conclusion. +□ +Proposition 2.12 implies a part of our main theorem. +Corollary 2.13. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. If p ̸≡ 1 mod 5, +we have +Fp(α) ≡ FIp(α)−1 or FIp(α)+1 +mod p. + +12 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Proof. +(1) In the case where p ≡ 2 mod 5, Proposition 2.8 and Proposition 2.12 imply +Fp(α) ≡ aIp(α) +mod p += + + + + + + + + + +FIp(α)+1 +if Ip(α) ≡ 1 +mod 5 +FIp(α)−1 +if Ip(α) ≡ 2 +mod 5 +FIp(α)−1 +if Ip(α) ≡ 3 +mod 5 +FIp(α)+1 +if Ip(α) ≡ 4 +mod 5. +(2) In the cases where p ≡ 3 mod 5 and p ≡ 4 mod 5, Proposition 2.8 and Proposition +2.12 imply +Fp(α) ≡ bIp(α) += + + + + + + + + + +FIp(α)−1 +if Ip(α) ≡ 1 +mod 5 +FIp(α)+1 +if Ip(α) ≡ 2 +mod 5 +FIp(α)+1 +if Ip(α) ≡ 3 +mod 5 +FIp(α)−1 +if Ip(α) ≡ 4 +mod 5. +□ +2.4. The case where p ≡ 1 mod 5. +In this subsection, we treat the case where p ≡ 1 mod 5. +Proposition 2.14. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. If p ≡ 1 mod 5 +and ordp(α) ̸≡ 0 mod 5, then +Fp(α) ≡ + + + + + + + + + +(−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 1 +�� +mod p +if ordp(α) ≡ ±2 +mod 5 +(−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 2 +�� +mod p +if ordp(α) ≡ ±1 +mod 5. +hold. +Proof. First we prove +(2.5) +Fp(α) ≡ (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +� +Ip(α) +k + 2 ordp(α)3 +�� +mod p. +We note that +Sp,1(α) = {k ∈ Z | k ≡ 0 +mod 5} +Sp,2(α) = {k ∈ Z | 2k ordp(α) ≡ −1 +mod 5} += {k ∈ Z | k ≡ 2 ordp(α)3 +mod 5} +and +Fp(α) ≡ +� +k∈5Z +α +p−1−2k ordp(α) +10 +�Ip(α) +k +� +− +�α +p +� � +k∈5Z +� +Ip(α) +k + 2 ordp(α)3 +� +≡ +� +k∈5Z +� +α +p−1 +10 − k +5 ordp(α) +�Ip(α) +k +� +− α +p−1 +2 +� +Ip(α) +k + 2 ordp(α)3 +�� +≡ +� +k∈5Z +� +α +p−1 +10 +�Ip(α) +k +� +− α +p−1 +2 +� +Ip(α) +k + 2 ordp(α)3 +�� + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +13 +hold. +i) If 2|Ip(α), then we have +Fp(α) ≡ +� +k∈5Z +� +αordp(α) +Ip(α) +10 +�Ip(α) +k +� +− αordp(α) +Ip(α) +2 +� +Ip(α) +k + 2 ordp(α)3 +�� +≡ +� +k∈5Z +��Ip(α) +k +� +− +� +Ip(α) +k + 2 ordp(α)3 +�� +. +ii) If 2 ̸ |Ip(α), we have +Fp(α) ≡ +� +k∈5Z +� +α +ordp(α) +2 +Ip(α) +5 +�Ip(α) +k +� +− α +ordp(α) +2 +Ip(α) +� +Ip(α) +k + 2 ordp(α)3 +�� +≡ − +� +k∈5Z +��Ip(α) +k +� +− +� +Ip(α) +k + 2 ordp(α)3 +�� +. +Hence the equation (2.5) is proved. Second, we prove the conclusion of this proposition. Since +Ip(α) ≡ 0 mod 5, +� +k∈5Z +�Ip(α) +k + a +� += +� +k∈5Z +� +Ip(α) +Ip(α) − k − a +� += +� +k∈5Z +�Ip(α) +k − a +� +hold for every a ∈ Z. +The equation (2.5) can be written as +Fp(α) ≡ + + + + + + + + + +(−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 1 +�� +mod p +if ordp(α) ≡ ±2 +mod 5 +(−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 2 +�� +mod p +if ordp(α) ≡ ±1 +mod 5. +□ +Remark 2.15. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. If ordp(α) ≡ 0 +mod 5, then we have +Fp(α) = +� +α− +ordp(α) +10 ++ α +ordp(α) +10 +�Ip(α) +mod p. +The proof is given as follows. +Proof. Since +Sp,1 = {k ∈ Z | 0 ≡ 0 +mod 5} = Z +Sp,2 = {k ∈ Z | 0 ≡ −1 +mod 5} = ∅ +holds, we have +Fp(α) ≡ +� +k∈Z +α +p−1 +10 −k +ordp(α) +5 +�Ip(α) +k +� += α +p−1 +10 +� +α− +ordp(α) +5 ++ 1 +�Ip(α) += α +p−1 +10 − p−1 +10 +� +α− +ordp(α) +10 ++ α +ordp(α) +10 +�Ip(α) += +� +α− +ordp(α) +10 ++ α +ordp(α) +10 +�Ip(α) +mod p. +□ +Corollary 2.16. Let α ∈ Q \ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0. If p ≡ 1 mod 5 +and ordp(α) ̸≡ 0 mod 5, then +Fp(α) ≡ FIp(α)−1 or FIp(α)+1 +mod p +holds. + +14 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Proof. +i) The case where ordp(α) ≡ ±2 mod 5. By Lemma 2.4, Proposition 2.14, Ip(α) ≡ 0 +mod 5 and Pascal’s rule, we have +Fp(α) ≡ (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 1 +�� +mod p += (−1)Ip(α) � +k∈5Z +��Ip(α) − 1 +k + 4 +� ++ +�Ip(α) − 1 +k +� +− +�Ip(α) − 1 +k +� +− +�Ip(α) − 1 +k + 1 +�� += (−1)Ip(α)−1 � +k∈5Z +��Ip(α) − 1 +k + 1 +� +− +�Ip(α) − 1 +k + 4 +�� += (−1)Ip(α)−1 � +k∈5Z +�� +Ip(α) − 1 +(Ip(α) − k) − 2 +� +− +�Ip(α) − 1 +k + 4 +�� += (−1)Ip(α)−1 � +k∈5Z +��Ip(α) − 1 +k + 3 +� +− +�Ip(α) − 1 +k + 4 +�� += (−1)Ip(α)−1 � +k∈5Z +��Ip(α) − 2 +k + 2 +� ++ +�Ip(α) − 2 +k + 3 +� +− +�Ip(α) − 2 +k + 3 +� +− +�Ip(α) − 2 +k + 4 +�� += (−1)Ip(α)−2 � +k∈5Z +��Ip(α) − 2 +k + 4 +� +− +�Ip(α) − 2 +k + 2 +�� += (−1)Ip(α)−2 � +k∈5Z +��Ip(α) − 2 +k − 6 +� +− +�Ip(α) − 2 +k − 8 +�� += (−1)Ip(α)−2 � +k∈5Z +�� +Ip(α) − 2 +(k + 3Ip(α)) − 6 +� +− +� +Ip(α) − 2 +(k + 4Ip(α)) − 8 +�� += (−1)Ip(α)−2 � +k∈5Z +�� +Ip(α) − 2 +k + 3(Ip(α) − 2) +� +− +� +Ip(α) − 2 +k + 4(Ip(α) − 2) +�� += bIp(α)−2 = FIp(α)−1. +ii) The case where ordp(α) ≡ 1 mod 5. By the similar way, we have +Fp(α) ≡ (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� +− +�Ip(α) +k + 2 +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� ++ +�Ip(α) +k + 1 +� +− +�Ip(α) +k + 3 +� +− +�Ip(α) +k + 1 +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� ++ +�Ip(α) +k + 1 +� +− +�Ip(α) +k + 3 +� +− +�Ip(α) +k − 1 +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) +k +� ++ +�Ip(α) +k + 1 +� +− +�Ip(α) +k + 3 +� +− +�Ip(α) +k + 4 +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) + 1 +k + 1 +� +− +�Ip(α) + 1 +k + 4 +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) + 1 +k + 1 +� +− +�Ip(α) + 1 +k + 4 +� ++ +�Ip(α) + 1 +k +� +− +�Ip(α) + 1 +k +�� += (−1)Ip(α) � +k∈5Z +��Ip(α) + 1 +k +� ++ +�Ip(α) + 1 +k + 1 +� +− +�Ip(α) + 1 +k + 4 +� +− +�Ip(α) + 1 +k + 5 +�� + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +15 += (−1)Ip(α) � +k∈5Z +��Ip(α) + 2 +k + 1 +� +− +�Ip(α) + 2 +k + 5 +�� += (−1)Ip(α)+2 � +k∈5Z +��Ip(α) + 2 +k + 1 +� +− +�Ip(α) + 2 +k +�� += (−1)Ip(α)+2 � +k∈5Z +�� +Ip(α) + 2 +(k + 3Ip(α)) + 6 +� +− +�Ip(α) + 2 +k +�� += (−1)Ip(α)+2 � +k∈5Z +�� +Ip(α) + 2 +k + 3(Ip(α) + 2) +� +− +�Ip(α) + 2 +k +�� += aIp(α)+2 ≡ FIp(α)+1 +mod p. +□ +By Corollary 2.13 and Corollary 2.16, we conclude as follow: +Theorem 2.17 (Theorem 1.1). For α ∈ Q× and p ∈ P satisfying vp(α) = vp(α − 1) = 0 and +ordp(α) ̸≡ 0 mod 5, +Fp(α) ≡ FIp(α)−1 or FIp(α)+1 +mod p +holds. +3. Finite algebraic numbers and finite transcendence of q-Fibonacci sequence +In this section, we show that, under GRH, the element of A associated with the q-Fibonacci +sequence is a finite transcendental number (in Theorem 3.15). +3.1. Review on finite algebraic numbers. +This subsection discusses on finite algebraic numbers introduced in [8] which is defined by +recurrent sequences. +For (αp)p ∈ � +p Z/pZ and n ∈ Z>0, we define +Pn((αp)) := {p | αp ≡ n +mod p} ⊂ P. +By abuse of notation, the image of α = (αp)p ∈ +� +p +Z/pZ under the natural projection to A is +denoted to be α. We say n occurs infinitely often in α if |Pn((αp)p)| is infinite (we note that it +is independent of any choice of representatives in +� +p +Z/pZ). +A sequence (an)n ⊂ Q is called recurrent if there exists a monic polynomial f(x) := xd + +c1xd−1 + · · · + cd ∈ Q[x] such that +an+d + c1an+d−1 + · · · + cdan = 0 (on Q) +for every n ∈ N. Such f and (a0, . . . , ad−1) is called a characteristic polynomial and an initial +value of (an)n respectively. +Definition 3.1 ([8]). An element (αp)p ∈ A is called a finite algebraic number if there exists +a recurrent sequence (an)n such that +αp ≡ ap +mod p +for sufficiently every large p. Let P0 +A ⊂ A be the set of finite algebraic numbers. +A characterization of the finite algebraic numbers is given in [8]. +Theorem 3.2 ([8, Theorem 1.1]). Let α := (αp)p ∈ A . Then the following conditions are +equivalent: +(1) The element α ∈ A is a finite algebraic number. + +16 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +(2) There exists a Galois extension L/Q and a map φ : Gal(L/Q) → L satisfying φ(στσ−1) = +σφ(τ) for σ, τ ∈ Gal(L/Q) such that +(αp)p = (φ(Fp) +mod p)p, +where Fp is a Frobenius map of L at prime p. +Remark 3.3. +(1) The set P0 +A is a Q-subalgebra of A in [8]. +(2) [8, §4] explains that the Q-algebra P0 +A is regarded as a finite analogue of an algebraic +closure Q of Q. +Let C0 +A := {α ∈ A | ∃f(X) ∈ Q[X] \ {0} such that f(α) = 0}. +Theorem 3.4 ([8, Theorem 1.4]). +(1) For each α ∈ P0 +A , there exists a polynomial f(x) ∈ +Q[x]× such that f(α) = 0 in A . +(2) There is a sequence of inclusions of subsets: +Q ⊊ P0 +A ⊊ C0 +A ⊂ A . +Remark 3.5. Note that P0 +A is countable and C0 +A is uncountable ([8]). +Definition 3.6. An element of A \ C0 +A is called a finite transcendental number. +Proposition 3.7. Let α ∈ A . If there exists a sequence of distinct integers (an)n such that an +occurs infinitely often in α for every n ∈ N, then α is a finite transcendental number. +Proof. Suppose that α ∈ C0 +A . By the definition of C0 +A , we can take f(x) ∈ Q[x] \ {0} with +f(α) = 0 as an element in A . Since the components of (an)n is distinct, we can take n ∈ N +such that +f(an) ̸= 0 +in Q. This implies that +f(an) ̸≡ 0 +mod p +for sufficiently every large p. Since an occurs infinitely often in α for every n ∈ N, +f(α) ̸= 0 +holds on A . It contradicts that f(α) = 0 in A . +□ +Example 3.8. Let α ∈ A be represented by ( +1 +���� +1 , +2 +���� +1, 2 , +3 +� �� � +1, 2, 3, . . .). +Then every m ∈ Z>0 +occurs infinitely often in α. By Proposition 3.7, this α is a finite transcendental number. Hence +C0 +A ⊊ A holds. +3.2. Finite transcendence of q-Fibonacci sequence. +This subsection gives the proof of Theorem 1.2 (Theorem 3.15). +By Theorem 1.1, the q-Fibonacci sequence is related with the residual indices. Several results +on the residual indices are obtained under the following conjecture called generalized Riemann +hypothesis (GRH, in short). +Conjecture 3.9. The real part of every non-trivial zero of the Dedekind zeta function of an +algebraic field K is equal to 1 +2. +Remark 3.10. In this paper, we assume the GRH for all fields Kg +s,r = Q(ζs, g +1 +r ), where g is a +positive square-free integer, s and r are integers satisfying r|s and ζu is a u-th root of the unity. + +CONGRUENCES OF THE q-FIBONACCI SEQUENCE +17 +Under GRH, Hooley ([2]) calculated the density of primes p satisfying Ip(α) = 1 for a square- +free positive integer α. Under GRH, Lenstra ([5]) calculated the density of primes satisfying +Ip(α) = k for a square-free integer α and a positive integer k. He also calculated the condition +that the density of primes satisfying Ip(α) = k is equal to 0. Murata ([7]) showed the asymptotic +formulae on such primes less than a given positive real number x for a given square-free integer +α. +We prepare some notations to prove Corollary ?? (Corollary ??). Let v and s ∈ Z>0. Let ζs +be a root of unity and σb : Q(ζs) → Q(ζs), ζs �→ ζb +s for b ∈ Z satisfying (s, b) = 1. +• Let [a, b] be the least common multiple of a and b and (a, b) be the greatest common +divisor of a and b. +• Cg(b, f, v) = +� +1 +if σb|Q(ζf )∩Kg +v,v = id +0 +otherwise. +• Let µ : Z>0 → {0, ±1} be the M¨obius function, i.e. +µ(n) := +� +0 +if n has a squared prime factor +(−1)k +if n is the product of k distinct primes. +Moree showed the following lemma under GRH. +Lemma 3.11 ([6, Lemma 11]). Let a, d, t ∈ Z>0, let g ∈ Z\{0, ±1} be square-free and x ∈ R>0. +Put +Vg(a, d; t)(x) := |{p ≤ x | p ∈ P, Ip(g) = t, p ≡ 1 + ta +mod dt}|. +For sufficiently large x, +Vg(a, d; t)(x) = +x +log xδ(a, d; t) + Og,d +� x log log x +ϕ(t) log2 x + +x +log2 x +� +holds under GRH, where +δ(a, d; t) := +∞ +� +n=1 +(n,d)|a +µ(n)Cg(1 + ta; dt; nt) +[Kg +[d,n]t,nt : Q] +, +ϕ is Euler’s totient function and Og,d is the Landau notation with respect to g and d. +Proposition 3.12. Under GRH for all such fields Kg +s,r, if t ∈ (5Z>0 + 1) ∩ P, a ∈ Z>0 and g is +a square-free integer, then there are infinitely many primes p satisfying Ip(g) = t and p ≡ 1 + ta +mod 5t. +We prepare the following auxiliary lemmas to prove above proposition. +Lemma 3.13 ([2, Equation (12)]). Let g be a square free integer. Let s and r be positive +integers satisfying r|s. Then we have +[Kg +s,r : Q] = rϕ(s) +εg(s) , +(3.1) +where +εg(s) := +� +2 +if 2g|s and g ≡ 1 +mod 4 +1 +otherwise. +Lemma 3.14. Let a ≥ 2, b ≥ 2 and p be an odd prime. Let g ∈ Z \ {0, ±1} be a square-free +integer. +(1) If b|a, then we have [Q(ζap, g +1 +b ) : Q(ζa, g +1 +b )] ≥ p − 1 +2 +. +(2) If bp|a, then [Q(ζa, g +1 +bp ) : Q(ζa, g +1 +b )] = p holds. + +18 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +Proof. (1) By (3.1), we have +[Q(ζap, g +1 +b ) : Q(ζa, g +1 +b )] = [Q(ζap, g +1 +b ) : Q]/[Q(ζa, g +1 +b ) : Q] = +bϕ(ap) +εg(ap) +bϕ(a) +εg(a) += ϕ(ap) +ϕ(a) · εg(a) +εg(ap) ≥ ϕ(ap) +2ϕ(a). +• Suppose (a, p) = 1. Since ϕ(ap) +ϕ(a) = ϕ(p) = p − 1, we have [Q(ζap, g +1 +b ) : Q(ζa, g +1 +b )] ≥ +p − 1 +2 +. +• Suppose (a, p) = p and take a = cpk (c ∈ Z>0). +Since ϕ(c)ϕ(pk+1) +ϕ(c)ϕ(pk) += p, we have +[Q(ζap, g +1 +b ) : Q(ζa, g +1 +b )] = p +2 ≥ p − 1 +2 +. +(2) The equation (3.1) shows +[Q(ζa, g +1 +bp ) : Q(ζa, g +1 +b )] = [Q(ζa, g +1 +bp ) : Q(ζa)]/[Q(ζa, g +1 +b ) : Q(ζa)] = +bp +εg(a) · εg(a) +b += p. +□ +Proof of Proposition 3.12. By Lemma 3.11, it is sufficient to show δ(1, 5; t) > 0. +If we consider the prime factorization of n and that (n, 5)|1 and (n, 5) = 1 are equivalent, +then δ(1, 5; t) can be written as follows: +δ(1, 5; t) = +� +k≥0 +(−1)k +� +p1<···0 and a prime +number p. This implies if Cg(1 + t, 5t, p1 · · · pkt) = 0, then Cg(1 + t, 5t, p1 · · · pk+1t) = 0 holds. +Moreover, Cg(1+t, 5t, p1 · · · pkt)Cg(1+t, 5t, p1 · · · pk+1t) = Cg(1+t, 5t, p1 · · · pk+1t) holds. Thus +we have +δ(1, 5; t) += +� +k∈2Z≥0 + + + +� +p1<··· 0, we have +� +pk0 +1 +4l2 ≤ π2 +12 < 5 +6 < 1. +(ii) In the case where k = 0, since we have +[Q(ζ5pt, g +1 +pt ) : Q(ζ5t, g +1 +t )] = +ptϕ(5pt) +εg(5pt) +tϕ(5t) +εg(5t) +≥ 5pϕ(pt) +2 +≥ 5p(p − 1) +2 +≥ 5(p − 1)2 +2 +by (3.1), we have +� +p∈P +p̸=5 +Cg(1 + t, 5t, pt) +[Q(ζ5pt, g +1 +pt ) : Q(ζ5t, g +1 +t )] +≤ 2 +5 +� +p∈P +1 +(p − 1)2 ≤ π2 +15 < 2 +3. +So (3.2) is obtained. Hence we show the claim. +□ +Using the above lemma, we obtain the result on finite transcendental numbers. +Theorem 3.15 (Theorem 1.2). Let g ∈ Z>1 be a square-free integer. Then F = (Fp(g))p is a +finite algebraic transcendental number under GRH for all such fields Kg +s,r. +Proof. Let t ∈ (5Z>0 + 1) ∩ P. By Proposition 3.12, there are infinitely many primes p which +satisfy Ip(g) = t and p ≡ 1 + t mod 5t. Therefore there are infinitely many primes p which +satisfy Ip(g) = t and p ≡ 2 mod 5 by t ≡ 1 mod 5. By Theorem 1.1, at least one of Ft−1 +or Ft+1 occurs infinitely often in F. That means there exists a subsequence (an) of Fibonacci +sequence such that an occurs infinitely on F for every n ∈ N. Hence F /∈ C0 +A by Proposition +3.7. +□ +Acknowledgements + +20 +TAKUMI ANZAWA AND HIDETAKA FUNAKURA +This work was financially supported by JST SPRING, Grant Number JPMJSP2125. T.A. +would like to take this opportunity to thank the “Interdisciplinary Frontier Next-Generation +Researcher Program of the Tokai Higher Education and Research System”. +We are deeply +grateful to Hidekazu Furusho; without his profound instruction and continuous encouragement, +this paper would never be accomplished. We are grateful to Henrik Bachmann, Minoru Hirose, +Toshiki Matsusaka, Leo Murata, Shin-ichiro Seki, Koji Tasaka and Shuji Yamamoto to answering +our questions. We would like to thank Jun Ueki for giving us many advices of the structure of +this paper. +References +1. George E. Andrews, A polynomial identity which implies the rogers-ramanujan identities, Scripta Math 28 +(1970), 297–305. +2. Christopher Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220. +3. Masanobu Kaneko, Finite multiple zeta values, Various aspects of multiple zeta values, RIMS Kˆokyˆuroku +Bessatsu, B68, Res. Inst. Math. Sci. (RIMS), Kyoto, 2017, pp. 175–190. +4. Maxim Kontsevich, Holonomic D-modules and positive characteristic, Jpn. J. Math. 4 (2009), no. 1, 1–25. +5. Hendrik Willem Lenstra, Jr., On Artin’s conjecture and Euclid’s algorithm in global fields, Invent. Math. 42 +(1977), 201–224. +6. Pieter Moree, On the distribution of the order and index of g (mod p) over residue classes. II, J. Number +Theory 117 (2006), no. 2, 330–354. +7. Leo Murata, A problem analogous to Artin’s conjecture for primitive roots and its applications, Arch. Math. +(Basel) 57 (1991), no. 6, 555–565. +8. Julian Rosen, A finite analogue of the ring of algebraic numbers, J. Number Theory 208 (2020), 59–71. +9. Julian Rosen, Yoshihiro Takeyama, Koji Tasaka, and Shuji Yamamoto, The ring of finite algebraic numbers +and its application to the law of decomposition of primes, arXix math.NT 2208.11381 (2022). +10. Issai Schur, Ein beitrag additiven zahalentheorie und zur theorie der kettenbr¨uche, S.-B. Preuss. Akad. Wiss. +Phys.-Math. Kl (1917), 302–321. +11. Shin-ichiro Seki, The p-adic duality for the finite star-multiple polylogarithms, Tohoku Math. J. (2) 71 +(2019), no. 1, 111–122. MR 3920792 +Email address: m20001s@math.nagoya-u.ac.jp , hidetaka.funakura@gmail.com +graduate school of mathematics, nagoya university, furo-cho, chikusa-ku, nagoya, 464-8602, japan + diff --git a/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/load_file.txt b/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d13a63d9eedaf9001f39b567e1f024ef5bb0474 --- /dev/null +++ b/3dE1T4oBgHgl3EQfAQLV/content/tmp_files/load_file.txt @@ -0,0 +1,1534 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf,len=1533 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='02838v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='NT] 7 Jan 2023 CONGRUENCES OF THE q-FIBONACCI SEQUENCE RELATED WITH ITS FINITE TRANSCENDENCE TAKUMI ANZAWA AND HIDETAKA FUNAKURA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By using Andrews’s explicit formulae of q-Fibonacci sequence introduced by Schur, we prove certain congruences of the q-Fibonacci sequence which relate the sequence with the original Fibonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' As a corollary, we show that it yields a transcendental element in the Q-algebra A of integers modulo infinitely large primes under the generalized Riemann hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Congruences of q-Fibonacci sequence 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Review on q-analogues 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Preliminaries 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The case where p ≡ 2, 3, 4 mod 5 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The case where p ≡ 1 mod 5 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Finite algebraic numbers and finite transcendence of q-Fibonacci sequence 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Review on finite algebraic numbers 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Finite transcendence of q-Fibonacci sequence 16 References 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Introduction In 1917, Schur ([10]) introduced the so-called the q-Fibonacci sequence {Fn(q)} which is the sequence of Q[q] defined by the initial value (F0(q), F1(q)) = (0, 1) and the recurrence relation Fn+2(q) − Fn+1(q) − qnFn(q) = 0 for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' It recovers the ordinary Fibonacci sequence {Fn} when q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Andrews ([1]) gave an explicit formula (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1) of the q-Fibonacci sequence to prove some kind of the Rogers-Ramanujan identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let P be the set of prime numbers and let vp(α) denote the p-adic valuation of α for α ∈ Q× and p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For a pair (α, p) ∈ Q× × P with vp(α) = 0, ordp(α) denotes the order of α in the multiplicative group (Z/pZ)× and Ip(α) := (p − 1)/ ordp(α), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Ip(α) is the index of the subgroup of (Z/pZ)× generated by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' So when α is a primitive root, we have ordp(α) = p−1 and Ip(α) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The values ordp(α) and Ip(α) are called the residual order of α and the residual index of α respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Our main theorem is on congruence which relates the q-Fibonacci sequence with the ordinary one: Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 1 2 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For α ∈ Q× and p ∈ P satisfying vp(α) = vp(α − 1) = 0 and ordp(α) ̸≡ 0 mod 5, Fp(α) ≡ FIp(α)−1 or FIp(α)+1 mod p holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The following quotient Q-algebra A := \uf8eb \uf8ed� p∈P Z/pZ \uf8f6 \uf8f8 � \uf8eb \uf8ed� p∈P Z/pZ \uf8f6 \uf8f8 appeared in [4], has been studied in several literatures ([8], [9], [11], etc) in relation with the study of finite multiple zeta values (FMZVs, in short) introduced by Kaneko and Zagier ([3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Rosen ([8]) introduced the notion of finite algebraic numbers in A by using recurrent sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' It should be noted that solutions of Q-polynomials in A are not always finite algebraic numbers in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In this paper we consider finite transcendental numbers, elements in A which are not roots of non-zero Q-polynomials in A (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' It is expected that non-zero FMZVs are finite transcendental, however so far no single example has been otained as far as authors know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Our interests in this paper is to construct examples of finite transcendental number in the algebra A We show that the (Fp(α))p ∈ A is a finite transcendental number by combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1 and Moree’s result ([6]) on the density of certain primes related to residual index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Under the generalized Riemann hypothesis (GRH, in short), (Fp(g))p ∈ A is a finite transcendental number when g ∈ Z>1 is square-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Congruences of q-Fibonacci sequence In this section we prove our main theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Review on q-analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We recall the following standard notation: For n ∈ Z>0, the q-integer [n]q is defined by [n]q := 1 − qn 1 − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For n ∈ Z>0, the q-factorial [n]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' is defined by [n]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' := [n]q[n − 1]q · · · [1]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For a pair of integers n, m, the q-binomial coefficient � n m � q is defined as follows: � n m � q := � [n]q···[n−m+1]q [m]q···[1]q if 0 ≤ m ≤ n 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Andrews gave a general explicit formula of q-Fibonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1 ([1, Theorem]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For any non-negative integer n, Fn+1(q) = ∞ � j=−∞ (−1)jqj(5j+1)/2 � n ⌊(n − 5j)/2⌋ � q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1) holds, where ⌊x⌋ denotes the greatest integer not exceeding x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' CONGRUENCES OF THE q-FIBONACCI SEQUENCE 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In this subsection, we prepare some lemmas of q-integers and q-binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We show the prime congruence of the q-Fibonacci sequence by the lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let Z(p) be the localization of Z with respect to the prime ideal (p) generated by a prime number p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Note that the isomorphism of field Z/pZ ∼= Z(p)/pZ(p) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let l and k ∈ Z be integers satisfying 1 ≤ l ≤ p − 1 and k ≡ l mod ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then [k]α [l]α ∈ Z(p) \\ {0} and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2) [k]α [l]α ≡ � k l mod p if k ≡ 0 mod ordp(α) 1 mod p if k ̸≡ 0 mod ordp(α) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Note that for n ∈ Z, α ∈ Q \\ {0, 1} and p be a prime number satisfying vp(α) = vp(α − 1) = 0, [n]α ≡ 0 mod p holds if and only if n ≡ 0 mod ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let i be an integer satisfying 0 ≤ i ≤ ordp(α)−1 and k ≡ i mod ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If i ̸= 0, then [l]α ̸≡ 0 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We have [k]α [l]α = αk−iαi − 1 αl−iαi − 1 ≡ αi − 1 αi − 1 = 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If i = 0, we have [k]q [l]q ���� q=α = qk − 1 ql − 1 ���� q=α = (1 − qordp(α)) �k/ ordp(α)−1 s=0 qs ordp(α) (1 − qordp(α)) �l/ ordp(α)−1 t=0 qt ordp(α) ����� q=α = �k/ ordp(α)−1 s=0 αs ordp(α) �l/ ordp(α)−1 t=0 αt ordp(α) ≡ k/ ordp(α) l/ ordp(α) mod p = k l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q\\{0, 1} and p ∈ P satisfying vp(α) = vp(α−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let k be a positive integer with 0 ≤ k ≤ p − 1 − ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Put Ck := [p − k − 1]α · · · [p − k − ordp(α)]α [k + ordp(α)]α · · · [k + 1]α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then Ck ∈ Z(p) \\ {0} and � p − 1 k + ordp(α) � α ≡ Ck �p − 1 k � α mod p holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Especially if there exists l ∈ Z such that k = l ordp(α), then we have Cl ordp(α) ≡ Ip(α) − l l + 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By the definition, we have � p − 1 k + ordp(α) � α = [p − 1]α · · · [p − k − ordp(α)]α [k + ordp(α)]α · · · [1]α = [p − k − 1]α · · · [p − k − ordp(α)]α [k + ordp(α)]α · · · [k + 1]α × [p − 1]α · · · [p − k]α [k]α · · · [1]α = Ck �p − 1 k � α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 4 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Note that for any 1 ≤ i ≤ ordp(α), there exists uniquely ji ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' , ordp(α)} such that k + i ≡ p − k − ji mod ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' This correspondence is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We have Ck = [p − k − 1]α · · · [p − k − ordp(α)]α [k + ordp(α)]α · · · [k + 1]α = [p − k − j1]α [k + 1]α [p − k − j2]α [k + 2]α · · [p − k − jordp(α)]α [k + ordp(α)]α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By 1 ≤ k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' , k + ordp(α) ≤ p − 1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2 implies [p − k − ji]α [k + i]α ∈ Z(p) \\ {0} for every 1 ≤ i ≤ ordp(α) and Ck ∈ Z(p) \\ {0} holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In tha case where k is given by k = l ordp(α), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2 implies Cl ordp(α) ≡ [p − k − 1]α [k + ordp(α)]α × [p − k − 2]α [k + ordp(α) − 1]α × · · · × [p − k − ordp(α)]α [k + 1]α ≡ p − k − 1 k + ordp(α) ≡ Ip(α) − l l + 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q\\{0, 1} and p ∈ P satisfying vp(α) = vp(α−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let k be a positive integer satisfying 0 ≤ k < ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then �p − 1 k � α ≡ � 1 mod p if k = 0 0 mod p if k ̸= 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We have a conclusion immediately if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Assume 1 ≤ k < ordp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Note that [k]α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ̸≡ 0 mod p holds since k ̸≡ 0 mod ordp(α) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By [p − 1]α ≡ 0 mod p, we have �p − 1 k � α ≡ 0 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0, �p − 1 k � α ≡ � � Ip(α) k/ ordp(α) � mod p if k ∈ (ordp(α)) 0 mod p else holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The case of k ̸∈ (ordp(α)) is clear by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If we suppose k = l ordp(α), then we have �p − 1 k � α ≡ (Ip(α) − (l − 1))(Ip(α) − (l − 2)) · · · (Ip(α) − 0) (l − 1 + 1)(l − 2 + 1) · · · (0 + 1) 1 ≡ �Ip(α) l � mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ The following proposition is prime congruences of the q-Fibonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0, Fp(α) ≡ � k∈Sp,1(α) α p−1−2k ordp(α) 10 �Ip(α) k � − �α p � � k∈Sp,2(α) �Ip(α) k � mod p CONGRUENCES OF THE q-FIBONACCI SEQUENCE 5 holds, where Sp,i(α) = {k ∈ Z | 2k ordp(α) ≡ p − i mod 5} for 1 ≤ i ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For k ∈ Z, we have ⌊(p − 1 − 5j)/2⌋ = k ordp(α) for some j ⇔ k ordp(α) ≤ (p − 1 − 5j)/2 < k ordp(α) + 1 for some j ⇔ 2k ordp(α) = p − 1 − 5j or p − 2 − 5j for some j ⇔ k ∈ Sp,1(α) or k ∈ Sp,2(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We note that j = p − i − 2k ordp(α) 5 ∈ Z holds for k ∈ Sp,i(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since we have α j(5j+1) 2 = α (p−1−2k ordp(α))(p−2k ordp(α)) 10 = � αp−2k ordp(α)� p−1−2k ordp(α) 10 = α p−1−2k ordp(α) 10 for every k ∈ Sp,1(α) and α j(5j+1) 2 = α (p−2−2k ordp(α))(p−1−2k ordp(α)) 10 = � α p−1−2k ordp(α) 2 � p−2−2k ordp(α) 5 = �α p � p−2−2k ordp(α) 5 = �α p � for every k ∈ Sp,2(α), we obtain Fp(α) ≡ � k∈Sp,1(α) α p−1−2k ordp(α) 10 �Ip(α) k � − �α p � � k∈Sp,2(α) �Ip(α) k � mod p by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The case where p ≡ 2, 3, 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In this subsection, we show the part of our main theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1) which is the case where p ≡ 2, 3, 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ̸≡ 0, 1 mod 5 holds, then Fp(α) ≡ � aIp(α) mod p if p ≡ 2 mod 5 bIp(α) mod p if p ≡ 3, 4 mod 5 holds, where we define the rational sequence (am)m and (bm)m by am = (−1)m � k∈5Z �� m k + 3m � − �m k �� bm = (−1)m � k∈5Z �� m k + 3m � − � m k + 4m �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' It is noted that am and bm are finite sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 6 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since Ip(α) ordp(α) = p − 1 ̸≡ 0 mod 5, we have ordp(α) ̸≡ 0 mod 5 and Ip(α) ̸≡ 0 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Hence we get Sp,i(α) = {k ∈ Z | 2k(p − 1) ≡ Ip(α)(p − 1) + Ip(α)(1 − i) mod 5} = {k ∈ Z | (2k − Ip(α))(p − 1) ≡ Ip(α)(1 − i) mod 5} = {k ∈ Z | k ≡ 3Ip(α){1 + (1 − i)/(p − 1)} mod 5} and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3) 3Ip(α) � 1 + (1 − i) p − 1 � ≡ \uf8f1 \uf8f2 \uf8f3 3Ip(α) mod 5 if i = 1 0 mod 5 if i = 2 and p ≡ 2 mod 5 3pIp(α) mod 5 if i = 2 and p ≡ 3, 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='7, we have Fp(α) ≡ � k∈Sp,1(α) α p−1−2k ordp(α) 10 �Ip(α) k � − �α p � � k∈Sp,2(α) �Ip(α) k � ≡ � k∈Sp,1(α) α ordp(α)(Ip(α)−2k) 10 �Ip(α) k � − α ordp(α)Ip(α) 2 � k∈Sp,2(α) �Ip(α) k � mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' i) If the case where 2|Ip(α), we have Fp(α) ≡ � k∈Sp,1(α) αordp(α) Ip(α)−2k 10 �Ip(α) k � − αordp(α) Ip(α) 2 � k∈Sp,2(α) �Ip(α) k � ≡ � k∈Sp,1(α) �Ip(α) k � − � k∈Sp,2(α) �Ip(α) k � mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ≡ 2 mod 5, then the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3) implies Fp(α) ≡ � k∈5Z �� Ip(α) k + 3Ip(α) � − �Ip(α) k �� = aIp(α) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ≡ 3 mod 5, by 3 · 3Ip(α) ≡ 4Ip(α) mod p, we have Fp(α) ≡ � k∈Sp,1(α) �Ip(α) k � − � k∈Sp,2(α) �Ip(α) k � ≡ � k∈5Z �� Ip(α) k + 3Ip(α) � − � Ip(α) k + 4Ip(α) �� mod p = bIp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' CONGRUENCES OF THE q-FIBONACCI SEQUENCE 7 If p ≡ 4 mod 5, by 3 · 4Ip(α) ≡ 2Ip(α) mod p, we have Fp(α) ≡ � k∈Sp,1(α) �Ip(α) k � − � k∈Sp,2(α) �Ip(α) k � ≡ � k∈5Z �� Ip(α) k + 3Ip(α) � − � Ip(α) k + 2Ip(α) �� ≡ � k∈5Z �� Ip(α) k + 3Ip(α) � − � Ip(α) −k − Ip(α) �� ≡ � k∈5Z �� Ip(α) k + 3Ip(α) � − � Ip(α) k + 4Ip(α) �� mod p = bIp(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) In the case where 2 ̸ |Ip(α), we have Fp(α) ≡ � k∈Sp,1(α) α ordp(α) 2 Ip(α)−2k 5 �Ip(α) k � − α ordp(α) 2 Ip(α) � k∈Sp,2(α) �Ip(α) k � ≡ � k∈Sp,1(α) (−1) Ip(α)−2k 5 �Ip(α) k � − (−1)Ip(α) � k∈Sp,2(α) �Ip(α) k � ≡ − � k∈Sp,1(α) �Ip(α) k � + � k∈Sp,2(α) �Ip(α) k � mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In a similar way as (i), we have Fp(α) ≡ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 − � k∈5Z �� Ip(α) k + 3Ip(α) � − �Ip(α) k �� = aIp(α) mod p if p ≡ 2 mod 5 − � k∈5Z �� Ip(α) k + 3Ip(α) � − � Ip(α) k + 4Ip(α) �� = bIp(α) mod p if p ≡ 3, 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ To prove part of our main theorem, we show that the sequences {an}n and {bn} describe the ordinary Fibonacci sequence in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' To prove it, we define a dummy sequence c5(i+1) as follows: c5(i+1) = a5i+3 + b5i+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For any non-negative integer m, we have the following relations: i) b1 = F0 and a2 = F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) If m ≡ 1 mod 5, then bm + am+1 = am+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iii) If m ≡ 2 mod 5, then am + am+1 = bm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iv) If m ≡ 4 mod 5, then bm + cm+1 = bm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' v) If m ≡ 0 mod 5, then cm + bm+1 = am+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' i) They are verified directly by b1 = −(0 − 0) = 0 = F0 and a2 = 2 − 1 = 1 = F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) By Pascal’s rule and a property of the following finite summation � k∈5Z � m k − i � = � k∈5Z � m k + 5 − i � 8 TAKUMI ANZAWA AND HIDETAKA FUNAKURA for i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='bm + am+1 − am+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)m � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='+ (−1)m+1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��m + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�m + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− (−1)m+2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��m + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�m − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�m − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For any non-negative integer m ≥ 0, we have following relations: i) If m ≡ 1 mod 5, then am = am+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) If m ≡ 2 mod 5, then bm = bm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iii) If m ≡ 3 mod 5, then bm = cm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iv) If m ≡ 4 mod 5, then am = bm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' i) It can be proved by direct calculation: am − am+2 = (−1)m � k∈5Z �� m k + 3 � − �m k � − �m + 2 k + 4 � + �m + 2 k �� = (−1)m � k∈5Z �� m k + 3 � − �m k � − � m k + 2 � − 2 � m k + 3 � − � m k + 4 � + � m k + 3 � + 2 � m k + 4 � + �m k �� = (−1)m � k∈5Z �� m k + 4 � − � m k + 2 �� = (−1)m � k∈5Z �� m k + 4 � − � m k + 4 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) Similarly, we obtain bm − bm+2 = (−1)m � k∈5Z �� m k + 1 � − � m k + 3 � − �m + 2 k + 2 � + �m + 2 k + 1 �� = (−1)m � k∈5Z �� m k + 1 � − � m k + 3 � − �m k � − 2 � m k + 1 � − � m k + 2 � + � m k + 4 � + 2 �m k � + � m k + 1 �� = (−1)m � k∈5Z �� m k + 4 � + �m k � − � m k + 3 � − � m k + 2 �� CONGRUENCES OF THE q-FIBONACCI SEQUENCE 11 = (−1)m � k∈5Z �� m k + 4 � + �m k � − � m k + 4 � − �m k �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iii) Similarly, we obtain bm − cm+2 = bm − am − bm+1 = (−1)m � k∈5Z � − � m k + 2 � + �m k � + �m + 1 k + 2 � − �m + 1 k + 1 �� = (−1)m � k∈5Z � − � m k + 2 � + �m k � + � m k + 1 � + � m k + 2 � − �m k � − � m k + 1 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' iv) Similarly, we obtain a5i+4 − b5(i+1)+2 = (−1)m � k∈5Z �� m k + 2 � − �m k � − �m + 2 k + 3 � + �m + 2 k + 4 �� = (−1)m � k∈5Z �� m k + 2 � − �m k � − � m k + 1 � − 2 � m k + 2 � − � m k + 3 � + � m k + 2 � + 2 � m k + 3 � + � m k + 4 �� = (−1)m � k∈5Z �� m k + 3 � + � m k + 4 � − �m k � − � m k + 1 �� = (−1)m � k∈5Z �� m k + 3 � + � m k + 4 � − � m k + 4 � − � m k + 3 �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ By the above lemmas, we have a part of conclusion: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For any positive integer m ≥ 1, the following holds: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4) Fm−1 = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 cm = bm−2 if m ≡ 0 mod 5 bm = am−2 if m ≡ 1 mod 5 am if m ≡ 2 mod 5 am = am−2 if m ≡ 3 mod 5 bm = bm−2 if m ≡ 4 mod 5, where a−1 = b−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='10 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='11, we have the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12 implies a part of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ̸≡ 1 mod 5, we have Fp(α) ≡ FIp(α)−1 or FIp(α)+1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 12 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (1) In the case where p ≡ 2 mod 5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='8 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12 imply Fp(α) ≡ aIp(α) mod p = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 FIp(α)+1 if Ip(α) ≡ 1 mod 5 FIp(α)−1 if Ip(α) ≡ 2 mod 5 FIp(α)−1 if Ip(α) ≡ 3 mod 5 FIp(α)+1 if Ip(α) ≡ 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (2) In the cases where p ≡ 3 mod 5 and p ≡ 4 mod 5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='8 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12 imply Fp(α) ≡ bIp(α) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 FIp(α)−1 if Ip(α) ≡ 1 mod 5 FIp(α)+1 if Ip(α) ≡ 2 mod 5 FIp(α)+1 if Ip(α) ≡ 3 mod 5 FIp(α)−1 if Ip(α) ≡ 4 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The case where p ≡ 1 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In this subsection, we treat the case where p ≡ 1 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ≡ 1 mod 5 and ordp(α) ̸≡ 0 mod 5, then Fp(α) ≡ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (−1)Ip(α) � k∈5Z ��Ip(α) k � − �Ip(α) k + 1 �� mod p if ordp(α) ≡ ±2 mod 5 (−1)Ip(α) � k∈5Z ��Ip(α) k � − �Ip(α) k + 2 �� mod p if ordp(α) ≡ ±1 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' First we prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5) Fp(α) ≡ (−1)Ip(α) � k∈5Z ��Ip(α) k � − � Ip(α) k + 2 ordp(α)3 �� mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We note that Sp,1(α) = {k ∈ Z | k ≡ 0 mod 5} Sp,2(α) = {k ∈ Z | 2k ordp(α) ≡ −1 mod 5} = {k ∈ Z | k ≡ 2 ordp(α)3 mod 5} and Fp(α) ≡ � k∈5Z α p−1−2k ordp(α) 10 �Ip(α) k � − �α p � � k∈5Z � Ip(α) k + 2 ordp(α)3 � ≡ � k∈5Z � α p−1 10 − k 5 ordp(α) �Ip(α) k � − α p−1 2 � Ip(α) k + 2 ordp(α)3 �� ≡ � k∈5Z � α p−1 10 �Ip(α) k � − α p−1 2 � Ip(α) k + 2 ordp(α)3 �� CONGRUENCES OF THE q-FIBONACCI SEQUENCE 13 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' i) If 2|Ip(α), then we have Fp(α) ≡ � k∈5Z � αordp(α) Ip(α) 10 �Ip(α) k � − αordp(α) Ip(α) 2 � Ip(α) k + 2 ordp(α)3 �� ≡ � k∈5Z ��Ip(α) k � − � Ip(α) k + 2 ordp(α)3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) If 2 ̸ |Ip(α), we have Fp(α) ≡ � k∈5Z � α ordp(α) 2 Ip(α) 5 �Ip(α) k � − α ordp(α) 2 Ip(α) � Ip(α) k + 2 ordp(α)3 �� ≡ − � k∈5Z ��Ip(α) k � − � Ip(α) k + 2 ordp(α)3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Hence the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Second, we prove the conclusion of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since Ip(α) ≡ 0 mod 5, � k∈5Z �Ip(α) k + a � = � k∈5Z � Ip(α) Ip(α) − k − a � = � k∈5Z �Ip(α) k − a � hold for every a ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5) can be written as Fp(α) ≡ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (−1)Ip(α) � k∈5Z ��Ip(α) k � − �Ip(α) k + 1 �� mod p if ordp(α) ≡ ±2 mod 5 (−1)Ip(α) � k∈5Z ��Ip(α) k � − �Ip(α) k + 2 �� mod p if ordp(α) ≡ ±1 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If ordp(α) ≡ 0 mod 5, then we have Fp(α) = � α− ordp(α) 10 + α ordp(α) 10 �Ip(α) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The proof is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since Sp,1 = {k ∈ Z | 0 ≡ 0 mod 5} = Z Sp,2 = {k ∈ Z | 0 ≡ −1 mod 5} = ∅ holds, we have Fp(α) ≡ � k∈Z α p−1 10 −k ordp(α) 5 �Ip(α) k � = α p−1 10 � α− ordp(α) 5 + 1 �Ip(α) = α p−1 10 − p−1 10 � α− ordp(α) 10 + α ordp(α) 10 �Ip(α) = � α− ordp(α) 10 + α ordp(α) 10 �Ip(α) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ Q \\ {0, 1} and p ∈ P with vp(α) = vp(α − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If p ≡ 1 mod 5 and ordp(α) ̸≡ 0 mod 5, then Fp(α) ≡ FIp(α)−1 or FIp(α)+1 mod p holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 14 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' i) The case where ordp(α) ≡ ±2 mod 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Ip(α) ≡ 0 mod 5 and Pascal’s rule,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Fp(α) ≡ (−1)Ip(α) � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='mod p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α) � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='+ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='(Ip(α) − k) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='��Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k − 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='(k + 3Ip(α)) − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='(k + 4Ip(α)) − 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)−2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3(Ip(α) − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 4(Ip(α) − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= bIp(α)−2 = FIp(α)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ii) The case where ordp(α) 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='(k + 3Ip(α)) + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= (−1)Ip(α)+2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k∈5Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='Ip(α) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k + 3(Ip(α) + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�Ip(α) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='= aIp(α)+2 ≡ FIp(α)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='13 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='16, we conclude as follow: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='17 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For α ∈ Q× and p ∈ P satisfying vp(α) = vp(α − 1) = 0 and ordp(α) ̸≡ 0 mod 5, Fp(α) ≡ FIp(α)−1 or FIp(α)+1 mod p holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Finite algebraic numbers and finite transcendence of q-Fibonacci sequence In this section, we show that, under GRH, the element of A associated with the q-Fibonacci sequence is a finite transcendental number (in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Review on finite algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' This subsection discusses on finite algebraic numbers introduced in [8] which is defined by recurrent sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For (αp)p ∈ � p Z/pZ and n ∈ Z>0, we define Pn((αp)) := {p | αp ≡ n mod p} ⊂ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By abuse of notation, the image of α = (αp)p ∈ � p Z/pZ under the natural projection to A is denoted to be α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We say n occurs infinitely often in α if |Pn((αp)p)| is infinite (we note that it is independent of any choice of representatives in � p Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' A sequence (an)n ⊂ Q is called recurrent if there exists a monic polynomial f(x) := xd + c1xd−1 + · · · + cd ∈ Q[x] such that an+d + c1an+d−1 + · · · + cdan = 0 (on Q) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Such f and (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' , ad−1) is called a characteristic polynomial and an initial value of (an)n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' An element (αp)p ∈ A is called a finite algebraic number if there exists a recurrent sequence (an)n such that αp ≡ ap mod p for sufficiently every large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let P0 A ⊂ A be the set of finite algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' A characterization of the finite algebraic numbers is given in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2 ([8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α := (αp)p ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then the following conditions are equivalent: (1) The element α ∈ A is a finite algebraic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 16 TAKUMI ANZAWA AND HIDETAKA FUNAKURA (2) There exists a Galois extension L/Q and a map φ : Gal(L/Q) → L satisfying φ(στσ−1) = σφ(τ) for σ, τ ∈ Gal(L/Q) such that (αp)p = (φ(Fp) mod p)p, where Fp is a Frobenius map of L at prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (1) The set P0 A is a Q-subalgebra of A in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (2) [8, §4] explains that the Q-algebra P0 A is regarded as a finite analogue of an algebraic closure Q of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let C0 A := {α ∈ A | ∃f(X) ∈ Q[X] \\ {0} such that f(α) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4 ([8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (1) For each α ∈ P0 A , there exists a polynomial f(x) ∈ Q[x]× such that f(α) = 0 in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (2) There is a sequence of inclusions of subsets: Q ⊊ P0 A ⊊ C0 A ⊂ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Note that P0 A is countable and C0 A is uncountable ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' An element of A \\ C0 A is called a finite transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If there exists a sequence of distinct integers (an)n such that an occurs infinitely often in α for every n ∈ N, then α is a finite transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Suppose that α ∈ C0 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By the definition of C0 A , we can take f(x) ∈ Q[x] \\ {0} with f(α) = 0 as an element in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since the components of (an)n is distinct, we can take n ∈ N such that f(an) ̸= 0 in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' This implies that f(an) ̸≡ 0 mod p for sufficiently every large p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since an occurs infinitely often in α for every n ∈ N, f(α) ̸= 0 holds on A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' It contradicts that f(α) = 0 in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let α ∈ A be represented by ( 1 ���� 1 , 2 ���� 1, 2 , 3 � �� � 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then every m ∈ Z>0 occurs infinitely often in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='7, this α is a finite transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Hence C0 A ⊊ A holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Finite transcendence of q-Fibonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' This subsection gives the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='2 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1, the q-Fibonacci sequence is related with the residual indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Several results on the residual indices are obtained under the following conjecture called generalized Riemann hypothesis (GRH, in short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' The real part of every non-trivial zero of the Dedekind zeta function of an algebraic field K is equal to 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' In this paper, we assume the GRH for all fields Kg s,r = Q(ζs, g 1 r ), where g is a positive square-free integer, s and r are integers satisfying r|s and ζu is a u-th root of the unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' CONGRUENCES OF THE q-FIBONACCI SEQUENCE 17 Under GRH, Hooley ([2]) calculated the density of primes p satisfying Ip(α) = 1 for a square- free positive integer α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Under GRH, Lenstra ([5]) calculated the density of primes satisfying Ip(α) = k for a square-free integer α and a positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' He also calculated the condition that the density of primes satisfying Ip(α) = k is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Murata ([7]) showed the asymptotic formulae on such primes less than a given positive real number x for a given square-free integer α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We prepare some notations to prove Corollary ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (Corollary ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let v and s ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let ζs be a root of unity and σb : Q(ζs) → Q(ζs), ζs �→ ζb s for b ∈ Z satisfying (s, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let [a, b] be the least common multiple of a and b and (a, b) be the greatest common divisor of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Cg(b, f, v) = � 1 if σb|Q(ζf )∩Kg v,v = id 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let µ : Z>0 → {0, ±1} be the M¨obius function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' µ(n) := � 0 if n has a squared prime factor (−1)k if n is the product of k distinct primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Moree showed the following lemma under GRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='11 ([6, Lemma 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let a, d, t ∈ Z>0, let g ∈ Z\\{0, ±1} be square-free and x ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Put Vg(a, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t)(x) := |{p ≤ x | p ∈ P, Ip(g) = t, p ≡ 1 + ta mod dt}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' For sufficiently large x, Vg(a, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t)(x) = x log xδ(a, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t) + Og,d � x log log x ϕ(t) log2 x + x log2 x � holds under GRH, where δ(a, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t) := ∞ � n=1 (n,d)|a µ(n)Cg(1 + ta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' nt) [Kg [d,n]t,nt : Q] , ϕ is Euler’s totient function and Og,d is the Landau notation with respect to g and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Under GRH for all such fields Kg s,r, if t ∈ (5Z>0 + 1) ∩ P, a ∈ Z>0 and g is a square-free integer, then there are infinitely many primes p satisfying Ip(g) = t and p ≡ 1 + ta mod 5t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' We prepare the following auxiliary lemmas to prove above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='13 ([2, Equation (12)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let g be a square free integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let s and r be positive integers satisfying r|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Then we have [Kg s,r : Q] = rϕ(s) εg(s) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1) where εg(s) := � 2 if 2g|s and g ≡ 1 mod 4 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let a ≥ 2, b ≥ 2 and p be an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Let g ∈ Z \\ {0, ±1} be a square-free integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (1) If b|a, then we have [Q(ζap, g 1 b ) : Q(ζa, g 1 b )] ≥ p − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (2) If bp|a, then [Q(ζa, g 1 bp ) : Q(ζa, g 1 b )] = p holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' 18 TAKUMI ANZAWA AND HIDETAKA FUNAKURA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (1) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1), we have [Q(ζap, g 1 b ) : Q(ζa, g 1 b )] = [Q(ζap, g 1 b ) : Q]/[Q(ζa, g 1 b ) : Q] = bϕ(ap) εg(ap) bϕ(a) εg(a) = ϕ(ap) ϕ(a) · εg(a) εg(ap) ≥ ϕ(ap) 2ϕ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Suppose (a, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since ϕ(ap) ϕ(a) = ϕ(p) = p − 1, we have [Q(ζap, g 1 b ) : Q(ζa, g 1 b )] ≥ p − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Suppose (a, p) = p and take a = cpk (c ∈ Z>0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' Since ϕ(c)ϕ(pk+1) ϕ(c)ϕ(pk) = p, we have [Q(ζap, g 1 b ) : Q(ζa, g 1 b )] = p 2 ≥ p − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' (2) The equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='1) shows [Q(ζa, g 1 bp ) : Q(ζa, g 1 b )] = [Q(ζa, g 1 bp ) : Q(ζa)]/[Q(ζa, g 1 b ) : Q(ζa)] = bp εg(a) · εg(a) b = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content='11, it is sufficient to show δ(1, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' If we consider the prime factorization of n and that (n, 5)|1 and (n, 5) = 1 are equivalent, then δ(1, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t) can be written as follows: δ(1, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE1T4oBgHgl3EQfAQLV/content/2301.02838v1.pdf'} +page_content=' t) = � k≥0 (−1)k � p1<··· 𝑡𝑎𝑟𝑔𝑒𝑡’s value then +15: +𝑡𝑎𝑟𝑔𝑒𝑡 ← 𝑡 +⊲ Update the chosen tree +16: +end if +17: +end for +18: +if 𝑡𝑎𝑟𝑔𝑒𝑡 ≠ 𝑛𝑢𝑙𝑙 then +19: +delete trees from 𝑆 that intersect with 𝑡𝑎𝑟𝑔𝑒𝑡 +20: +end if +21: +end while +22: +return +23: end procedure +flow graph can be regarded as a directed graph, and the edge +weight represents the frequency of execution which indicates +the cold/hot information. Partitioning the code region can be +regarded as cutting the graph, where the weight of the cut +edge is the cost of performance and the obfuscation effect is +the number of the nodes in the sub-graph. +The region identifying algorithm. Based on the above +idea, we design the region identifying algorithm (algorithm +1) on top of the directed weighted graph cut algorithm [62] to +balance the performance overhead and the obfuscation effect. +The algorithm takes function code as input and performs +dominator tree analysis [40] (line 2) at first. To avoid sepa- +rating the whole function body into a sepFunc, we remove +the dominator tree of function itself (line 3) and identify the +regions from the rest of the trees. To indicate the effect of the +fission on obfuscation, we use the number of basic blocks in +the tree to represent it (line 7). To indicate the effect of the +fission on performance, we use the execution frequency of +the root node of the dominator tree by using block frequency +analysis [43] (line 8) and the loop count (if the region is in +a loop, the call to sepFunc will increase) as the cost of the +cut (lines 8-12). We iteratively select the most cost-effective +(i.e., maximum the ratio of effect and cost) dominator tree to +separate until the tree set is empty (lines 13-16). +3.2.2 +Data-flow Rebuild. In addition to identifying re- +gions as the function bodies of sepFuncs, we also need to +identify the inputs and the outputs of these regions to con- +struct the parameters and return value of sepFuncs. For each +variable used in a region, it should be an input if its point is +outside the region; Similarly, for each variable defined in a +region, it should be an output if it has a use point outside +the region. For example, as shown in Figure 2, the fd and n +variables are inputs because the defined points are outside +the region, and the value variable has a use point outside the +region, so it is an output. For the variables whose define-use +relationship are across regions, we use the function param- +eters to pass the pointer to them. We don’t pass a region’s +output variables by using the return value of sepFunc because +a region may have multiple output variables. +Data-flow reduction. In general, the local variables of a +function are defined at the entry basic block. Therefore, if an +identified region needs to use local variables, these variables +need to be passed into the sepFunc through parameters. In +fact, if some local variables are only used by a sepFunc, then +these variables do not need to be passed into the sepFunc, +they can be defined directly in the sepFunc. This can shorten +the length of the sepFunc parameter list, save unnecessary +variable transmission, and further improve performance. To +5 + +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang. +BB1 +BB4 +BB3 +BB5 +BB8 +BB6 +BB7 +BB2 +BB9 +exit 0 +exit 1 +value += cal(...) +return value +n = read(fd, …) +int fd = -1, n = 0 +define +use +region-1 +region-2 +control flow +data flow +basic block +region to split +statement +Figure 2. The control-flow and data-flow graphs of cal_file() in +Figure 1 +achieve this, we propose a lazy allocation strategy — if a +local variable is only used in the region, we will move the +variable definition to the sepFunc. For example, the n variable +in Figure 2 is initially defined in the oriFunc but redefined +and only used in the region-2, which becomes sepFunc-2 +function, so the definition point of the variable can be delayed +in the sepFunc-2 function. +3.2.3 +Control-flow Rebuild. We extract the basic blocks +of each identified region into a sepFunc. The jump relation- +ship between the regions in the oriFunc is transformed into +the function call-return relationship after fission. The cre- +ation of a function call is simple, we only need to insert the +function call at the location of the entry basic block of the +region before extraction and set the parameters that need to +be passed into the sepFunc. +The handling of function returns is relatively complex +due to: If a region has multiple exits, the corresponding +sepFunc needs to encode this information into the return +value, so that the remFunc can use this information to select +the corresponding code to execute. As Figure 2 shows, for the +two exits (0 and 1) in region-1, when sepFunc-1 returns +from exit 0, the control flow should go to BB5, and when +returns from exit 1, it should go to BB9. +We use the return value of sepFunc to indicate the remFunc +to determine the execution direction: We first number each +exit of the sepFunc, uses the number as its return value, and +then insert a basic block at the call-site of this sepFunc in the +remFunc (e.g., a○ in Figure 1) to transfer control flow based +on the return value. +3.2.4 +Handling the Exception Control-flows. During +program execution, there are some exception control flows +that deviate from the usual function call and return, including +the setjmp/longjmp and the C++ exception handling (EH +in short). The fission requires special handles of them. +Handling the setjmp/longjmp. Programmers could use +the setjmp() in a function to record the current context into +a jmpbuf structure. And then, they could use the longjmp() +in any subroutines on the call chain of this function to go +back to place the jmpbuf is pointing to, i.e., the call-site of +the setjmp(). There is a requirement here that the setjmp() +and the longjmp() using the same jmpbuf must be in the +same call chain. Therefore, the call-site of the setjmp() can- +not be separated into any sepFunc, because the stack frame of +the function that calling the setjmp() cannot be freed when +the corresponding longjmp() is executed. Otherwise, the +longjmp() will direct control flow to an unknown location. +Handling the C++ exception. The EH mechanism is a fea- +ture of the C++ that developers can capture exceptions in the +try block by writing the catch statements. Since the fission +moves part of the code into a sepFunc, the try-catch pair +may be broken, making EH information inconsistent. Simply +skipping the exception-relevant function would reduce the +obfuscation effect. Therefore, when identifying the code re- +gion, if it contains any code that may generate an exception, +we will locate the corresponding catch code and divide it +into the same region. +3.3 +The Fusion Primitive +The fusion selects functions to form fusFunc, and rebuilds +the control and the data flow to ensure the correctness. In +theory, the fusion can aggregate any number of functions. To +balance the performance overhead and the obfuscation effect, +we choose to aggregate two functions to form a fusFunc. +3.3.1 +Selecting Functions to Form fusFunc. The fusion +cannot arbitrarily select functions, it needs to select functions +with compatible types of the return values. The definition +of incompatibility is that if converting between two types +loses precision, the two types are incompatible. For example, +when the return value of one function is an integer and the +other is a float, these two functions cannot be aggregated. +In fact, there are other conditions that limit the selection of +functions: 1) The variadic functions, e.g., the printf(...); 2) +Two functions with incompatible types of the return values; +3) Two functions that have a direct calling relationship. The +first two constraints are designed for correctness, and the +last is designed for performance to avoid generating a lot of +recursive fusFuncs. Functions that meet the above constraints +will be randomly aggregated in pairs. +3.3.2 +Data-flow Rebuild. Once the two functions to be +aggregated are determined, the function prototype of the +corresponding fusFunc can be determined immediately. For +example, as shown in Figure 3 (a) and (b), the bar() and +the foo() are aggregated into int bar_foo_fusion(). The +ctrl parameter is used to select the function bodies aggre- +gated from the bar() and the foo(). Determining the func- +tion prototype of fusFunc is crucial to the rebuild of the data +flow, which involves setting the parameter list and return +value. +6 + +Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +void bar(short a, float b) { +// bar's code +printf("bar: %d, %f\n", a, b); +} +int foo(int m) { +// foo's code +printf("foo: %d\n", m); +return m; +} +int main() { +bar(0x1234, 0.1); +int res = foo(1); +... +} +(a) Before fusion +(b) Fusion w/o parameter compression +(c) Fusion w/ parameter compression +int bar_foo_fusion(int ctrl, short a, float b, int m) { +if (ctrl) { // bar's code +printf("bar: %d, %f\n", a, b); +return 0; +} else { // foo's code +printf("foo: %d\n", m); +return m; } +} +int main() { +// ctrl is 1, executing bar +bar_foo_fusion(1, 0x1234, 0.1, 0); +// ctrl is 0, executing foo +int res = bar_foo_fusion(0, 0, 0.0, 1); +} +int bar_foo_fusion(int ctrl, int x, float b) { +if (ctrl) { // bar's code +printf("bar: %d, %f\n", (short)x, b); +return 0; +} else { // foo's code +printf("foo: %d\n", x); +return x; } +} +int main() { +// ctrl is 1, executing bar +bar_foo_fusion(1, 0x1234, 0.1); +// ctrl is 0, executing foo +int res = bar_foo_fusion(0, 1, 0.0); +} +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +Figure 3. An example of performing the fusion on two functions. +Parameter list compression. Simply merging the param- +eter lists of the two functions makes the parameter list of +fusFunc too long, which will degrade the performance of +calling fusFunc. This is because in the X86_64 calling con- +vention, the first six parameters are passed in registers, and +the rest of the parameters are passed on the stack, which is +an inefficient way. To achieve efficient parameter passing, +we propose a parameter list compression mechanism — if +the types of the two parameters from the two functions are +compatible, we compress them into one. The reason why +we can do this is that when a fusFunc is called, only the +parameter list of one of the functions participating in the +aggregation is used. For example, as Figure 3(c) shows, both +the bar() and the foo() have an integer parameter (short +a and int m), we compress them into one integer parameter +(int x). +If a parameter can not participate in the compression, it +is copied into the parameter list of the fusFunc. The number +of parameters after the fusion will increase. In the worst +case, it is the sum of the parameters of the two functions, +which means none of the parameters can be compressed. To +avoid using the stack to pass parameters as much as possible, +we preferentially select functions with the total number of +parameters less than six for the fusion. +Return value determination. Determining the return type +of fusFunc is relatively simple: 1) If the return type of one +function is void, then the return type of the fusFunc is the +return type of another; 2) If the return types of the two +functions are both not void, the compressed type is used +as the return type of the fusFunc, which is similar to the +parameter list compression mechanism. +3.3.3 +Control-flow Rebuild. Once the fusFunc is created, +the two involved oriFuncs need to be removed, and all call- +sites to the oriFuncs need to be replaced to call the fusFunc. +As mentioned before, a ctrl parameter will be added into +the parameter list of the fusFunc to select the code block +aggregated from the oriFuncs. The value of this parameter is +int bar() { +// bar's code +} +int foo() { +// foo's code +} +int (*fptr)(); +int main(int argc) { +if (...) +fptr = &bar; +else +fptr = &foo; +int res = fptr(); +... +} +int (*fptr)() +int res = fptr() +fptr = (&bar_foo) | tag +value +tag +if (extract_tag(fptr)) +res = tmp +fptr = &bar +fptr = &foo +tmp = fptr() +val = clear_tag(fptr) +tmp = val(extract_tag()) +(a) +(b) +(c) +Figure 4. Function reference and indirect call processing. +0 or 1, which is set according to the original call-site of the +oriFunc. Since the fusFunc parameter list includes the param- +eters of both oriFuncs, we only need to pass the parameters +required by the oriFunc to the fusFunc at the call-site of this +oriFunc. Unused parameters are set to be 0. +Handling Indirect function calls. Indirect function calls +are more difficult to handle than direct function calls because +we do not know where the oriFunc will be called. Figure 4 (a) +shows an example that calls two functions by de-referencing +the function pointer. The corresponding data flow is given +in Figure 4 (b). When aggregating the bar() and the foo(), +we need to change the function pointer points to the fusFunc +and then replace the function call to call this fusFunc. But, +we encounter a problem that we do not know what the value +of the ctrl parameter should be set to. This is because at the +compile time, we don’t know whether the original function +pointer fptr points to the bar() or the foo(). +To address the above problem, we propose a tagged pointer +mechanism, which is similar to the low-fat pointer [39]. The +core idea is to encode the information (called tag) of which +oriFunc pointed to by the original function pointer into the +new function pointer, and when the new function pointer is +de-referenced to make a call, the value of the ctrl parameter +can be dynamically determined by parsing the new function +pointer. In detail, when the operation of taking the address of +7 + +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang. +the function participating in the aggregation occurs, we need +to perform the encoding operation. Since the tag is encoded +into the function pointer, it can be propagated along with the +function pointer. When the function pointer is de-referenced +to make a call, we will extract the tag in the pointer as the +and set the ctrl parameter according to the tag. +The tag requires two extra bits, where a bit indicates +whether the pointer points to a fusFunc, and the other bit +records the value of the ctrl parameter. For example, as +shown in Figure 4 (c), if the pointer fptr points to the bar(), +the value of the tag will be set to 11b. When the pointer fptr +is dereferenced to make a call, we insert code to first check +whether the tag is empty. If not, the code will extract the +ctrl parameter and call the fusFunc. Otherwise, no addi- +tional operations are required. +We choose the 2nd bit and the 3rd bit of function pointers +to place the tag. This is because the functions are usually +16-bytes aligned with the performance consideration, so +the lowest 4 bits of the function pointer can be used (more +reasons and considerations are detailed in A.1). +Handling function calls across modules. There are two +cases of cross-module function calls, one is the function +pointer of a module is propagated to other modules, and +the other is a module directly calls functions exported by +other modules. If any case happens on a fusFunc, we needs +to process all involved modules to ensure the fusFunc can be +called correctly. But in some cases, we can not process all +the modules (e.g., some libraries may have no source code). +To address this problem, we propose a trampoline mecha- +nism so that all modules do not need to be processed. In detail, +we transverse the data flow conservatively and identify all +function pointers that may propagate outside the module. +And then, we modify these function pointers to point to a +piece of trampoline code instead of the fusFunc. So that when +the external module calls these function pointers, the con- +trol flow will transfer to the trampoline code first, and the +trampoline code will help the function outside the module to +reorganize the function parameters and call the fusFunc. For +the exported oriFuncs, the method is similar to the replacing +the oriFunc’s function body with the trampoline code. +3.3.4 +The Deep Fusion. To further improve the obfusca- +tion effect, we propose a deep fusion method to aggregate +as many basic blocks as possible between the two parts of +the code during the fusion process. +We have observed that some basic blocks can be executed +many times without affecting the normal function. The char- +acteristic of these basic blocks is that their execution does +not affect the global memory state, and they are called the in- +nocuous basic block in this paper. The concept is very similar +to the reentrant function [56] that it can be re-executed with- +out affecting the functionality of the program. For innocuous +basic blocks from different oriFuncs, they can be aggregated +together within the fusFunc. The innocuous analysis of each +Current = tmp1; +oldtr = tmp2; +return; +delta = tr - oldtr; +if (delta < -10) delta += 256; +tmp1 = Current + delta * 1000; +tmp2 = tr; +Update(int tr) +int delta; +static int oldtr = -1; +int tmp1 = 0, tmp2 = 0; +tmp1 = 0; +tmp2 = 0; +oldtr == -1? +1 +2 +3 +4 +5 +UMV(int y, int x, int height, int width) +int width4 = ((width+2*4-1)<<2); +int height4 = ((height+2*4-1)<<2); +x = x + IMG_PAD_SIZE*4; +y = y + IMG_PAD_SIZE*4; +7 +6 +Fusion(ctrl,…) +7 +1 +2 +8 +4 +5 +0 +control flow +basic block +function +basic blocks +Figure 5. A real-world example of the deep fusion method. +basic block is conservative. For example, 1) if a memory write +operation in a basic block cannot be determined whether +the modified data is local or global, then this basic block is +not innocuous; 2) if there is a function call to an external +function in a basic block, this basic block is not innocuous. +We give a simplified example of 464.h264ref in SPEC +CPU 2006 benchmark. As shown in Figure 5, the Update() +and UMV() are aggregated into the Fusion(). The basic block +(BB) 3○ of the Update() firstly redefines the local variable +delta, and then loads the value of global variable Current, +and writes two local variables tmp1 and tmp2 at last. Since +these operations do not affect the global memory state, the +BB 3○ is determined to be innocuous, and so as the BB 6○ of +UMV(), thus we aggregate them into one — the BB 8○. +This deep fusion method modifies the control flow graph +and data flow graph of the fusFunc at the same time, adding +data dependency and control dependency so that the fusFunc +cannot be simply separated back to the two functions. +3.4 +Combining the Fission and the Fusion +The fission and the fusion can be used together to further +enhance the obfuscation effect. There are three combinations +as follows: +• FuFi.sep: Only aggregating the sepFuncs generated by +the fission. In this case, the issue of handling indirect +function calls no longer exists; +• FuFi.ori: Only aggregating the oriFuncs that are not pro- +cessed by the fission, e.g., the functions with only one +basic block. This combination could balance the obfus- +cation effect and the performance overhead well, and is +suitable for software in most real-world scenarios; +• FuFi.all: Aggregating the fission-generated sepFuncs and +the fission-unprocessed oriFuncs uniformly and randomly. +In this combination, the obfuscation effect is prioritized, +followed by the performance overhead. It is suitable for +programs that require a high obfuscation effect. +4 +Evaluation +We implemented Khaos based on the LLVM-9.0.1. The fis- +sion and the fusion are implemented as the middle-end +passes, and the fission pass is scheduled before the fusion +8 + +Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +-5 +15 +35 +55 +75 +400.perlbench +401.bzip2 +403.gcc +429.mcf +433.milc +444.namd +445.gobmk +447.dealll +450.soplex +453.povray +456.hmmer +458.sjeng +462.libquantum +464.h264ref +470.lbm +471.omnetpp +473.astar +482.sphinx3 +483.xalancbmk +GEOMEAN +overhead(%) +Fission +Fusion +FuFi.sep +FuFi.ori +FuFi.all +-5 +15 +35 +55 +75 +500.perlbench_r +502.gcc_r +505.mcf_r +508.named_r +510.parest_r +511.povray_r +519.lbm_r +520.omnetpp_r +523.xalancbmk_r +525.x264_r +526.blender_r +531.deepsjeng_r +538.imagick_r +541.leela_r +544.nab_r +557.xz_r +600.perlbench_s +602.gcc_s +605.mcf_s +619.lbm_s +620.omnetpp_s +623.xalancbmk_s +625.x264_s +631.deepsjeng_s +638.imagick_s +641.leela_s +644.nab_s +657.xz_s +GEOMEAN +overhead(%) +111 131 +138 160 +Figure 6. Runtime overhead of SPEC CPU 2006 (upper part) and 2017 (lower part) C/C++ programs. +pass. We run Khaos on Ubuntu 20.04 (Kernel v5.4.0) that +runs on an Intel(R) Xeon(R) Gold 5218 CPU with 128G mem- +ory. This section evaluates Khaos in terms of effectiveness +and performance, and answers the following questions: +• (Q1) How is the performance of the obfuscated programs? +• (Q2) How does Khaos work against the state-of-the-art +binary diffing techniques? +• (Q3) How good is Khaos at hiding real vulnerable code? +Test Suites. We used three test suites to evaluate Khaos: 1) +All C/C++ programs in SPEC CPU 2006/2017 benchmarks +with the ref input (denoted as the T-I); 2) All 108 programs +in the CoreUtils 8.32 (denoted as the T-II); 3) Five commonly +used programs in embedded devices with at least one vul- +nerability, including two popular IoT JavaScript engines (Jer- +ryScript and QuickJS), OpenSSL-1.1.1, BusyBox-1.33.1 and +libcurl-7.34.0 (denoted as the T-III). The performance eval- +uation was performed on the T-I (Q1); The effectiveness +against binary diffing techniques was evaluated on the T-I +and the T-II (Q2); The ability to hide vulnerable code was +evaluated on the T-III (Q3). Since software developers typi- +cally link programs into a single binary in embedded devices, +we compiled and obfuscated these test suites in the same +way under O2 with the link-time optimization (LTO). +Comparison targets. To compare with existing obfusca- +tor, we choose the popular compiler-level obfuscation tool +O-LLVM [36] as our comparison target because it is open- +sourced and compiler-based (same as Khaos). O-LLVM [36] +contains three obfuscation methods: instruction substitu- +tion (Sub), bogus control flow (Bog), and control flow flatten- +ing (Fla). Literatures [5, 20, 57, 69] in software engineering, +systems security, and programming languages fields all use +it in their experiments. To ensure the consistency of the +evaluation environment, we upgrade the LLVM version of O- +LLVM [36] to 9.0.1, which is same as Khaos. We also choose +BinTuner [57], which is an iterative compiler tool that uses +compiler options to transform the code to enlarge the differ- +ence of binaries, as another target to compare Khaos with +compiler’s options. +Confrontation targets. We use five state-of-the-art binary +diffing techniques, i.e., Google BinDiff [81], VulSeeker [26], +Asm2Vec [20], SAFE [45], DeepBinDiff [21], to evaluate the +effectiveness of Khaos. Among them, Google BinDiff is an +industry-standard binary diffing tool. Asm2Vec, SAFE, Deep- +BinDiff, and VulSeeker are the state-of-the-art methods for +learning the semantic similarity in different granularity (e.g., +function, basic block, control flow graph, call graph). +4.1 +Performance Overhead after Obfuscation +We separately evaluated the performance overhead of the +fission and the fusion, and the three combination modes in- +troduced in subsection 3.4 on the T-I. As shown in Figure 6, +the geometric performance overhead of the fission and the fu- +sion are 5% and 6%, respectively. The reason why some cases +(e.g., 456.hmmer) have a negative performance overhead is +that after the fission separates part of the code, the remFunc +can be further inlined to its callers, and the fusion improves +the code locality of the aggregated functions. The results +demonstrated that obfuscations compliant with the compiler +optimizations can have good performance advantages. +Compared with the FuFi.ori, the other two combinations +have a higher overhead because the fission generates many +sepFuncs, aggregating them all incurs non-negligible per- +formance overhead. For example, the 502.gcc_r contains +many recursive functions, the sepFuncs generated by these +functions are aggregated to the fusFuncs which are also the +recursive functions. Since the stack frames of fusFuncs are +larger, they will bring much pressure to the stack. +Compared with O-LLVM[36]. We compared the perfor- +mance overhead of Khaos with Sub, Bog, Fla. As shown +9 + +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang. +6 +3 +282 +22 +3 +5 +8 +5 +14 +6 +9 +277 +39 +5 +6 +15 +7 +23 +6 +7 +279 +32 +5 +6 +12 +6 +19 +0 +20 +40 +60 +80 +Sub +Bog +Fla +Fla-10 +Fission +Fusion +FuFi.sep FuFi.ori +FuFi.all +overhead(%) +SPEC CPU 2006 +SPEC CPU 2017 +GEOMEAN +Figure 7. Runtime overhead of O-LLVM and Khaos. +Table 1. Summarize of chosen diffing works. +diffing +symbol +time +memory +call-graph +granularity +relying +consuming +consuming +lacking +BinDiff [81] +function +Y +N +N +N +VulSeeker [26] +function +N +Y +Y +Y +Asm2Vec [20] +function +N +N +N +Y +Safe [45] +function +N +N +N +Y +DeepBinDiff [21] +basic block +N +Y +Y +N +in Figure 7, Khaos has comparable overhead with the Sub +and the Bog. Due to the high overhead of Fla, we reduce its +obfuscation ratio to 10% (Fla-10), and others are all at 100%. +4.2 +The Effectiveness against Binary Diffing +Comparing binary diffing works is challenging due to their +measurements of similarity are very different [57], such as +graph edit distance or statistical significance. Simply com- +paring their similarity scores does not provide accurate infor- +mation. For the commercial binary diffing tool BinDiff [81], +we normalized its similarity score to [0, 1]. For other tools +open-sourced in academia, we normalized their results by +computing the ratio of true matching function pairs that are +also the top-ranked matching candidates (i.e. Precision@1). +Paring success judgment method. Since Khaos changes +the number of functions, we relax the requirements for Pre- +cision@1. For the fission, if the oriFunc is paired with any +sepFuncs generated from it or the remFunc, this pairing is rec- +ognized as successful. For the fusion, if the fusFunc is paired +with any function before the fusion, this pairing is recog- +nized as successful. For the DeepBinDiff [21], since its result +is basic block to basic block, the pairing is recognized as +successful as long as their belonging functions are matched, +even if the two basic blocks are not truly matched. It is worth +noting that the above setting is looser than originally used +in these tools but is more challenging for Khaos. +Test suite adjustment adaptability. The characteristics +of used binary diffing tools were summarized in Table 1. +The column “symbol relying” means the un-stripped binaries +whether have side-effects or not, for example, BinDiff usually +uses function names to reduce the searching space; The col- +umn “time consuming” or the column “memory consuming” +means the diffing process takes a long time or requires a lot +of memory; The column “call-graph lacking” means whether +using the call-graph as the feature or not. +The test suites for VulSeeker [26] and DeepBinDiff [21] +need to be adjusted due to unable to run results. VulSeeker [26] +takes more than 1 day to diff two large binaries and often +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BinDiff +VulSeeker +Asm2Vec +Safe +DeepBinDiff +precision@1 +Sub +Bog +Fla-10 +Fission +Fusion +FuFi.sep +FuFi.ori +FuFi.all +Figure 8. Precision@1 result of chosen binary diffing work. +gets killed due to memory limit. To speed up VulSeeker, we +group the related functions into small groups (30 functions +per group) to manually reduce the searching space, which is +unfavorable to Khaos because the smaller the group size, the +easier to diff. DeepBinDiff [21] requires too much memory +(sometimes more than 10 TB) due to its representation of +basic blocks. Since its diffing process is tightly coupled with +binary size, we decide not to modify it and only use pro- +grams less than 40k lines. Even with the reduced test suite, +it is still time consuming (e.g., over 1 week to diff binaries of +508.namd_r). It’s worth mentioning that this is also unfavor- +able to Khaos because it uses original functions to obfuscate +each other, lacking material reduces the obfuscation effect. +Other binary diffing tools still use the normal test suites. +Results. We evaluated the accuracy of these tools by com- +paring obfuscated and un-obfuscated (un-stripped) binaries +on the T-I and T-II. As Figure 8 shows, higher accuracy +means lower adversarial effect. Since BinDiff [81] takes the +advantage of function names, its result is much higher than +others. Although DeepBinDiff [21] uses the basic block level +instead of the function level as its granularity, the feature +vector of the basic block still encodes the control flow graph +and call graph, which have been changed by Khaos, and +that’s why Khaos can defeat it. With comparable overhead, +Khaos can achieve a much better adversarial effect than +O-LLVM [36]. +Compared with compiler options. We follow the com- +pare method of BinTuner[57] to calculate the similarity score +of BinDiff[81] under different compiler settings. For the Bin- +Tuner part, we set O0’s binary code (same setting in the +paper[57]) as the baseline during its iterative compilation. +For the Khaos part, we use binaries generated by FuFi.all. As +shown in Figure 9, Khaos has a much lower similarity score +in different compiler options. We also compared the over- +head of programs generated by BinTuner with the baseline +of Khaos (O2 with LTO). The overhead is 30.35%. +4.3 +The Ability of Hiding Vulnerable Code +We use the T-III to further evaluate the ability of hiding +real world vulnerable code. Each program contains at least +one vulnerability (detailed in A.2). In this experiment, we +only used VulSeeker [26], Asm2Vec [20], and SAFE [45] to +calculate the escape@n ratio (the rank of truly matched pair +10 + +Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +400.perlbench +401.bzip2 +429.mcf +445.gobmk +456.hmmer +458.sjeng +462.libquantum +464.h264ref +473.astar +483.xalancbmk +600.perlbench_s +605.mcf_s +620.omnetpp_s +623.xalancbmk_s +625.x264_s +631.deepsjeng_s +641.leela_s +657.xz_s +GEOMEAN +BinDiff Similarity Score +BinTuner vs. O0 +BinTuner vs. O1 +BinTuner vs. O2 +BinTuner vs. O3 +Khaos vs. O0 +Khaos vs. O1 +Khaos vs. O2 +Khaos vs. O3 +Figure 9. BinDiff similarity score of SPECint 2006, SPECspeed 2017 C/C++ programs. +Table 2. Statistics of the fission and the fusion. +SPEC CPU 2006 +SPEC CPU 2017 +CoreUtils +Fission Ratio +116% +145% +152% +#BB +5.89 +6.46 +5.35 +RR +34% +42% +44% +Fusion Ratio +98% +97% +99% +#RP +1.27 +1.43 +1.47 +#HBB +1.08 +1.02 +1.89 +in the matched result) of vulnerable functions. The reason +why BinDiff and DeepBinDiff were not used is that they only +give top-1 matched result. We calculated escape@1/10/50 ratio +of vulnerable functions. For example, as shown in Figure 10, +the escape@50 ratio of FuFi.all on Asm2Vec is over 0.8, which +means more than 80% of vulnerable functions can not be +found within top-50 ranked functions. Moreover, this time +we set the obfuscation ratio of Fla in O-LLVM to 100%, which +would bring unacceptable overhead in the real scenario. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +VulSeeker Asm2Vec +Safe +VulSeeker Asm2Vec +Safe +VulSeeker Asm2Vec +Safe +Escape@1 +Escape@10 +Escape@50 +escape ratio +Sub +Bog +Fla +FuFi.sep +FuFi.ori +Fufi.all +Figure 10. Escape ratio for top@1/10/50 of vulnerable functions. +Higher means stronger hiding ability. +The escape ratio could reflect the ability of hiding the +vulnerable code with different obfuscations. With the same +precision and binary diffing tool (e.g., escape@50- Asm2Vec), +the FuFi.sep and the FuFi.all are better than the FuFi.ori, and +all of them are better than the Sub, the Bog, and the Fla in +O-LLVM. This ratio could also reflect the diffing ability of +binary diffing tools. With the same precision and the settings +of obfuscators, e.g., escape@50-FuFi.all, Asm2Vec is more +accurate than Safe, and both of them outperform VulSeeker. +The experimental results show that Khaos can not only +fight against binary diffing tools, but also reduce the pairing +ranking of vulnerable functions significantly, achieving the +purpose of hiding vulnerable code. +4.4 +The Statistics of Khaos Internals +We collected some internal information on the T-I and T- +II to demonstrate the effectiveness of Khaos. We used the +objdump tool to disassemble all the binaries and calculated +the histogram of opcodes. After that, we calculated the vector +distance between the origin and obfuscated binaries. Since +different programs contain different amounts of codes, we +used the max distance of all obfuscated programs as the +baseline to normalize these distances. As shown in Figure 11, +the opcode distance of FuFi.all is the largest, followed by +FuFi.sep and FuFi.ori. +We also calculated the statistics of the fission and the fu- +sion individually without the combination. For the fission, +we counted the fission ratio (#sepFuncs / #oriFuncs), and the +average number of basic blocks in sepFuncs (#BB), the re- +duced ratio of oriFuncs after the fission (RR). For the fusion, +we counted the fusion ratio (ratio of functions aggregated +successfully), the reduced parameter number (#RP) by param- +eter lists compression, and the number of innocuous basic +blocks of each function (#HBB). The statistical results are +shown in Table 2. +These internal statistics proved that Khaos can obfuscate +the oriFuncs with full force. For example, the Fusion Ratio +is 97-99%, which means almost all functions are aggregated. +It also proved that both optimizations for runtime overhead +(e.g., data-flow reduction) and obfuscation enhancement (e.g., +innocuous analysis) have worked effectively. +5 +Discussion and Future Work +Aside from obfuscation techniques, we found that existing +obfuscators have limitations on their implementation. In +O-LLVM[36], Sub can be optimized back under LLVM O3 +option, which leads us to choose O2 as our baseline. Bog and +Fla skip the exception-relevant functions. For Tigress[12], +we were unable to evaluate it in the same way as O-LLVM +due to compilation errors. +The diffing process can be seen as a feature searching +process. After we separate and aggregate these features, +the searching difficulty increases and searching accuracy +decreases. From our conclusion in table 1, the lacking of +11 + +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang. +0 +0.2 +0.4 +0.6 +0.8 +1 +400.perlbench +401.bzip2 +403.gcc +429.mcf +433.milc +444.namd +445.gobmk +447.dealII +450.soplex +453.povray +456.hmmer +458.sjeng +462.libquantum +464.h264ref +470.lbm +471.omnetpp +473.astar +482.sphinx3 +483.xalancbmk +GEOMEAN +Opcode Histogram Distance (Normalized) for SPEC CPU 2006 & 2017 C/C++ +0 +0.2 +0.4 +0.6 +0.8 +1 +500.perlbench_r +502.gcc_r +505.mcf_r +508.namd_r +510.parest_r +511.povray_r +519.lbm_r +520.omnetpp_r +523.xalancbmk_r +525.x264_r +526.blender_r +531.deepsjeng_r +538.imagick_r +541.leela_r +544.nab_r +557.xz_r +600.perlbench_s +602.gcc_s +605.mcf_s +619.lbm_s +620.omnetpp_s +623.xalancbmk_s +625.x264_s +631.deepsjeng_s +638.imagick_s +641.leela_s +644.nab_s +657.xz_s +GEOMEAN +Sub +Bog +Fla-10 +BinTuner +Fission +Fusion +FuFi.sep +FuFi.ori +FuFi.all +Figure 11. Opcode Histogram Distance (Normalized) for SPEC CPU 2006 & 2017 C/C++ programs. +call-graph consideration makes them unable to adopt inter- +procedural obfuscation. We believe our study will raise aware- +ness of inter-procedural obfuscation on binary diffing. +Smaller diffing granularity brings higher diffing costs. One +way to reduce the cost is to use context information to nar- +row the searching space. Previous works pay much more +attention to control flow rather than data flow. From the diff- +ing perspective, data flow is harder to capture and encode. +But from the obfuscation perspective, data flow is harder to +change, too. Therefore, we predict the potential of data flow +representation can be further tapped. +6 +Conclusion +Binary diffing techniques can be used for 1-day/n-day vul- +nerability searching by attacker. In this paper, we propose +an inter-procedural obfuscation technique Khaos to protect +software against the state-of-the-art binary diffing. We de- +sign two obfuscation primitives — the fission and the fusion. +Experimental results show that Khaos is not only effective, +but also efficient. We wish our study could not only help +developers to protect their software, but also promote the +development of binary diffing techniques in turn. +Acknowledgments +This research was supported by the National Natural Sci- +ence Foundation of China (NSFC) under Grants 61902374, +62272442, U1736208, 61872386, and the Innovation Funding +of ICT, CAS under Grant No.E161040. +A +Appendix +A.1 +The tag bits choice. +As mentioned in subsection 3.3, the tagged pointer is used to +select the code block aggregated from different oriFuncs. On +the X86_64 architecture, only 48 bits of the virtual address +are effective, so the upper 16 bits of the function pointer +are unused and they can be used to place the tag informa- +tion for the fusion. But this approach is expensive when +handling statically initialized pointers, such as global static +function pointers and virtual function tables. For the position- +independent executable, the values of these pointers need +to be relocated to point to the actual function at load time. +To attach the tag information to these pointers, we need to +add an initialization code to rewrite these pointers after the +relocation which will make the program load slower. +To address the above problem, we choose to use the lowest +bits of function pointers. This is because the addresses of +functions are usually 16-bytes aligned with the performance +consideration, so the lowest 4 bits of the function pointer can +also be used to place the tag information. Actually, the clang +compiler has already used the least bit to identify whether +a function pointer points to a virtual function or not, so +currently, only the 3 bits are unused. Instead of rewriting +statically initialized pointers after the relocation, we utilize +the relocation mechanism directly by adding the tag’s value +to the addend field (which is used to add an offset when +relocating) of the relocation item, so the tag information can +be attached to the pointer during the relocation. This method +cannot be applied to support the upper bits tag because it +exceeds the range supported by the addend field, i.e., (−231, ++231]. +A.2 +CVE Detail +As discussed in subsection 4.4, we use the Test Suite III to +further evaluate the ability of hiding real world vulnerable +code. As shown in Table 3, each program contains at least +one vulnerability. +12 + +Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques +CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada +Table 3. Vulnerable functions of Test Suite III +Program +Function +CVE +JerryScript +opfunc_spread_arguments +2020-13991 +QuickJS +compute_stack_size_rec +2020-22876 +BusyBox1.33.1 +getvar_s +2021-42382 +handle_special +2021-42384 +OpenSSL 1.1.1 +init_sig_algs +2021-3449 +EC_GROUP_set_generator +2019-1547 +libcurl 7.34.0 +suboption +2021-22925,2021-22898 +init_wc_data +2020-8285 +conn_is_conn +2020-8231 +tftp_connect +2019-5482,2019-5436 +ftp_state_list +2018-1000120 +alloc_addbyter +2016-8618 +Curl_cookie_getlist +2016-8623 +ConnectionExists +2016-8616,2016-0755, +2014-0138,2015-3143 +Total +14 +19 +References +[1] Markus F.X.J. Oberhumer and László Molnár and John F. Reiser. 2022. +The Ultimate Packer for eXecutables. https://upx.github.io/. +[2] Robert B Allan and Renu Laskar. 1978. On domination and independent +domination numbers of a graph. 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The +Internet Society. https://doi.org/10.14722/ndss.2019.23492 +[81] zynamics GmbH and Google LLC. 2022. +BinDiff Manual. +http://www.zynamics.com/bindiff/manual/index.html. +15 + diff --git a/69FJT4oBgHgl3EQfmCy-/content/tmp_files/load_file.txt b/69FJT4oBgHgl3EQfmCy-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02f35f903324c567acda79b0f6d2d815ea440542 --- /dev/null +++ b/69FJT4oBgHgl3EQfmCy-/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf,len=1478 +page_content='Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques Peihua Zhang SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China zhangpeihua@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Chenggang Wu SKLP, Institute of Computing Technology, CAS & University of Chinese Academy of Sciences Zhongguancun Laboratory Beijing, China wucg@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Mingfan Peng SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China pengmingfan20g@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Kai Zeng SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China zengkai@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Ding Yu SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China yuding19s@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Yuanming Lai∗ SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China laiyuanming@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Yan Kang SKLP, Institute of Computing Technology, CAS University of Chinese Academy of Sciences Beijing, China kangyan@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Wei Wang SKLP, Institute of Computing Technology, CAS Beijing, China wangwei2021@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Zhe Wang SKLP, Institute of Computing Technology, CAS Zhongguancun Laboratory Beijing, China wangzhe12@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='cn Abstract Software obfuscation techniques can prevent binary diffing techniques from locating vulnerable code by obfuscating the third-party code, to achieve the purpose of protecting embedded device software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' With the rapid development of binary diffing techniques, they can achieve more and more accurate function matching and identification by extract- ing the features within the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This makes existing software obfuscation techniques, which mainly focus on the intra-procedural code obfuscation, no longer effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In this paper, we propose a new inter-procedural code ob- fuscation mechanism Khaos, which moves the code across functions to obfuscate the function by using compilation optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Two obfuscation primitives are proposed to separate and aggregate the function, which are called fis- sion and fusion respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' A prototype of Khaos is im- plemented based on the LLVM compiler and evaluated on a large number of real-world programs including SPEC CPU 2006 & 2017, CoreUtils, JavaScript engines, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Experimental results show that Khaos outperforms existing code obfus- cations and can significantly reduce the accuracy rates of five state-of-the-art binary diffing techniques (less than 19%) with lower runtime overhead (less than 7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' ∗Yuanming Lai is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Keywords: Software Protection, Obfuscation, Binary Diffing 1 Introduction Embedded devices have been widespread in many fields of modern life, such as wearables, traffic lights, and autonomous driving vision sensors and the total number are expected to reach 30 billion by 2025 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In recent years, the number of vulnerabilities disclosed in embedded device software has been on the rise, and attacks targeting embedded devices have increased more than fivefold in the past four years [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Once a vulnerability in an embedded device is exploited, it can lead to the collapse of the backbone network [4], while vulnerabilities in medical devices such as pacemakers are life-threatening [11, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In addition to directly writing flawed code to introduce vulnerabilities, the reuse of vulnerable third-party code is another important reason for the widespread existence of vulnerabilities in embedded devices [14, 33, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For exam- ple, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' [14] found that 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4% of LaserJet printers used third-party libraries with known vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' However, vulnerabilities in these embedded devices cannot be patched in time due to the fragment issues — similar code exists in multiple versions of various products due to the fast replace- ment of embedded devices [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, the QualPwn vulnerability [63] in the Qualcomm’s WiFi controller, which is equipped in millions of Android phones, took nearly 6 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='11586v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='CR] 27 Jan 2023 CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' months from the vulnerability disclose to the patch released by the Qualcomm, and OEMs took longer to patch all devices across all versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Unfortunately, the above problem favors attackers in which they could detect existing vulnerabilities instead of explor- ing 0-day vulnerabilities laboriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since most embedded device software is not open source, attackers usually uti- lize the binary diffing techniques [3, 6–10, 15–17, 20, 21, 23– 32, 34, 35, 42, 44, 47, 54, 55, 66, 67, 69, 71, 73, 75–81] to locate the vulnerable code reused in the binary by comparing the binary with the third-party code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' With the introduction of machine learning, binary differing techniques have made great progress in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This greatly facilitates attack- ers locating existing vulnerabilities in binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' [17] searched for common vulnerabilities in mo- bile devices, wearable devices, and medical devices, and were able to locate 373 existing vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Software obfuscation techniques [5, 12, 36, 41, 68, 72] can transform the program code to change the character- istics of the binary code even though the source code is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' They could be used against binary diffing tech- niques, preventing attackers from locating existing vulner- abilities and thus protecting software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Recent researches have shown that software obfuscation techniques are no longer effective against the state-of-the-art binary diffing techniques [20, 44, 47, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The main reason is that most soft- ware obfuscation techniques focus on the intra-procedural code obfuscation, which does not fundamentally change the semantics of functions, while binary diffing techniques can more and more accurately extract features within functions to obtain their semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Based on the above observations, we argue that inter- procedural code obfuscation should be emphasized due to their ability to change function semantics at the binary level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To this end, we propose an inter-procedural code obfuscation technique, Khaos, which moves the code across functions and utilizes the compiler’s optimizations to transform (obfus- cate) the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The core idea of Khaos is that once the code is restructured among functions, the generated binary code after compilation optimizations can be very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To achieve the inter-procedural code obfuscation, Khaos changes func- tion code across functions by separating a function into sub-functions and aggregating functions into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It is non-trivial that transform arbitrary functions in Khaos due to the challenges posed by performance, correctness, and obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, 1) To balance the obfusca- tion effect with the performance overhead, choosing which code blocks within a function (or functions) to be separated (or aggregated) is a problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) Rebuilding all control flow and data flow among functions after transformations (espe- cially the indirect function calls handling in the fusion) is difficult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3) Aggregating functions deeply without affecting the functionality of each function is a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To address these challenges, two obfuscation primitives are proposed in Khaos— the fission primitive and the fu- sion primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission is used to separate a function into sub-functions, and the fusion is used to aggregate several functions into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission and the fusion are two com- plementary primitives, in which the fission tries to obfuscate the function by itself and the fusion tries to obfuscate the function by other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Furthermore, these two primi- tives can also be used together to improve the obfuscation effect, that is, the sub-functions separated by the fission can be aggregated with other functions again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission partitions the code region to a sub-function on the control flow of the function with the dominator tree as the granularity, and also combines the static cold/hot code analysis technique to achieve lower performance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since the define-use relationships of variables are changed from within a function to cross functions, the fission needs to rebuild the data flow by passing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To minimize the performance degradation caused by parameters passing, we also propose a data-flow reduction mechanism to reduce the number of parameters of the sub-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The control flow (including the exception control flow) is also rebuilt by inserting the function calls that call to sub-functions and encoding the return values in the sub-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fusion selects two functions with compatible return values and no variadic parameters for the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Com- patible means converting between different data types with- out losing precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The parameter list of the post-aggregation function is merged from these two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To avoid the inefficient way of passing parameters through the stack, we propose a parameter list compression mechanism to re- duce the number of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To rebuild the control flow completely, we propose a tagged pointer mechanism, which attaches control bits on function pointers to decide the executed code when the aggregated functions are called indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We also propose a trampoline mechanism to han- dle the function calls across modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To further improve the obfuscation effect, the deep fusion method is proposed to aggregate innocuous basic blocks, whose execution does not affect the global memory state, from different functions together within the aggregated function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Khaos was implemented based on the LLVM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The experimental evaluations were conducted on the Lin- ux/X86_64 platform by using SPEC CPU 2006 & 2017 C/C++ programs, CoreUtils, and 5 common embedded device soft- ware containing vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Five state-of-the-art binary diffing tools [20, 21, 26, 45, 81] were used to evaluate the effectiveness of Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The results show that Khaos is not only effective but also efficient: the effectiveness experiments show that the accuracy of these binary diffing was reduced to be less than 19%, and the ranking of the vulnerable func- tions decreased significantly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' the performance experiments show that Khaos incurs less than 7% overhead on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In summary, our contributions are as follows: 2 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada A novel inter-procedural code obfuscation mecha- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We point out that the inter-procedural code obfus- cation is necessary against the binary diffing techniques, and propose a new obfuscation mechanism, Khaos, which could obfuscate the code across functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission and the fusion primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We propose two obfuscation primitives in Khaos to move the code across functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission separates a function into multiple functions, and the fusion aggregates multiple functions into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' New insights from implementation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We implement and evaluate a prototype of Khaos, and the results show that it outperforms the existing obfuscators against the state-of-the-art binary diffing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Our study suggests that binary diffing techniques should focus more on extracting the inter-procedural code features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2 Background and Motivation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 Binary Diffing Binary diffing is a technique for visualizing and identifying differences between two binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It can quantitatively mea- sure the differences between two given binaries and give matching result at predefined granularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It has been widely used in software vulnerability search [10, 15, 17, 18, 23–26, 54, 55, 73, 78, 80], security patch analy- sis [75, 76, 79], malware detection [9, 28, 34, 47, 71], code clone detection [3, 8, 30, 31, 44], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The workflow of binary diffing can be divided into two stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', the offline features extraction and the online code search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' On the offline stage, tools extract features from binaries, while what features should be extracted is the focus of recent research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' On the online stage, tools calculate the similarity of the given binaries by using extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, BinDiff [81] extracts the number of basic blocks, control flow edges, and function calls within a function as the function’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Then, it combines the control flow graph matching algorithm to search for similar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 Software Obfuscation Software obfuscation transforms the program without chang- ing its functionality to make it hard to be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It can be used to hide vulnerabilities, protect intellectual property, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Actually, there is an arm race between software obfusca- tion and binary diffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Software obfuscation does not want binary diffing techniques to match un-obfuscated with ob- fuscated code successfully, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In recent decades, there are various techniques proposed in software obfusca- tion, and they can be classified into data obfuscation, static code rewriting, and dynamic code rewriting [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Data obfuscation techniques [13] transform the format of data to prevent it from direct matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since most binary diffing techniques utilize the features of the code, obfuscating data is less effective against binary diffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Various dynamic code rewriting approaches follow the concept of packing [50, 58], which hides code by encoding or encrypting it as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' But, the packing techniques are easy to be automatically unpacked [1, 53] or be memory-dumped [19, 22, 60, 65], which would lose the effect of obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Code virtualization is another popular obfuscation technique [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It translates code into specific interpret representations (IRs) instead of the native instructions and then uses an engine to interpret the IRs at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This technique sacrifices much performance (10x slowdown at least [38]) in exchange for a more powerful obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, the dynamic code rewriting technique is not suitable for fighting against binary diffing due to less effectiveness or too much overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In contrast, static code rewriting is a promising technique against binary diffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It modifies program code during ob- fuscation without further runtime modifications, which is similar to compiler optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Researchers have proposed many techniques for static code rewriting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For ease of intro- duction, we categorize them by obfuscation granularity: Instruction level: Instruction substitution [12, 36] replaces the original instruction with equivalent instruction(s), such as replacing an “add” instruction with two “sub” instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O-LLVM [36] designed 10 different substitution strategies for arithmetic and logical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To increase the com- plexity of conditional branch instructions, opaque predicate techniques [12, 36, 48, 64, 72] were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' They add per- manent true or false (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', 𝑥2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='= -1) conditions that do not affect the original control flow, which are frequently used against analytical techniques such as symbolic execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Basic block level: Bogus control flow [12, 36, 52] inserts dead code into the original control flow and often utilizes opaque predicates to prevent these codes from being op- timized away and executed, thereby ensuring the original functionality of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Function level: Control flow flattening [12, 36] converts the control flow of the function into the “switch-case” form, which is hard to be analyzed, and maintains the original jump relationship by controlling the values of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To prevent being degraded back to the original control flow, the “case” relationship is also obfuscated (encrypted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 Motivation As binary diffing techniques continue to advance, many static code rewriting techniques (referred to as code obfuscation in the rest of the paper) with the intra-procedural gran- ularity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', instruction, basic block, and function) are no longer effective [20, 44, 47, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The main reason is that intra- procedure code obfuscations do not fundamentally change the semantics of each function, while most binary diffing techniques are increasingly capable of extracting features within functions to understand their semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, we argue that inter-procedural code obfuscations should be emphasized due to their ability to change function 3 CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' semantics at the binary level which is the key to defeating binary diffing techniques: 1) For the binary diffing works that only consider the intra-procedural information, the inter- procedural code obfuscation can fundamentally defeat them because the code structures along with the semantics are significantly changed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) For the binary diffing works that take inter-procedure information into account, the inter- procedural code obfuscation can also defeat them because the inter-procedural information extracted, such as the types of function calls, the numbers of function calls, and the call graph, are also significantly changed after the obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Our thinking is also hinted by the literature published from both the offensive and defensive sides: 1) most of binary diffing works have discussed the issues of function inline [3, 7, 10, 16, 18, 20, 23–25, 29–32, 44, 74], and many of them [3, 17, 18, 23–25, 30–32, 74] admitted that it would affect the accuracy of diffing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' [57] found the function inline could reduce the binary similarity by approximately 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3 Our Solution: Khaos 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 Overview To achieve the inter-procedural code obfuscation, Khaos changes the amount of code within a function by moving code across functions firstly and then utilizes the compiler’s optimizations to transform (obfuscate) the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The idea behind it is that once the code is restructured among functions, the generated binary code after compilation optimizations (es- pecially intra-procedural optimizations) can be very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In detail, we propose two obfuscation primitives — the fis- sion primitive and the fusion primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission primitive separates a function into multiple sub-functions thus making the function thinner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fusion primitive aggregates func- tions into one thus making the function fatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' These two primitives can also be used together to make more in-depth changes to the function, that is, the separated sub-functions can be aggregated with other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the convenience of discussion, we denote a function before the transformation as an oriFunc (short for original function), and denote the new function formed after the fusion as the fusFunc (short for fused function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The new function formed by the separated code during the fission is denoted as the sepFunc (short for separated function), and the function formed by the remaining code is denoted as the remFunc (short for remnant function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Figure 1 gives an example about how the fission and the fusion are performed on a function named cal_file().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The function is used to find the number of a special character in a given file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It first checks the file name and open the file (lines 4-7), then reads the content and counts the amount (lines 9-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We can see that the fission separates two basic blocks ( 2○ 3○) to sepFunc-1, and four basic blocks ( 5○- 8○) to sepFunc-2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To maintain the correctness, the fis- sion inserts three trampoline basic blocks in the remFunc-1 ( a○ b○ c○) to create the call relationship of two sepFuncs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Ba- sic block ( d○) is used to return different value of sepFunc-1 (detailed in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' On top of the fission, the fusion aggregates the log() function and the sepFunc-2 into a fusFunc-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The entry basic block ( e○) will be inserted into the fusFunc-1 to select the aggregated code blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Changing functions by recombining basic blocks from different functions is not trivial, and it still faces several chal- lenges from performance, correctness, and obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Challenge-1: Choosing which basic blocks (or functions) to be separated (or aggregated) will seriously affect the performance overhead and obfuscation effect, and how to balance them well is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, separating each basic block as a sepFunc would favor the obfuscation, but brings unacceptable overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Challenge-2: How to completely rebuild all control flow and data flow among functions after transformation (es- pecially the fusion) is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, once several functions participate in the fusion, we need to handle all pointers of the oriFuncs so that it can correctly jump to the fusFunc when de-referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Challenge-3: Simply merging functions makes they be- come each other’s junk code and has a limited obfuscation effect because the compiler will still optimize the code for different functions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Binding control flows and data flows belonging to different functions in the fusFunc can prevent that but is also challenging to avoid changing the functionality of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In the following subsections, we will detail the fission and the fusion design, and how we address the above challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 The Fission Primitive The fission first identifies the regions (each region is a basic block set) that need to be separated, then composes these regions into sepFuncs, and finally rebuilds the control flow and the data flow among sepFuncs and remFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 Partitioning Regions to Form sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In gen- eral, a function’s property is single entry and multiple exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Hence, as long as a certain code region satisfies this prop- erty, it can be separated to become a new function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' More precisely, as long as a code region is a dominator tree [2] on the control flow graph, it can be extracted into a sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission creates call relationship among sepFuncs and remFunc to ensure correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' If the fission generates too many sep- Funcs, the newly created function calls in remFunc will bring additional overhead (especially new function calls inside a loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' However, if the number or size of the sepFuncs is small, the oriFunc cannot be significantly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, de- signing a reasonable region identify algorithm is the key to reducing the overhead and improving the obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The core idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We abstract the code region partitioning problem as a graph cutting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The function’s control 4 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada int cal_file(char *file_name) { int fd = -1, n = 0, value = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' char buffer[130];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' if (file_name) { log(file_name);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' fd = open(file_name, …);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' if(fd == -1) return -1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } // other checks omitted while (n = read(fd, buffer, 128)) value += cal(buffer, n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' close(fd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } log fission fusion sepFunc-1 2 d 3 1 4 c 9 a b remFunc-1 1 4 3 5 9 8 6 7 2 log original 5 8 6 7 sepFunc-2 1 4 c 9 a b remFunc-1 log e 5 8 6 7 fusFunc-1 sepFunc-1 2 d 3 control flow call basic block ret function origin 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' An example of obfuscating a function by using Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='Algorithm 1 The region identifying algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1: procedure Identify(𝑓 )⊲ The function’s representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='get dominator tree set 𝑆 of 𝑓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑆 ← 𝑆 \\ 𝑓 ⊲ We won’t separate the whole function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='while 𝑆 is not empty do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑡𝑎𝑟𝑔𝑒𝑡 ← null ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='for dominator tree t in 𝑆 do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑒𝑓 𝑓 𝑒𝑐𝑡 ← basic block count of 𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑐𝑜𝑠𝑡 ← frequency of 𝑡’s head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='if 𝑡 is in loop then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑙 ← the innermost loop where𝑡 is located ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑐𝑜𝑠𝑡 ← loop count of 𝑙 × 𝑐𝑜𝑠𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑣𝑎𝑙𝑢𝑒 ← 𝑒𝑓 𝑓 𝑒𝑐𝑡 ÷ 𝑐𝑜𝑠𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='if 𝑣𝑎𝑙𝑢𝑒 > 𝑡𝑎𝑟𝑔𝑒𝑡’s value then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='𝑡𝑎𝑟𝑔𝑒𝑡 ← 𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='⊲ Update the chosen tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='end for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='if 𝑡𝑎𝑟𝑔𝑒𝑡 ≠ 𝑛𝑢𝑙𝑙 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='delete trees from 𝑆 that intersect with 𝑡𝑎𝑟𝑔𝑒𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='end while ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='23: end procedure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='flow graph can be regarded as a directed graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' and the edge weight represents the frequency of execution which indicates the cold/hot information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Partitioning the code region can be regarded as cutting the graph, where the weight of the cut edge is the cost of performance and the obfuscation effect is the number of the nodes in the sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The region identifying algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Based on the above idea, we design the region identifying algorithm (algorithm 1) on top of the directed weighted graph cut algorithm [62] to balance the performance overhead and the obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The algorithm takes function code as input and performs dominator tree analysis [40] (line 2) at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To avoid sepa- rating the whole function body into a sepFunc, we remove the dominator tree of function itself (line 3) and identify the regions from the rest of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To indicate the effect of the fission on obfuscation, we use the number of basic blocks in the tree to represent it (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To indicate the effect of the fission on performance, we use the execution frequency of the root node of the dominator tree by using block frequency analysis [43] (line 8) and the loop count (if the region is in a loop, the call to sepFunc will increase) as the cost of the cut (lines 8-12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We iteratively select the most cost-effective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', maximum the ratio of effect and cost) dominator tree to separate until the tree set is empty (lines 13-16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 Data-flow Rebuild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In addition to identifying re- gions as the function bodies of sepFuncs, we also need to identify the inputs and the outputs of these regions to con- struct the parameters and return value of sepFuncs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For each variable used in a region, it should be an input if its point is outside the region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Similarly, for each variable defined in a region, it should be an output if it has a use point outside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, as shown in Figure 2, the fd and n variables are inputs because the defined points are outside the region, and the value variable has a use point outside the region, so it is an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the variables whose define-use relationship are across regions, we use the function param- eters to pass the pointer to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We don’t pass a region’s output variables by using the return value of sepFunc because a region may have multiple output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Data-flow reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In general, the local variables of a function are defined at the entry basic block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, if an identified region needs to use local variables, these variables need to be passed into the sepFunc through parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In fact, if some local variables are only used by a sepFunc, then these variables do not need to be passed into the sepFunc, they can be defined directly in the sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This can shorten the length of the sepFunc parameter list, save unnecessary variable transmission, and further improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To 5 CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' BB1 BB4 BB3 BB5 BB8 BB6 BB7 BB2 BB9 exit 0 exit 1 value += cal(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=') return value n = read(fd, …) int fd = -1, n = 0 define use region-1 region-2 control flow data flow basic block region to split statement Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The control-flow and data-flow graphs of cal_file() in Figure 1 achieve this, we propose a lazy allocation strategy — if a local variable is only used in the region, we will move the variable definition to the sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, the n variable in Figure 2 is initially defined in the oriFunc but redefined and only used in the region-2, which becomes sepFunc-2 function, so the definition point of the variable can be delayed in the sepFunc-2 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 Control-flow Rebuild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We extract the basic blocks of each identified region into a sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The jump relation- ship between the regions in the oriFunc is transformed into the function call-return relationship after fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The cre- ation of a function call is simple, we only need to insert the function call at the location of the entry basic block of the region before extraction and set the parameters that need to be passed into the sepFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The handling of function returns is relatively complex due to: If a region has multiple exits, the corresponding sepFunc needs to encode this information into the return value, so that the remFunc can use this information to select the corresponding code to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As Figure 2 shows, for the two exits (0 and 1) in region-1, when sepFunc-1 returns from exit 0, the control flow should go to BB5, and when returns from exit 1, it should go to BB9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We use the return value of sepFunc to indicate the remFunc to determine the execution direction: We first number each exit of the sepFunc, uses the number as its return value, and then insert a basic block at the call-site of this sepFunc in the remFunc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', a○ in Figure 1) to transfer control flow based on the return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 Handling the Exception Control-flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' During program execution, there are some exception control flows that deviate from the usual function call and return, including the setjmp/longjmp and the C++ exception handling (EH in short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fission requires special handles of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Handling the setjmp/longjmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Programmers could use the setjmp() in a function to record the current context into a jmpbuf structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' And then, they could use the longjmp() in any subroutines on the call chain of this function to go back to place the jmpbuf is pointing to, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', the call-site of the setjmp().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' There is a requirement here that the setjmp() and the longjmp() using the same jmpbuf must be in the same call chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, the call-site of the setjmp() can- not be separated into any sepFunc, because the stack frame of the function that calling the setjmp() cannot be freed when the corresponding longjmp() is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Otherwise, the longjmp() will direct control flow to an unknown location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Handling the C++ exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The EH mechanism is a fea- ture of the C++ that developers can capture exceptions in the try block by writing the catch statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since the fission moves part of the code into a sepFunc, the try-catch pair may be broken, making EH information inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Simply skipping the exception-relevant function would reduce the obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, when identifying the code re- gion, if it contains any code that may generate an exception, we will locate the corresponding catch code and divide it into the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 The Fusion Primitive The fusion selects functions to form fusFunc, and rebuilds the control and the data flow to ensure the correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In theory, the fusion can aggregate any number of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To balance the performance overhead and the obfuscation effect, we choose to aggregate two functions to form a fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 Selecting Functions to Form fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fusion cannot arbitrarily select functions, it needs to select functions with compatible types of the return values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The definition of incompatibility is that if converting between two types loses precision, the two types are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, when the return value of one function is an integer and the other is a float, these two functions cannot be aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In fact, there are other conditions that limit the selection of functions: 1) The variadic functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', the printf(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=');' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) Two functions with incompatible types of the return values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3) Two functions that have a direct calling relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The first two constraints are designed for correctness, and the last is designed for performance to avoid generating a lot of recursive fusFuncs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Functions that meet the above constraints will be randomly aggregated in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 Data-flow Rebuild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Once the two functions to be aggregated are determined, the function prototype of the corresponding fusFunc can be determined immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, as shown in Figure 3 (a) and (b), the bar() and the foo() are aggregated into int bar_foo_fusion().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The ctrl parameter is used to select the function bodies aggre- gated from the bar() and the foo().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Determining the func- tion prototype of fusFunc is crucial to the rebuild of the data flow, which involves setting the parameter list and return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 6 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada void bar(short a, float b) { // bar\'s code printf("bar: %d, %f\\n", a, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } int foo(int m) { // foo\'s code printf("foo: %d\\n", m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } int main() { bar(0x1234, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' int res = foo(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } (a) Before fusion (b) Fusion w/o parameter compression (c) Fusion w/ parameter compression int bar_foo_fusion(int ctrl, short a, float b, int m) { if (ctrl) { // bar\'s code printf("bar: %d, %f\\n", a, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } else { // foo\'s code printf("foo: %d\\n", m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } } int main() { // ctrl is 1, executing bar bar_foo_fusion(1, 0x1234, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' // ctrl is 0, executing foo int res = bar_foo_fusion(0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } int bar_foo_fusion(int ctrl, int x, float b) { if (ctrl) { // bar\'s code printf("bar: %d, %f\\n", (short)x, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } else { // foo\'s code printf("foo: %d\\n", x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } } int main() { // ctrl is 1, executing bar bar_foo_fusion(1, 0x1234, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' // ctrl is 0, executing foo int res = bar_foo_fusion(0, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' An example of performing the fusion on two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Parameter list compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Simply merging the param- eter lists of the two functions makes the parameter list of fusFunc too long, which will degrade the performance of calling fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This is because in the X86_64 calling con- vention, the first six parameters are passed in registers, and the rest of the parameters are passed on the stack, which is an inefficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To achieve efficient parameter passing, we propose a parameter list compression mechanism — if the types of the two parameters from the two functions are compatible, we compress them into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The reason why we can do this is that when a fusFunc is called, only the parameter list of one of the functions participating in the aggregation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, as Figure 3(c) shows, both the bar() and the foo() have an integer parameter (short a and int m), we compress them into one integer parameter (int x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' If a parameter can not participate in the compression, it is copied into the parameter list of the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The number of parameters after the fusion will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In the worst case, it is the sum of the parameters of the two functions, which means none of the parameters can be compressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To avoid using the stack to pass parameters as much as possible, we preferentially select functions with the total number of parameters less than six for the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Return value determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Determining the return type of fusFunc is relatively simple: 1) If the return type of one function is void, then the return type of the fusFunc is the return type of another;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) If the return types of the two functions are both not void, the compressed type is used as the return type of the fusFunc, which is similar to the parameter list compression mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 Control-flow Rebuild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Once the fusFunc is created, the two involved oriFuncs need to be removed, and all call- sites to the oriFuncs need to be replaced to call the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As mentioned before, a ctrl parameter will be added into the parameter list of the fusFunc to select the code block aggregated from the oriFuncs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=" The value of this parameter is int bar() { // bar's code } int foo() { // foo's code } int (*fptr)();" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' int main(int argc) { if (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=') fptr = &bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' else fptr = &foo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' int res = fptr();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' } int (*fptr)() int res = fptr() fptr = (&bar_foo) | tag value tag if (extract_tag(fptr)) res = tmp fptr = &bar fptr = &foo tmp = fptr() val = clear_tag(fptr) tmp = val(extract_tag()) (a) (b) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Function reference and indirect call processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 0 or 1, which is set according to the original call-site of the oriFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since the fusFunc parameter list includes the param- eters of both oriFuncs, we only need to pass the parameters required by the oriFunc to the fusFunc at the call-site of this oriFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Unused parameters are set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Handling Indirect function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Indirect function calls are more difficult to handle than direct function calls because we do not know where the oriFunc will be called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Figure 4 (a) shows an example that calls two functions by de-referencing the function pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The corresponding data flow is given in Figure 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' When aggregating the bar() and the foo(), we need to change the function pointer points to the fusFunc and then replace the function call to call this fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' But, we encounter a problem that we do not know what the value of the ctrl parameter should be set to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This is because at the compile time, we don’t know whether the original function pointer fptr points to the bar() or the foo().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To address the above problem, we propose a tagged pointer mechanism, which is similar to the low-fat pointer [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The core idea is to encode the information (called tag) of which oriFunc pointed to by the original function pointer into the new function pointer, and when the new function pointer is de-referenced to make a call, the value of the ctrl parameter can be dynamically determined by parsing the new function pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In detail, when the operation of taking the address of 7 CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' the function participating in the aggregation occurs, we need to perform the encoding operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since the tag is encoded into the function pointer, it can be propagated along with the function pointer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' When the function pointer is de-referenced to make a call, we will extract the tag in the pointer as the and set the ctrl parameter according to the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The tag requires two extra bits, where a bit indicates whether the pointer points to a fusFunc, and the other bit records the value of the ctrl parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, as shown in Figure 4 (c), if the pointer fptr points to the bar(), the value of the tag will be set to 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' When the pointer fptr is dereferenced to make a call, we insert code to first check whether the tag is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' If not, the code will extract the ctrl parameter and call the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Otherwise, no addi- tional operations are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We choose the 2nd bit and the 3rd bit of function pointers to place the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This is because the functions are usually 16-bytes aligned with the performance consideration, so the lowest 4 bits of the function pointer can be used (more reasons and considerations are detailed in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Handling function calls across modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' There are two cases of cross-module function calls, one is the function pointer of a module is propagated to other modules, and the other is a module directly calls functions exported by other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' If any case happens on a fusFunc, we needs to process all involved modules to ensure the fusFunc can be called correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' But in some cases, we can not process all the modules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', some libraries may have no source code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To address this problem, we propose a trampoline mecha- nism so that all modules do not need to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In detail, we transverse the data flow conservatively and identify all function pointers that may propagate outside the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' And then, we modify these function pointers to point to a piece of trampoline code instead of the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' So that when the external module calls these function pointers, the con- trol flow will transfer to the trampoline code first, and the trampoline code will help the function outside the module to reorganize the function parameters and call the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the exported oriFuncs, the method is similar to the replacing the oriFunc’s function body with the trampoline code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 The Deep Fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To further improve the obfusca- tion effect, we propose a deep fusion method to aggregate as many basic blocks as possible between the two parts of the code during the fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We have observed that some basic blocks can be executed many times without affecting the normal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The char- acteristic of these basic blocks is that their execution does not affect the global memory state, and they are called the in- nocuous basic block in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The concept is very similar to the reentrant function [56] that it can be re-executed with- out affecting the functionality of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For innocuous basic blocks from different oriFuncs, they can be aggregated together within the fusFunc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The innocuous analysis of each Current = tmp1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' oldtr = tmp2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' return;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' delta = tr - oldtr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' if (delta < -10) delta += 256;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' tmp1 = Current + delta * 1000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' tmp2 = tr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Update(int tr) int delta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' static int oldtr = -1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' int tmp1 = 0, tmp2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' tmp1 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' tmp2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' oldtr == -1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 1 2 3 4 5 UMV(int y, int x, int height, int width) int width4 = ((width+2*4-1)<<2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' int height4 = ((height+2*4-1)<<2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' x = x + IMG_PAD_SIZE*4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' y = y + IMG_PAD_SIZE*4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 7 6 Fusion(ctrl,…) 7 1 2 8 4 5 0 control flow basic block function basic blocks Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' A real-world example of the deep fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' basic block is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, 1) if a memory write operation in a basic block cannot be determined whether the modified data is local or global, then this basic block is not innocuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) if there is a function call to an external function in a basic block, this basic block is not innocuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We give a simplified example of 464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='h264ref in SPEC CPU 2006 benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown in Figure 5, the Update() and UMV() are aggregated into the Fusion().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The basic block (BB) 3○ of the Update() firstly redefines the local variable delta, and then loads the value of global variable Current, and writes two local variables tmp1 and tmp2 at last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since these operations do not affect the global memory state, the BB 3○ is determined to be innocuous, and so as the BB 6○ of UMV(), thus we aggregate them into one — the BB 8○.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This deep fusion method modifies the control flow graph and data flow graph of the fusFunc at the same time, adding data dependency and control dependency so that the fusFunc cannot be simply separated back to the two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 Combining the Fission and the Fusion The fission and the fusion can be used together to further enhance the obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' There are three combinations as follows: FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep: Only aggregating the sepFuncs generated by the fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In this case, the issue of handling indirect function calls no longer exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori: Only aggregating the oriFuncs that are not pro- cessed by the fission, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', the functions with only one basic block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This combination could balance the obfus- cation effect and the performance overhead well, and is suitable for software in most real-world scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all: Aggregating the fission-generated sepFuncs and the fission-unprocessed oriFuncs uniformly and randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In this combination, the obfuscation effect is prioritized, followed by the performance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It is suitable for programs that require a high obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 4 Evaluation We implemented Khaos based on the LLVM-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The fis- sion and the fusion are implemented as the middle-end passes, and the fission pass is scheduled before the fusion 8 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada 5 15 35 55 75 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='lbm 471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='omnetpp 473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='astar 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sphinx3 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='xalancbmk GEOMEAN overhead(%) Fission Fusion FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep FuFi.' metadata={'source': 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605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='mcf_s 619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='lbm_s 620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='omnetpp_s 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='xalancbmk_s 625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='x264_s 631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='deepsjeng_s 638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='imagick_s 641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='leela_s 644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='nab_s 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='xz_s GEOMEAN overhead(%) 111 131 138 160 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Runtime overhead of SPEC CPU 2006 (upper part) and 2017 (lower part) C/C++ programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We run Khaos on Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='04 (Kernel v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0) that runs on an Intel(R) Xeon(R) Gold 5218 CPU with 128G mem- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This section evaluates Khaos in terms of effectiveness and performance, and answers the following questions: (Q1) How is the performance of the obfuscated programs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' (Q2) How does Khaos work against the state-of-the-art binary diffing techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' (Q3) How good is Khaos at hiding real vulnerable code?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Test Suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We used three test suites to evaluate Khaos: 1) All C/C++ programs in SPEC CPU 2006/2017 benchmarks with the ref input (denoted as the T-I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2) All 108 programs in the CoreUtils 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='32 (denoted as the T-II);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 3) Five commonly used programs in embedded devices with at least one vul- nerability, including two popular IoT JavaScript engines (Jer- ryScript and QuickJS), OpenSSL-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1, BusyBox-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 and libcurl-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 (denoted as the T-III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The performance eval- uation was performed on the T-I (Q1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The effectiveness against binary diffing techniques was evaluated on the T-I and the T-II (Q2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The ability to hide vulnerable code was evaluated on the T-III (Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since software developers typi- cally link programs into a single binary in embedded devices, we compiled and obfuscated these test suites in the same way under O2 with the link-time optimization (LTO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Comparison targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To compare with existing obfusca- tor, we choose the popular compiler-level obfuscation tool O-LLVM [36] as our comparison target because it is open- sourced and compiler-based (same as Khaos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O-LLVM [36] contains three obfuscation methods: instruction substitu- tion (Sub), bogus control flow (Bog), and control flow flatten- ing (Fla).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Literatures [5, 20, 57, 69] in software engineering, systems security, and programming languages fields all use it in their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To ensure the consistency of the evaluation environment, we upgrade the LLVM version of O- LLVM [36] to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1, which is same as Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We also choose BinTuner [57], which is an iterative compiler tool that uses compiler options to transform the code to enlarge the differ- ence of binaries, as another target to compare Khaos with compiler’s options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Confrontation targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We use five state-of-the-art binary diffing techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', Google BinDiff [81], VulSeeker [26], Asm2Vec [20], SAFE [45], DeepBinDiff [21], to evaluate the effectiveness of Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Among them, Google BinDiff is an industry-standard binary diffing tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Asm2Vec, SAFE, Deep- BinDiff, and VulSeeker are the state-of-the-art methods for learning the semantic similarity in different granularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', function, basic block, control flow graph, call graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 Performance Overhead after Obfuscation We separately evaluated the performance overhead of the fission and the fusion, and the three combination modes in- troduced in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 on the T-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown in Figure 6, the geometric performance overhead of the fission and the fu- sion are 5% and 6%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The reason why some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='hmmer) have a negative performance overhead is that after the fission separates part of the code, the remFunc can be further inlined to its callers, and the fusion improves the code locality of the aggregated functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The results demonstrated that obfuscations compliant with the compiler optimizations can have good performance advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Compared with the FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori, the other two combinations have a higher overhead because the fission generates many sepFuncs, aggregating them all incurs non-negligible per- formance overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, the 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='gcc_r contains many recursive functions, the sepFuncs generated by these functions are aggregated to the fusFuncs which are also the recursive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since the stack frames of fusFuncs are larger, they will bring much pressure to the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Compared with O-LLVM[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We compared the perfor- mance overhead of Khaos with Sub, Bog, Fla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown 9 CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada P Zhang, C Wu, M Peng, K Zeng, D Yu, Y Lai, Y Kang, W Wang, and Z Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 6 3 282 22 3 5 8 5 14 6 9 277 39 5 6 15 7 23 6 7 279 32 5 6 12 6 19 0 20 40 60 80 Sub Bog Fla Fla-10 Fission Fusion FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all overhead(%) SPEC CPU 2006 SPEC CPU 2017 GEOMEAN Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Runtime overhead of O-LLVM and Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Summarize of chosen diffing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' diffing symbol time memory call-graph granularity relying consuming consuming lacking BinDiff [81] function Y N N N VulSeeker [26] function N Y Y Y Asm2Vec [20] function N N N Y Safe [45] function N N N Y DeepBinDiff [21] basic block N Y Y N in Figure 7, Khaos has comparable overhead with the Sub and the Bog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Due to the high overhead of Fla, we reduce its obfuscation ratio to 10% (Fla-10), and others are all at 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 The Effectiveness against Binary Diffing Comparing binary diffing works is challenging due to their measurements of similarity are very different [57], such as graph edit distance or statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Simply com- paring their similarity scores does not provide accurate infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the commercial binary diffing tool BinDiff [81], we normalized its similarity score to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For other tools open-sourced in academia, we normalized their results by computing the ratio of true matching function pairs that are also the top-ranked matching candidates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Precision@1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Paring success judgment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since Khaos changes the number of functions, we relax the requirements for Pre- cision@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the fission, if the oriFunc is paired with any sepFuncs generated from it or the remFunc, this pairing is rec- ognized as successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the fusion, if the fusFunc is paired with any function before the fusion, this pairing is recog- nized as successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the DeepBinDiff [21], since its result is basic block to basic block, the pairing is recognized as successful as long as their belonging functions are matched, even if the two basic blocks are not truly matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It is worth noting that the above setting is looser than originally used in these tools but is more challenging for Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Test suite adjustment adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The characteristics of used binary diffing tools were summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The column “symbol relying” means the un-stripped binaries whether have side-effects or not, for example, BinDiff usually uses function names to reduce the searching space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The col- umn “time consuming” or the column “memory consuming” means the diffing process takes a long time or requires a lot of memory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The column “call-graph lacking” means whether using the call-graph as the feature or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The test suites for VulSeeker [26] and DeepBinDiff [21] need to be adjusted due to unable to run results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' VulSeeker [26] takes more than 1 day to diff two large binaries and often 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 BinDiff VulSeeker Asm2Vec Safe DeepBinDiff precision@1 Sub Bog Fla-10 Fission Fusion FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Precision@1 result of chosen binary diffing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' gets killed due to memory limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To speed up VulSeeker, we group the related functions into small groups (30 functions per group) to manually reduce the searching space, which is unfavorable to Khaos because the smaller the group size, the easier to diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' DeepBinDiff [21] requires too much memory (sometimes more than 10 TB) due to its representation of basic blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since its diffing process is tightly coupled with binary size, we decide not to modify it and only use pro- grams less than 40k lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Even with the reduced test suite, it is still time consuming (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', over 1 week to diff binaries of 508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='namd_r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It’s worth mentioning that this is also unfavor- able to Khaos because it uses original functions to obfuscate each other, lacking material reduces the obfuscation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Other binary diffing tools still use the normal test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We evaluated the accuracy of these tools by com- paring obfuscated and un-obfuscated (un-stripped) binaries on the T-I and T-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As Figure 8 shows, higher accuracy means lower adversarial effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since BinDiff [81] takes the advantage of function names, its result is much higher than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Although DeepBinDiff [21] uses the basic block level instead of the function level as its granularity, the feature vector of the basic block still encodes the control flow graph and call graph, which have been changed by Khaos, and that’s why Khaos can defeat it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' With comparable overhead, Khaos can achieve a much better adversarial effect than O-LLVM [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Compared with compiler options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We follow the com- pare method of BinTuner[57] to calculate the similarity score of BinDiff[81] under different compiler settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the Bin- Tuner part, we set O0’s binary code (same setting in the paper[57]) as the baseline during its iterative compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the Khaos part, we use binaries generated by FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown in Figure 9, Khaos has a much lower similarity score in different compiler options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We also compared the over- head of programs generated by BinTuner with the baseline of Khaos (O2 with LTO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The overhead is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='35%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 The Ability of Hiding Vulnerable Code We use the T-III to further evaluate the ability of hiding real world vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Each program contains at least one vulnerability (detailed in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In this experiment, we only used VulSeeker [26], Asm2Vec [20], and SAFE [45] to calculate the escape@n ratio (the rank of truly matched pair 10 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O0 BinTuner vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O1 BinTuner vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O2 BinTuner vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O3 Khaos vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O0 Khaos vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O1 Khaos vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O2 Khaos vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' O3 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' BinDiff similarity score of SPECint 2006, SPECspeed 2017 C/C++ programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Statistics of the fission and the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' SPEC CPU 2006 SPEC CPU 2017 CoreUtils Fission Ratio 116% 145% 152% #BB 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='35 RR 34% 42% 44% Fusion Ratio 98% 97% 99% #RP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='47 #HBB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='89 in the matched result) of vulnerable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The reason why BinDiff and DeepBinDiff were not used is that they only give top-1 matched result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We calculated escape@1/10/50 ratio of vulnerable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, as shown in Figure 10, the escape@50 ratio of FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all on Asm2Vec is over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='8, which means more than 80% of vulnerable functions can not be found within top-50 ranked functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Moreover, this time we set the obfuscation ratio of Fla in O-LLVM to 100%, which would bring unacceptable overhead in the real scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 VulSeeker Asm2Vec Safe VulSeeker Asm2Vec Safe VulSeeker Asm2Vec Safe Escape@1 Escape@10 Escape@50 escape ratio Sub Bog Fla FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori Fufi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Escape ratio for top@1/10/50 of vulnerable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Higher means stronger hiding ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The escape ratio could reflect the ability of hiding the vulnerable code with different obfuscations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' With the same precision and binary diffing tool (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', escape@50- Asm2Vec), the FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep and the FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all are better than the FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori, and all of them are better than the Sub, the Bog, and the Fla in O-LLVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This ratio could also reflect the diffing ability of binary diffing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' With the same precision and the settings of obfuscators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', escape@50-FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all, Asm2Vec is more accurate than Safe, and both of them outperform VulSeeker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The experimental results show that Khaos can not only fight against binary diffing tools, but also reduce the pairing ranking of vulnerable functions significantly, achieving the purpose of hiding vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4 The Statistics of Khaos Internals We collected some internal information on the T-I and T- II to demonstrate the effectiveness of Khaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We used the objdump tool to disassemble all the binaries and calculated the histogram of opcodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' After that, we calculated the vector distance between the origin and obfuscated binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Since different programs contain different amounts of codes, we used the max distance of all obfuscated programs as the baseline to normalize these distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown in Figure 11, the opcode distance of FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all is the largest, followed by FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep and FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We also calculated the statistics of the fission and the fu- sion individually without the combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the fission, we counted the fission ratio (#sepFuncs / #oriFuncs), and the average number of basic blocks in sepFuncs (#BB), the re- duced ratio of oriFuncs after the fission (RR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the fusion, we counted the fusion ratio (ratio of functions aggregated successfully), the reduced parameter number (#RP) by param- eter lists compression, and the number of innocuous basic blocks of each function (#HBB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The statistical results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' These internal statistics proved that Khaos can obfuscate the oriFuncs with full force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For example, the Fusion Ratio is 97-99%, which means almost all functions are aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' It also proved that both optimizations for runtime overhead (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', data-flow reduction) and obfuscation enhancement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', innocuous analysis) have worked effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 5 Discussion and Future Work Aside from obfuscation techniques, we found that existing obfuscators have limitations on their implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In O-LLVM[36], Sub can be optimized back under LLVM O3 option, which leads us to choose O2 as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Bog and Fla skip the exception-relevant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For Tigress[12], we were unable to evaluate it in the same way as O-LLVM due to compilation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' The diffing process can be seen as a feature searching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' After we separate and aggregate these features, the searching difficulty increases and searching accuracy decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 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638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='imagick_s 641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='leela_s 644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='nab_s 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='xz_s GEOMEAN Sub Bog Fla-10 BinTuner Fission Fusion FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='sep FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='ori FuFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='all Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Opcode Histogram Distance (Normalized) for SPEC CPU 2006 & 2017 C/C++ programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' call-graph consideration makes them unable to adopt inter- procedural obfuscation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We believe our study will raise aware- ness of inter-procedural obfuscation on binary diffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Smaller diffing granularity brings higher diffing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' One way to reduce the cost is to use context information to nar- row the searching space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Previous works pay much more attention to control flow rather than data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' From the diff- ing perspective, data flow is harder to capture and encode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' But from the obfuscation perspective, data flow is harder to change, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Therefore, we predict the potential of data flow representation can be further tapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 6 Conclusion Binary diffing techniques can be used for 1-day/n-day vul- nerability searching by attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In this paper, we propose an inter-procedural obfuscation technique Khaos to protect software against the state-of-the-art binary diffing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We de- sign two obfuscation primitives — the fission and the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Experimental results show that Khaos is not only effective, but also efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' We wish our study could not only help developers to protect their software, but also promote the development of binary diffing techniques in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Acknowledgments This research was supported by the National Natural Sci- ence Foundation of China (NSFC) under Grants 61902374, 62272442, U1736208, 61872386, and the Innovation Funding of ICT, CAS under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='E161040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 The tag bits choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As mentioned in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='3, the tagged pointer is used to select the code block aggregated from different oriFuncs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' On the X86_64 architecture, only 48 bits of the virtual address are effective, so the upper 16 bits of the function pointer are unused and they can be used to place the tag informa- tion for the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' But this approach is expensive when handling statically initialized pointers, such as global static function pointers and virtual function tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' For the position- independent executable, the values of these pointers need to be relocated to point to the actual function at load time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To attach the tag information to these pointers, we need to add an initialization code to rewrite these pointers after the relocation which will make the program load slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' To address the above problem, we choose to use the lowest bits of function pointers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This is because the addresses of functions are usually 16-bytes aligned with the performance consideration, so the lowest 4 bits of the function pointer can also be used to place the tag information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Actually, the clang compiler has already used the least bit to identify whether a function pointer points to a virtual function or not, so currently, only the 3 bits are unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Instead of rewriting statically initialized pointers after the relocation, we utilize the relocation mechanism directly by adding the tag’s value to the addend field (which is used to add an offset when relocating) of the relocation item, so the tag information can be attached to the pointer during the relocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' This method cannot be applied to support the upper bits tag because it exceeds the range supported by the addend field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=', (−231, +231].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='2 CVE Detail As discussed in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='4, we use the Test Suite III to further evaluate the ability of hiding real world vulnerable code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' As shown in Table 3, each program contains at least one vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 12 Khaos: The Impact of Inter-procedural Code Obfuscation on Binary Diffing Techniques CGO ’23, February 25 – March 1, 2023, Montréal, QC, Canada Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Vulnerable functions of Test Suite III Program Function CVE JerryScript opfunc_spread_arguments 2020-13991 QuickJS compute_stack_size_rec 2020-22876 BusyBox1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 getvar_s 2021-42382 handle_special 2021-42384 OpenSSL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='1 init_sig_algs 2021-3449 EC_GROUP_set_generator 2019-1547 libcurl 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='0 suboption 2021-22925,2021-22898 init_wc_data 2020-8285 conn_is_conn 2020-8231 tftp_connect 2019-5482,2019-5436 ftp_state_list 2018-1000120 alloc_addbyter 2016-8618 Curl_cookie_getlist 2016-8623 ConnectionExists 2016-8616,2016-0755, 2014-0138,2015-3143 Total 14 19 References [1] Markus F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Oberhumer and László Molnár and John F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Reiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2022.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' Similarity of bina- ries through re-optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' In Proceedings of the 38th ACM SIGPLAN Conference on Programming Language Design and Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 79– 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='org/10.' metadata={'source': 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+page_content='com/bindiff/manual/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FJT4oBgHgl3EQfmCy-/content/2301.11586v1.pdf'} diff --git a/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/2301.05507v1.pdf.txt b/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/2301.05507v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4e2f0ee06602582b00d8d5e5d43806fa6933c9a --- /dev/null +++ b/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/2301.05507v1.pdf.txt @@ -0,0 +1,1130 @@ +arXiv:2301.05507v1 [stat.OT] 13 Jan 2023 +Correlation-Based And-Operations Can Be +Copulas: A Proof +Enrique Miralles-Dolz1,2, Ander Gray1,2, Edoardo Patelli3, +Scott Ferson2, Vladik Kreinovich4, and Olga Kosheleva5 +1Institute for Risk and Uncertainty, University of Liverpool, +Liverpool, UK, {enmidol,akgray,ferson}@liverpool.ac.uk +2United Kingdom Atomic Energy Authority, Abingdon, UK +3Centre for Intelligent Infrastructure, University of Strathclyde, +Glasgow, UK, edoardo.patelli@strath.ac.uk +4Department of Computer Science, University of Texas at El Paso, +El Paso, Texas 79968, USA, vladik@utep.edu +5Department of Teacher Education, University of Texas at El Paso, +El Paso, Texas 79968, USA, olgak@utep.edu +Abstract +In many practical situations, we know the probabilities a and b of two +events A and B, and we want to estimate the joint probability Prob(A & B). +The algorithm that estimates the joint probability based on the known +values a and b is called an and-operation. An important case when such a +reconstruction is possible is when we know the correlation between A and +B; we call the resulting and-operation correlation-based. On the other +hand, in statistics, there is a widely used class of and-operations known +as copulas. +Empirical evidence seems to indicate that the correlation- +based and-operation derived in [4] is a copula, but until now, no proof of +this statement was available. In this paper, we provide such a proof. +1 +Formulation of the problem +Correlation-based “and”-operation. In many practical situations, we know +the probabilities a and b of two events A and B, and we need to estimate the +joint probability Prob(A & B). An algorithm f&(a, b) that transforms the known +values a and b into such an estimate is usually called an and-operation. +One important case when such an estimate is possible is when, in addition +to the probabilities a and b, we also know the correlation ρ between the corre- +sponding two random events. It is known (see, e.g., [3, 4]) that in this case, we +can uniquely determine the probability of Prob(A & B) as +a · b + ρ · +� +a · (1 − a) · b · (1 − b). +(1) +1 + +While this formula is true whenever the correlation is known, this formula +does not lead to an everywhere defined and-operation. For example, for a = +b = 0.1 and ρ = −1, this formula leads to a meaningless negative probability +0.1 · 0.1 + (−1) · +√ +0.1 · 0.9 · 0.1 · 0.9 = 0.01 − 0.09 = −0.08 < 0. +To avoid such meaningless estimates, we need to take into account that the joint +probability Prob(A & B) must satisfy Fr´echet inequalities (see, e.g., [2]): +max(a + b − 1, 0) ≤ Prob(A & B) ≤ min(a, b). +(2) +So, if an expert claims to know the correlation ρ and the estimate for Prob(A & B) +based on this value ρ is smaller than the lower bound max(a + b − 1, 0) – +which cannot be – a reasonable idea is to take the closest possible value of the +joint probability, i.e., the value max(a + b − 1, 0). Similarly, if the estimate +for Prob(A & B) based on the expert-provided value ρ is larger than the up- +per bound min(a, b) – which also cannot be – a reasonable idea is to take the +closest possible value of the joint probability, i.e., the value min(a, b). Thus, +we arrive at the following and-operation – which we will call correlation-based +and-operation: +fρ(a, b) = Ta,b +� +a · b + ρ · +� +a · (1 − a) · b · (1 − b) +� +, +(3) +where +Ta,b(c) = max(a + b − 1, 0) if c < max(a + b − 1, 0); +Ta,b(c) = c if max(a + b − 1, 0) ≤ c ≤ min(a, b); and +(4) +Ta,b(c) = min(a, b) if min(a, b) < c. +Question: is this and-operation a copula? In probability theory, there +is a known class of and-operations known as copulas (see, e.g., [5, 6]). These +are functions C(a, b) for which, for some random 2-D vector (X, Y ), the joint +cumulative distribution function FXY (x, y) +def += Prob(X ≤ x & Y +≤ y) has +the form FXY (x, y) = C(FX(x), FY (y)), where FX(x) +def += Prob(X ≤ x) and +FY (y) +def += Prob(Y ≤ y) are known as marginals. +One important aspect of (3)-(4) is that these formulas can be expressed as +a copula (2-copula) family as described in [4], allowing us to operate not only +with precise probabilities, but also with interval probabilities and probability +boxes. A 2-copula must satisfy the following properties: +1. Grounded: C(0, b) = C(a, 0) = 0 +2. Uniform margins: C(a, 1) = a; C(1, b) = b +3. 2-increasing: C(a, b) + C(a, b) − C(a, b) − C(a, b) ≥ 0 for all a < a and +b < b +2 + +It is easy to see that (3)-(4) satisfies the two first properties. +In [4] the +third property was checked for a dense set of tuples (a, a, b, b, ρ), and for all +these tuples, the inequality was satisfied. However, at that moment, we could +not prove that the correlation-based and-operation is indeed a 2-copula. In this +paper we provide the missing proof. +2 +Main result +Proposition. For every ρ ∈ [−1, 1], the correlation and-operation fρ(a, b) de- +scribed by the formulas (3)-(4) is a copula. +Proof. +1◦. It is known that the desired inequality has the following property – if we +represent a box [a, a] × [b, b] as a union of several sub-boxes, then the left- +hand side of the desired inequality is equal to the sum of the left-hand sides +corresponding to sub-boxes. +Indeed, as one can easily check, there is the following additivity property: +for each box consisting of several sub-boxes, the left-hand side of the inequality +(4a) that corresponds to the larger box is equal to the sum of expressions (4a) +corresponding to sub-boxes. Thus, if the expressions corresponding to sub-boxes +are non-negative, then the expression (4a) corresponding to the larger box is +also non-negative. +In general, the and-operation described by the formula (4) has three different +expressions. So, to prove that the expression (4a) corresponding to this expres- +sion is also non-negative, we need to consider cases when at different vertices of +the box, we may have different expressions. Good news is that every box whose +vertices are described by different expressions can be represented as the union +of sub-boxes in which: +• either all vertices are described by the same expression +• or two vertices are on the boundary between the areas of different expres- +sions. +This is easy to see visually: the following box, in which the slanted line represents +the boundary between the areas +� +� +� +� +� +can be represented as the union of sub-boxes with the desired property: +3 + +� +� +� +� +� +Thus, to prove that our and-operation is a copula, it is sufficient to consider +only boxes of the following type: +• boxes for which all four vertices belong to the same area, and +• boxes for which two vertices belong to the boundary between two areas. +The functions max(a + b − 1, 0) and min(a, b) are known to be copulas, so if +all four vertices belong to one of these areas, then the desired inequality (4a) is +satisfied. So, it is sufficient to consider: +• boxes for which all four vertices belong to the new area, in which the and- +operation is described by the expression (1); we will consider such boxes +in Parts 2–4 of this proof, and +• boxes for which two vertices belong to the boundary between two areas; +these boxes will be considered in the following Parts of the proof. +2◦. Let us start by considering boxes for which all four vertices belongs to the +area in which the and-operation is described by the formula (1). +It is known [1] – and it is easy to prove by considering infinitesimal differences +x − x and y − y – that for smooth functions, the desired inequality is equivalent +to the fact that the partial derivative +∂C +∂a +is non-decreasing in b, i.e., equivalently, that the mixed derivative is non- +negative: +d +def += +∂2C +∂a ∂b ≥ 0. +Thus, to prove that fρ(a, b) is a copula, it is sufficient to prove that its mixed +derivative is non-negative everywhere where the new formula is applied. +Indeed, at the points where the formula (1) is applied, the derivative of +fρ(a, b) with respect to a has the has the form +∂fρ +∂a = b + ρ · +1 − 2 · a +2 · +� +a · (1 − a) +· +� +b · (1 − b), +(4b) +and thus, the mixed derivative has the following form: +d = ∂ +∂b +�∂fρ +∂a +� += 1 + ρ · +(1 − 2 · a) · (1 − 2 · b) +4 · +� +a · (1 − a) · b · (1 − b) +. +(5) +4 + +Since the expression (1) does not change if we swap a and b, it is sufficient to +consider the case when a ≤ b. +When ρ = 0, we get a known copula f0(a, b) = a · b. So, it is sufficient to +consider cases when ρ ̸= 0. This can happen when ρ > 0 and when ρ < 0. Let +us consider these cases one by one. +3◦. Let us first consider the case when ρ > 0. +In this case, since a ≤ b, we have min(a, b) = a and thus, the condition (4) +takes the form +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≤ a, +(6) +i.e., equivalently, +ρ · +� +a · (1 − a) · b · (1 − b) ≤ a − a · b = a · (1 − b) +(7) +and thus, +ρ ≤ +a · (1 − b) +� +a · (1 − a) · b · (1 − b) += +� +a · (1 − b) +� +(1 − a) · b +. +(8) +For all such ρ, we need to prove that the expression (5) is non-negative. +When both a and b are larger than 0.5 or both are smaller than 0.5, the +differences 1 − 2a and 1 − 2b have the same sign and thus, their product is non- +negative and the expression (5) is non-negative. So, the only case when we need +to check that d ≥ 0 is when one of the two values a and b is smaller than 0.5 +and another one is larger than 0.5. Since a ≤ 0.5, this means that a < 0.5 < b. +In this case, the condition d ≥ 0 takes the form +1 − ρ · +(1 − 2 · a) · (2 · b − 1) +4 · +� +a · (1 − a) · b · (1 − b) +≥ 0, +(9) +i.e., equivalently, +ρ · +(1 − 2 · a) · (2 · b − 1) +4 · +� +a · (1 − a) · b · (1 − b) +≤ 1, +(10) +and +ρ ≤ 4 · +� +a · (1 − a) · b · (1 − b) +(1 − 2 · a) · (2 · b − 1) +. +(11) +So, to prove that we always have d ≥ 0, we need to prove that every ρ that +satisfies the inequality (8) also satisfies the inequality (11). Clearly, if some +value ρ satisfies the inequality (11), then every smaller value ρ also satisfies this +inequality. Thus, to prove the desired implication, it is sufficient to check that +the inequality (11) is satisfied for the largest possible value ρ that satisfies the +inequality (8), i.e., for the value ρ which is equal to the right-hand side of the +inequality (8). For this ρ, the desired inequality (11) takes the form +� +a · (1 − b) +� +(1 − a) · b +≤ 4 · +� +a · (1 − a) · b · (1 − b) +(1 − 2 · a) · (2 · b − 1) +. +(12) +5 + +Dividing both sides by +� +a · (1 − b), we get an equivalent inequality +1 +� +(1 − a) · b +≤ +4 · +� +(1 − a) · b +(1 − 2 · a) · (2 · b − 1). +(13) +Multiplying both sides by both denominators, we get the following equivalent +inequality: +(1 − 2 · a) · (2 · b − 1) ≤ 4 · (1 − a) · b. +(14) +If we open parentheses, this inequality takes the equivalent form +2 · b − 4 · a · b − 1 + 2 · a ≤ 4 · b − 4 · a · b, +(15) +i.e., by adding 4 · a · b − 2 · b to both sides, the form +−1 + 2 · a ≤ 2 · b. +(16) +We are considering the case when a ≤ b – since, as we have mentioned earlier, +it is sufficient to only consider this case. Thus, the equivalent inequality (12) is +also true and hence, for the case when ρ > 0, we indeed have d ≥ 0. +4◦. To complete the proof, it is now sufficient to consider the case when ρ < 0. +In this case, if one of the values a and b is smaller than 0.5 and another +one is larger than 0.5, then the differences 1 − 2 · a and 1 − 2 · b have different +signs, so the right-hand side of the expression (5) for d is larger than 1 and thus, +non-negative. Thus, it is sufficient to consider the cases when: +• either both a and b are larger than 0.5 +• or both a and b are smaller than 0.5. +Let us consider these two cases one by one. +4.1◦. Let us first consider the case when a > 0.5 and b > 0.5. +In this case, a + b − 1 > 0, so the inequality (4) takes the form +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≥ a + b − 1, +(17) +i.e., equivalently, that +|ρ| · +� +a · (1 − a) · b · (1 − b) ≤ a · b − a − b + 1 = (1 − a) · (1 − b), +(18) +or that +|ρ| ≤ +(1 − a) · (1 − b) +� +a · (1 − a) · b · (1 − b) += +� +(1 − a) · (1 − b) +√ +a · b +. +(19) +In this case, the condition d ≥ 0 that the value (5) is non-negative takes the +form +1 − |ρ| · +(2 · a − 1) · (2 · b − 1) +4 · +� +a · (1 − a) · b · (1 − b) +, +(20) +6 + +i.e., equivalently, +|ρ| · +(2 · a − 1) · (2 · b − 1) +4 · +� +a · (1 − a) · b · (1 − b) +≤ 1 +(21) +and +|ρ| ≤ 4 · +� +a · (1 − a) · b · (1 − b) +(2 · a − 1) · (2 · b − 1) +. +(22) +Similarly to the case when ρ > 0, to check that all values |ρ| satisfying the +inequality (19) also satisfies the inequality (22), it is sufficient to check that the +largest possible value |ρ| satisfying the inequality (19) satisfies the inequality +(22), i.e., that +� +(1 − a) · (1 − b) +√ +a · b +≤ 4 · +� +a · (1 − a) · b · (1 − b) +(2 · a − 1) · (2 · b − 1) +. +(23) +If we divide both sides by +� +(1 − a) · (1 − b), we get the following equivalent +inequality +1 +√ +a · b +≤ +4 · +√ +a · b +(2 · a − 1) · (2 · b − 1). +(24) +Multiplying both sides by both denominators, we get the following equivalent +inequality +(2 · a − 1) · (2 · b − 1) ≤ 4 · a · b. +(25) +Opening parentheses, we get +4 · a · b − 2 · a − 2 · b + 1 ≤ 4 · a · b. +(26) +Adding 2 · a + 2 · b − 4 · a · b to both sides, we get an equivalent inequality +1 ≤ 2 · a + 2 · b, +(27) +which is true since we consider the case when a + b > 1. So, in this case, we +indeed have d ≥ 0. +4.2◦. Let us now consider the case when a < 0.5 and b < 0.5. +In this case, a + b − 1 < 0, so the inequality (4) takes the form +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≥ 0, +(28) +i.e., equivalently, that +|ρ| · +� +a · (1 − a) · b · (1 − b) ≤ a · b, +(29) +or that +|ρ| ≤ +a · b +� +a · (1 − a) · b · (1 − b) += +√ +a · b +� +(1 − a) · (1 − b) +. +(30) +7 + +In this case, the condition d ≥ 0 that the value (5) is non-negative takes the +form +1 − |ρ| · +(1 − 2 · a) · (1 − 2 · b) +4 · +� +a · (1 − a) · b · (1 − b) +, +(31) +i.e., equivalently, +|ρ| · +(1 − 2 · a) · (1 − 2 · b) +4 · +� +a · (1 − a) · b · (1 − b) +≤ 1 +(32) +and +|ρ| ≤ 4 · +� +a · (1 − a) · b · (1 − b) +(1 − 2 · a) · (1 − 2 · b) +. +(33) +Similarly to the cases when ρ > 0 and when a + b > 1, to check that all values +|ρ| satisfying the inequality (30) also satisfies the inequality (33), it is sufficient +to check that the largest possible value |ρ| satisfying the inequality (30) satisfies +the inequality (33), i.e., that +√ +a · b +� +(1 − a) · (1 − b) +≤ 4 · +� +a · (1 − a) · b · (1 − b) +(1 − 2 · a) · (1 − 2 · b) +. +(34) +If we divide both sides by +√ +a · b, we get the following equivalent inequality +1 +� +(1 − a) · (1 − b) +≤ 4 · +� +(1 − a) · (1 − b) +(1 − 2 · a) · (1 − 2 · b). +(35) +Multiplying both sides by both denominators, we get the following equivalent +inequality +(1 − 2 · a) · (1 − 2 · b) ≤ 4 · (1 − a) · (1 − b). +(36) +Opening parentheses, we get +1 − 2 · a − 2 · b + 4 · a · b ≤ 4 − 4 · a − 4 · b + 4 · a · b. +(37) +Adding 4 · a + 4 · b − 4 · a · b − 1 to both sides, we get an equivalent inequality +2 · a + 2 · b ≤ 3, +(38) +which is true since we consider the case when a + b < 1. So, in this case, we +indeed have d ≥ 0. +In all cases when have d ≥ 0, thus, the and-operation fρ(a, b) is indeed a +copula. Thus, for boxes in which all four vertices belong to the area described +by the expression (1), the inequality (4a) is always satisfied. +5◦. Let us now consider the boxes in which two vertices belong to the boundary +between two areas. First, we will consider the case when ρ > 0 and then, we +will consider the case when ρ < 0. +6◦. Let us first consider the case when ρ > 0. For this case, let us first describe +the boundaries between the areas. +8 + +6.1◦. Let us analyze which of the three areas listed in formula (4) are possible +in this case. +When ρ > 0, we have +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≥ a · b, +and since it is known that we always have a · b ≥ max(a + b − 1, 0), we have +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≥ max(a + b − 1, 0). +So, for ρ > 0, we cannot have the first of the three cases described by the +formula (4). So, we only have two areas: +• the area where the and-operation is described by the formula (1), and +• the area where the and-operation is described by the formula min(a, b). +6.2◦. Let us describe the two possible areas and the boundary between these +two areas. +The first area is characterized by the inequality +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≤ min(a, b). +(39) +Similarly to the previous part of the proof, without losing generality, we can +consider the case when a ≤ b. In this case, the inequality (39) describing the +first area takes the following form: +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≤ a. +(40) +If we subtract a · b from both sides of this inequality, we get the following +equivalent inequality: +ρ · +� +a · (1 − a) · b · (1 − b) ≤ a · (1 − b). +(41) +Both sides of this inequality are non-negative, so we can get an equivalent +inequality is we square both sides: +ρ2 · a · (1 − a) · b · (1 − b) ≤ a2 · (1 − b)2. +(42) +The cases when a or b are equal to 0 or 1 can be obtained by taking a limit +from the cases when both a and b are located insyed the interval (0, 1). For such +values, we can divide both side of the inequality by positive numbers a2, b, and +1 − b, and get the following equivalent inequality: +ρ2 · 1 − a +a +≤ 1 − b +b +, +(43) +i.e., equivalently, +ρ2 · 1 − a +a +≤ 1 +b − 1. +(44) +9 + +By adding 1 to both sides of this inequality, we get +a + ρ2 · (1 − a) +a +≤ 1 +b , +(45) +i.e., equivalently, that +b ≤ +a +a + ρ2 · (1 − a). +(46) +This inequality describes the first area, in which the and-operation is described +by the formula (1). Thus, the boundary between the two areas is described by +the equality +b = +a +a + ρ2 · (1 − a). +(47) +Comment. One can see that for a = 0 we get b = 0, for a = 1, we get b = 1. +6.3◦. Let us prove that for all a, the corresponding boundary value b is greater +than or equal to a – i.e., that for all the points (a, b) on this boundary, we have +a ≤ b. +Indeed, for the expression (47), the desired inequality a ≤ b takes the form +a ≤ +a +a + ρ2 · (1 − a). +(48) +If we divide both sides by a and multiply both sides by the denominator of the +right-hand side, we get the following equivalent inequality +a + ρ2 · (1 − a) ≤ 1. +(49) +If we move all the terms to the right-hand side, we get an equivalent inequality +0 ≤ 1 − a − ρ2 · (1 − a) = (1 − ρ2) · (1 − a). +(50) +This inequality is always true, since ρ2 ≤ 1 and a ≤ 1, so indeed, for all boundary +points, we have a ≤ b. +6.4◦. Let us prove that the boundary describes b as an increasing function of a. +By applying, to the equality (47) that describes the boundary, the same +transformations that show the equivalent of inequalities (43) and (46), we can +conclude that the equality (47) is equivalent to +ρ2 · 1 − a +a += 1 − b +b +, +(51) +i.e., to +ρ2 · +�1 +a − 1 +� += 1 +b − 1. +(52) +10 + +The left-hand side is decreasing with respect to a, the right-hand side is a +decreasing function of b. Thus, as a increases, the left-hand side decreases, thus +the right-hand side also decreases and hence, the value b increases as well. +6.5◦. For ρ = 1 the condition (46) describing the first area takes the form b ≤ a. +Since we have a ≤ b, this means that this condition is only satisfies for a = b. +For these values, the expression (4a) is equal to +a · a + +� +a · (1 − a) · a · (1 − a) = a2 + a · (1 − a) = a2 + a − a2 = +a = min(a, b), +(53) +which means that our and-operation is always equal to min(a, b). The expression +min(a, b) is known to be a copula. +So, we only need to prove the fact that our and-operation is a copula for the +case when ρ < 1. This is the case we will consider from now on. +6.6◦. Let us prove that for ρ < 1, the only boundary points for which a = b are +points for which a = b = 0 and a = b = 1. +Indeed, as we have mentioned, the points (0, 0) and (1, 1) are boundary +points. Let us prove, by contradiction, that there are no other boundary points +for which a = b. +Indeed, when a = b, the equality (52) that describes the +boundary takes the form: +ρ2 · +�1 +a − 1 +� += 1 +a − 1. +(54) +Dividing both sides of this equality by the non-zero right-hand side, we get +ρ2 = 1. This contradicts to the fact that we are considering the case when +ρ < 1 and thus, ρ2 < 1. This contradiction shows that other boundary points +with a = b are not possible. +6.7◦. The boundary consists of a curved line that is separate from the line a = b – +except for the endpoints. So, if we limit ourselves to a sub-box [ε, 1−ε]×[ε, 1−ε] +for some small ε > 0, the boundary line is separated from the line a = b – there +is the smallest distance δ > 0 between points of these two lines. So, if we have +a box that includes both points with a ≤ b and with a ≥ b, we can divide this +box into sub-boxes of linear size < δ/2 and thus, make sure that every sub-box +that contains boundary points with a ≤ b cannot contain any points with a = b +– and therefore, only contains points with a ≤ b. +So, due to additivity, it is sufficient to prove the inequality (4a) for boxes +for which: +• two vertices lie on the boundary, and +• we have a ≤ b for all the points from this sub-box. +This will allow us to prove the inequality (4a) for all sub-boxes of the square +[ε, 1−ε]×[ε, 1−ε]. We can do it for any ε and thus, in the limit, get the desired +inequality for all sub-boxes of the original square [0, 1] × [0, 1] as well. +So, suppose that we have a box for which: +11 + +• two vertices lie on the boundary, and +• we have a ≤ b for all the points from this box. +Since the boundary describes the increasing function of a, the corresponding +box has the form +� +� +� +� +� +So, in the corresponding box: +• the two vertices (a, b) and (a, b) are on the boundary, +• the vertex (a, b) is in the first area, i.e., for this point, we have the expres- +sion (1), and +• the vertex (a, b) is in the second area, i.e., here C(a, b) = min(a, b). +The desired inequality (4a) has the form +C(a, b) − C(a, b) ≤ C(a, b) − C(a, b). +(55) +The points (a, b) and (a, b) are both in the second area for which C(a, b) = +min(a, b) – to be more precise, the second of these points is in the boundary, +which means it also satisfies the condition C(a, b) = min(a, b). For all the points +from the box, a ≤ b, so we have +C(a, b) − C(a, b) = min(a, b) − min(a, b) = a − a. +(56) +On the other hand, for the difference in the left-hand side of the formula (55), +we have +C(a, b) − C(a, b) = +� a +a +∂C +∂a da. +(57) +So, if we prove that the partial derivative ∂C/∂a is always smaller or equal than +1, we would indeed conclude that +C(a, b) − C(a, b) = +� a +a +1 da = a − a, +(58) +i.e., exactly, the desired inequality (55). +For the points (a, b) and (a, b) – and the points from the interval connecting +these two points – the expression C(a, b) is described by the formula (1). Thus, +the partial derivative of C(a, b) with respect to a is described by the formula +(4b). Thus, the inequality +∂C +∂a (a, b) ≤ 1, +(59) +12 + +takes the form +b + ρ · +1 − 2 · a +2 · +� +a · (1 − a) +· +� +b · (1 − b) ≤ 1. +(60) +Subtracting b from both sides of (60), we get an equivalent inequality +ρ · +1 − 2 · a +2 · +� +a · (1 − a) +· +� +b · (1 − b) ≤ 1 − b. +(61) +To separate the variables, we can divide both sides by +� +b · (1 − b), then we get +an equivalent inequality +ρ · +1 − 2 · a +2 · +� +a · (1 − a) +≤ +� +1 − b +b +. +(62) +By taking the square root of both sides of the inequality (46), we conclude that: +ρ · +� +1 − a +a +≤ +� +1 − b +b +. +(63) +Thus, if we prove that the left-hand side of the inequality (62) is smaller than +or equal to the left-hand side of the inequality (63), i.e., that +ρ · +1 − 2 · a +2 · +� +a · (1 − a) +≤ ρ · +� +1 − a +a +; +(64) +this will prove the inequality (62) and thus, the desired upper bound (60) on the +partial derivative. We can simplify the inequality (64) by dividing both sides +by ρ and multiplying both sides by 2 · +� +a · (1 − a). Then, we get an equivalent +inequality +1 − 2 · a ≤ 2 · (1 − a) = 2 − 2 · a, +(65) +which is equivalent to 1 ≤ 2 and is, thus, always true. Thus, (55) holds, so the +inequality (4a) is true for all the boxes in which two vertices are located on the +boundary. +This completes the proof of the Proposition for the case when ρ > 0. +7◦. Let us now consider the case when ρ < 0. For this case, let us first describe +the boundaries between the areas. +7.1◦. Let us analyze which of the three areas listed in formula (4) are possible +in this case. +When ρ < 0, we have +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≤ a · b, +and since it is known that we always have a · b ≤ min(a, b), we have +a · b + ρ · +� +a · (1 − a) · b · (1 − b) ≤ min(a, b). +So, for ρ < 0, we cannot have the third of the three cases described by the +formula (4). So, we only have two areas: +13 + +• the area where the and-operation is described by the formula (1), and +• the area where the and-operation is described by the formula +max(a + b − 1, 0). +7.2◦. Let us describe the two possible areas and the boundary between these +two areas. +The first area is characterized by the inequality C(a, b) ≥ max(a + b − 1, 0), +i.e., equivalently, by two inequalities +a · b − |ρ| · +� +a · (1 − a) · b · (1 − b) ≥ 0 +(66) +and +a · b − |ρ| · +� +a · (1 − a) · b · (1 − b) ≥ a + b − 1. +(67) +Let us consider these two inequalities one by one. +7.2.1◦. The inequality (66) is equivalent to: +a · b ≥ |ρ| · +� +a · (1 − a) · b · (1 − b). +(68) +To separate the variables, let us divide both sides of this inequality by +a · +� +b · (1 − b), +then we get an equivalent inequality +� +b +1 − b ≥ |ρ| · +� +1 − a +a +. +(69) +Both sides of this inequality are non-negative, thus if we square both sides, we +get an equivalent inequality +b +1 − b ≥ ρ2 · 1 − a +a +. +(70) +Reversing both sides, we get an equivalent inequality +1 − b +b +≤ +a +ρ2 · (1 − a), +(71) +i.e., equivalently, +1 +b − 1 ≤ +a +ρ2 · (1 − a). +(72) +By adding 1 to both sides, we get +1 +b ≤ ρ2 · (1 − a) + a +ρ2 · (1 − a) +, +(73) +14 + +i.e., equivalently, +b ≥ +ρ2 · (1 − a) +ρ2 · (1 − a) + a. +(74) +7.2.2◦. The inequality (67) is equivalent to +a · b − a − b + 1 ≥ |ρ| · +� +a · (1 − a) · b · (1 − b), +(75) +i.e., +(1 − a) · (1 − b) ≥ |ρ| · +� +a · (1 − a) · b · (1 − b). +(76) +To separate the variables, let us divide both sides by (1 − a) · +� +b · (1 − b), then +we get an equivalent inequality +� +1 − b +b +≥ |ρ| · +� +a +1 − a. +(77) +Both sides of this inequality are non-negative, thus if we square both sides, we +get an equivalent inequality +1 − b +b +≥ ρ2 · +a +1 − a, +(78) +i.e., equivalently, +1 +b − 1 ≥ ρ2 · +a +1 − a. +(79) +By adding 1 to both sides, we get +1 +b ≥ ρ2 · a + (1 − a) +1 − a +, +(80) +i.e., equivalently, +b ≤ +1 − a +ρ2 · a + (1 − a). +(81) +7.2.3◦. By combining the inequalities (74) and (81), we get the following de- +scription of the area in which the and-operation is described by the formula +(1): +ρ2 · (1 − a) +ρ2 · (1 − a) + a ≤ b ≤ +1 − a +ρ2 · a + (1 − a). +(82) +Thus, the boundary between the two areas consists of the following two curves: +b = +ρ2 · (1 − a) +ρ2 · (1 − a) + a +(83) +and +b = +1 − a +ρ2 · a + (1 − a). +(84) +7.3◦. Let us prove that: +15 + +• the curve (83) lies in the area where a + b ≤ 1, and +• the curve (84) lies in the area where a + b ≥ 1. +7.3.1◦. Let us first prove that for each value b described by the formula (83), +we have a + b ≤ 1. +We need to prove the inequality +a + +ρ2 · (1 − a) +ρ2 · (1 − a) + a ≤ 1. +(85) +Subtracting a from both sides, we get an equivalent inequality +ρ2 · (1 − a) +ρ2 · (1 − a) + a ≤ 1 − a. +(86) +Dividing both sides by 1 − a and multiplying both sides by the denominator of +the left-hand side, we get the following equivalent inequality: +ρ2 ≤ ρ2 · (1 − a) + a = ρ2 + (1 − ρ2) · a, +(87) +which is, of course, always true, since ρ2 ≤ 1 and a ≥ 0. The statement is +proven. +7.3.2◦. Let us now prove that for each value b described by the formula (84), +we have a + b ≥ 1. +We need to prove the inequality +a + +1 − a +ρ2 · a + (1 − a) ≥ 1. +(88) +Subtracting a from both sides, we get an equivalent inequality +1 − a +ρ2 · a + (1 − a) ≥ 1 − a. +(89) +Dividing both sides by 1 − a and multiplying both sides by the denominator of +the left-hand side, we get the following equivalent inequality: +1 ≥ ρ2 ≤ a + (1 − a) = 1 − (1 − ρ2) · a, +(90) +which is, of course, always true. The statement is proven. +7.4◦. Similarly to Part 6 of this proof, it is sufficient to prove the inequality (4a) +for boxes in which two vertices are on the boundary and for which: +• either we have a + b ≤ 1 for all the points from the box, +• or we have a + b ≥ 1 for all the points from the box. +16 + +Let us consider the two parts of the boundary one by one. +7.4.1◦. Let us first consider the case when we have a + b ≤ 1 for all the points +from the box. In this case, the corresponding part of the boundary is described +by the formula (83). By reformulating this expression in the equivalent form +b = +1 +1 + +a +ρ2 · (1 − a) += +1 +1 + 1 +ρ2 · +1 +1 +a − 1 +, +(91) +we can see that b is a decreasing function of a. Thus, the corresponding box has +the form +❅ +❅ +❅ +❅ +❅ +So, in the corresponding box: +• the two vertices (a, b) and (a, b) are on the boundary, +• the vertex (a, b) is in the first area, i.e., for this point, we have the expres- +sion (1), and +• the vertex (a, b) is in the second area, i.e., here C(a, b) = max(a+b−1, 0). +So, for three vertices, we have C(a, b) = max(a + b − 1, 0). Since for all the +points from the box, we have a + b ≤ 1, this means that for three vertices, we +have C(a, b) = 0. In this case, the inequality (4a) is clearly true. +7.4.2◦. Let us now consider the case when we have a + b ≥ 1 for all the points +from the box. In this case, the corresponding part of the boundary is described +by the formula (84). By reformulating this expression in the equivalent form +b = +1 +1 + ρ2 · +a +1 − a += +1 +1 + ρ2 · +1 +1 +a − 1 +, +(91) +we can see that b is also a decreasing function of a. Thus, the corresponding +box has the same form as in the case a + b ≤ 1: +❅ +❅ +❅ +❅ +❅ +17 + +So, in the corresponding box: +• the two vertices (a, b) and (a, b) are on the boundary, +• the vertex (a, b) is in the first area, i.e., for this point, we have the expres- +sion (1), and +• the vertex (a, b) is in the second area, i.e., here C(a, b) = max(a+b−1, 0). +Similarly to Part 6 of the proof, we can show that the desired inequality (4a) is +satisfied if we the corresponding partial derivatives is smaller than or equal to +1, i.e., if +∂C +∂a = b − |ρ| · +1 − 2 · a +2 · +� +a · (1 − a) +· +� +b · (1 − b) ≤ 1. +(92) +Subtracting b from both sides, we get an equivalent inequality +−|ρ| · +1 − 2 · a +2 · +� +a · (1 − a) +· +� +b · (1 − b) ≤ 1 − b. +(93) +We can separate the variable if we divide both sides by +� +b · (1 − b), then we +get an equivalent inequality +−|ρ| · +1 − 2 · a +2 · +� +a · (1 − a) +≤ +� +1 − b +b +. +(94) +We know a lower bound on the expression in the right-hand side – it is provided +by the inequality (77). Thus, to prove the inequality (94), it is sufficient to +prove that the left-hand side of the formula (94) is smaller than or equal to this +lower bound, i.e., that +−|ρ| · +1 − 2 · a +2 · +� +a · (1 − a) +≤ |ρ| · +� +a +1 − a. +(95) +Let us prove this inequality. Dividing both sides of (95) by |ρ| and multiplying +both sides by 2· +� +a · (1 − a), we get an equivalent inequality −(1−2·a) ≤ 2·a, +i.e., 2 · a − 1 ≤ 2 · a, which is always true. Thus, the inequality (94) holds, hence +the inequality (92) also holds, and therefore, in this case, the inequality (4a) +that describes a copula is also true. +8◦. We have considered all possible cases, and in all these cases, we have shown +that the inequality (4a) – that defines a copula – is true. Thus, our and-operation +is indeed a copula. The proposition is proven. +Acknowledgments +This research was partly funded by the EPSRC and ESRC CDT in Risk and +Uncertainty (EP/L015927/1), established within the Institute for Risk and +18 + +Uncertainty at the University of Liverpool. This work has been carried out +within the framework of the EUROfusion Consortium, funded by the European +Union via the Euratom Research and Training Programme (Grant Agreement +No 101052200 - EUROfusion). +Views and opinions expressed are however those of the author(s) only and +do not necessarily reflect those of the European Union or the European Com- +mission. Neither the European Union nor the European Commission can be +held responsible for them. +V.K. was supported in part by the National Science Foundation grants +1623190 (A Model of Change for Preparing a New Generation for Professional +Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI +Includes), and by the AT&T Fellowship in Information Technology. He was +also supported by the program of the development of the Scientific-Educational +Mathematical Center of Volga Federal District No. 075-02-2020-1478, and by +a grant from the Hungarian National Research, Development and Innovation +Office (NRDI). +References +[1] F. Durante and P. Jaworski, “A new characterization of bivariate copulas”, +Communications in Statistics – Theory and Methods, 2010, Vol. 39, No. 16, +pp. 2901–2912. +[2] M. Fr´echet, “G´en´eralisation du th´eoreme des probabilit´es totales”, Funda- +menta Mathematicae, 1935, Vol. 1(25), pp. 379–387. +[3] D. J. Lucas, “Default correlation and credit analysis”, Journal of Fixed +Income, 1995, Vol. 4, No. 4, pp. 76–87. +[4] E. Miralles-Dolz, A. Gray, E. Patelli, and S. Ferson, “Correlated boolean +operators for uncertainty logic”, In: D. Ciucci et al. (Eds.), Proceedings of +the 2022 International Conference on Processing and Management of Un- +certainty IPMU’2022, Milan, Italy, July 11–15, 2022, Springer, 2022, pp. +798–811. +[5] R. B. Nelsen, An Introduction to Copulas, Springer, New York, 2007. +[6] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Courier Corporation, +2011. +19 + diff --git a/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/load_file.txt b/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c359a99aef1fd8c79b4e83216782f41a6185e3fe --- /dev/null +++ b/6tE5T4oBgHgl3EQfPw5R/content/tmp_files/load_file.txt @@ -0,0 +1,422 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf,len=421 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='05507v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='OT] 13 Jan 2023 Correlation-Based And-Operations Can Be Copulas: A Proof Enrique Miralles-Dolz1,2, Ander Gray1,2, Edoardo Patelli3, Scott Ferson2, Vladik Kreinovich4, and Olga Kosheleva5 1Institute for Risk and Uncertainty, University of Liverpool, Liverpool, UK, {enmidol,akgray,ferson}@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='uk 2United Kingdom Atomic Energy Authority, Abingdon, UK 3Centre for Intelligent Infrastructure, University of Strathclyde, Glasgow, UK, edoardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='patelli@strath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='uk 4Department of Computer Science, University of Texas at El Paso, El Paso, Texas 79968, USA, vladik@utep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='edu 5Department of Teacher Education, University of Texas at El Paso, El Paso, Texas 79968, USA, olgak@utep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='edu Abstract In many practical situations, we know the probabilities a and b of two events A and B, and we want to estimate the joint probability Prob(A & B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The algorithm that estimates the joint probability based on the known values a and b is called an and-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' An important case when such a reconstruction is possible is when we know the correlation between A and B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' we call the resulting and-operation correlation-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' On the other hand, in statistics, there is a widely used class of and-operations known as copulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Empirical evidence seems to indicate that the correlation- based and-operation derived in [4] is a copula, but until now, no proof of this statement was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this paper, we provide such a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 1 Formulation of the problem Correlation-based “and”-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In many practical situations, we know the probabilities a and b of two events A and B, and we need to estimate the joint probability Prob(A & B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' An algorithm f&(a, b) that transforms the known values a and b into such an estimate is usually called an and-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' One important case when such an estimate is possible is when, in addition to the probabilities a and b, we also know the correlation ρ between the corre- sponding two random events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' It is known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', [3, 4]) that in this case, we can uniquely determine the probability of Prob(A & B) as a · b + ρ · � a · (1 − a) · b · (1 − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (1) 1 While this formula is true whenever the correlation is known, this formula does not lead to an everywhere defined and-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For example, for a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1 and ρ = −1, this formula leads to a meaningless negative probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1 + (−1) · √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='9 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='9 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='01 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='09 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='08 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' To avoid such meaningless estimates, we need to take into account that the joint probability Prob(A & B) must satisfy Fr´echet inequalities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', [2]): max(a + b − 1, 0) ≤ Prob(A & B) ≤ min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (2) So, if an expert claims to know the correlation ρ and the estimate for Prob(A & B) based on this value ρ is smaller than the lower bound max(a + b − 1, 0) – which cannot be – a reasonable idea is to take the closest possible value of the joint probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', the value max(a + b − 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Similarly, if the estimate for Prob(A & B) based on the expert-provided value ρ is larger than the up- per bound min(a, b) – which also cannot be – a reasonable idea is to take the closest possible value of the joint probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', the value min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, we arrive at the following and-operation – which we will call correlation-based and-operation: fρ(a, b) = Ta,b � a · b + ρ · � a · (1 − a) · b · (1 − b) � , (3) where Ta,b(c) = max(a + b − 1, 0) if c < max(a + b − 1, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Ta,b(c) = c if max(a + b − 1, 0) ≤ c ≤ min(a, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' and (4) Ta,b(c) = min(a, b) if min(a, b) < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Question: is this and-operation a copula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In probability theory, there is a known class of and-operations known as copulas (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', [5, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' These are functions C(a, b) for which, for some random 2-D vector (X, Y ), the joint cumulative distribution function FXY (x, y) def = Prob(X ≤ x & Y ≤ y) has the form FXY (x, y) = C(FX(x), FY (y)), where FX(x) def = Prob(X ≤ x) and FY (y) def = Prob(Y ≤ y) are known as marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' One important aspect of (3)-(4) is that these formulas can be expressed as a copula (2-copula) family as described in [4], allowing us to operate not only with precise probabilities, but also with interval probabilities and probability boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' A 2-copula must satisfy the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Grounded: C(0, b) = C(a, 0) = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Uniform margins: C(a, 1) = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' C(1, b) = b 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 2-increasing: C(a, b) + C(a, b) − C(a, b) − C(a, b) ≥ 0 for all a < a and b < b 2 It is easy to see that (3)-(4) satisfies the two first properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In [4] the third property was checked for a dense set of tuples (a, a, b, b, ρ), and for all these tuples, the inequality was satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' However, at that moment, we could not prove that the correlation-based and-operation is indeed a 2-copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this paper we provide the missing proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 2 Main result Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For every ρ ∈ [−1, 1], the correlation and-operation fρ(a, b) de- scribed by the formulas (3)-(4) is a copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' It is known that the desired inequality has the following property – if we represent a box [a, a] × [b, b] as a union of several sub-boxes, then the left- hand side of the desired inequality is equal to the sum of the left-hand sides corresponding to sub-boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Indeed, as one can easily check, there is the following additivity property: for each box consisting of several sub-boxes, the left-hand side of the inequality (4a) that corresponds to the larger box is equal to the sum of expressions (4a) corresponding to sub-boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, if the expressions corresponding to sub-boxes are non-negative, then the expression (4a) corresponding to the larger box is also non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In general, the and-operation described by the formula (4) has three different expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, to prove that the expression (4a) corresponding to this expres- sion is also non-negative, we need to consider cases when at different vertices of the box, we may have different expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Good news is that every box whose vertices are described by different expressions can be represented as the union of sub-boxes in which: either all vertices are described by the same expression or two vertices are on the boundary between the areas of different expres- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This is easy to see visually: the following box, in which the slanted line represents the boundary between the areas � � � � � can be represented as the union of sub-boxes with the desired property: 3 � � � � � Thus, to prove that our and-operation is a copula, it is sufficient to consider only boxes of the following type: boxes for which all four vertices belong to the same area, and boxes for which two vertices belong to the boundary between two areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The functions max(a + b − 1, 0) and min(a, b) are known to be copulas, so if all four vertices belong to one of these areas, then the desired inequality (4a) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, it is sufficient to consider: boxes for which all four vertices belong to the new area, in which the and- operation is described by the expression (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' we will consider such boxes in Parts 2–4 of this proof, and boxes for which two vertices belong to the boundary between two areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' these boxes will be considered in the following Parts of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us start by considering boxes for which all four vertices belongs to the area in which the and-operation is described by the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' It is known [1] – and it is easy to prove by considering infinitesimal differences x − x and y − y – that for smooth functions, the desired inequality is equivalent to the fact that the partial derivative ∂C ∂a is non-decreasing in b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, that the mixed derivative is non- negative: d def = ∂2C ∂a ∂b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, to prove that fρ(a, b) is a copula, it is sufficient to prove that its mixed derivative is non-negative everywhere where the new formula is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Indeed, at the points where the formula (1) is applied, the derivative of fρ(a, b) with respect to a has the has the form ∂fρ ∂a = b + ρ · 1 − 2 · a 2 · � a · (1 − a) � b · (1 − b), (4b) and thus, the mixed derivative has the following form: d = ∂ ∂b �∂fρ ∂a � = 1 + ρ · (1 − 2 · a) · (1 − 2 · b) 4 · � a · (1 − a) · b · (1 − b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (5) 4 Since the expression (1) does not change if we swap a and b, it is sufficient to consider the case when a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' When ρ = 0, we get a known copula f0(a, b) = a · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, it is sufficient to consider cases when ρ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This can happen when ρ > 0 and when ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us consider these cases one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us first consider the case when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, since a ≤ b, we have min(a, b) = a and thus, the condition (4) takes the form a · b + ρ · � a · (1 − a) · b · (1 − b) ≤ a, (6) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, ρ · � a · (1 − a) · b · (1 − b) ≤ a − a · b = a · (1 − b) (7) and thus, ρ ≤ a · (1 − b) � a · (1 − a) · b · (1 − b) = � a · (1 − b) � (1 − a) · b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (8) For all such ρ, we need to prove that the expression (5) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' When both a and b are larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 or both are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5, the differences 1 − 2a and 1 − 2b have the same sign and thus, their product is non- negative and the expression (5) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, the only case when we need to check that d ≥ 0 is when one of the two values a and b is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 and another one is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Since a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5, this means that a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, the condition d ≥ 0 takes the form 1 − ρ · (1 − 2 · a) · (2 · b − 1) 4 · � a · (1 − a) · b · (1 − b) ≥ 0, (9) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, ρ · (1 − 2 · a) · (2 · b − 1) 4 · � a · (1 − a) · b · (1 − b) ≤ 1, (10) and ρ ≤ 4 · � a · (1 − a) · b · (1 − b) (1 − 2 · a) · (2 · b − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (11) So, to prove that we always have d ≥ 0, we need to prove that every ρ that satisfies the inequality (8) also satisfies the inequality (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Clearly, if some value ρ satisfies the inequality (11), then every smaller value ρ also satisfies this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, to prove the desired implication, it is sufficient to check that the inequality (11) is satisfied for the largest possible value ρ that satisfies the inequality (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', for the value ρ which is equal to the right-hand side of the inequality (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For this ρ, the desired inequality (11) takes the form � a · (1 − b) � (1 − a) · b ≤ 4 · � a · (1 − a) · b · (1 − b) (1 − 2 · a) · (2 · b − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (12) 5 Dividing both sides by � a · (1 − b), we get an equivalent inequality 1 � (1 − a) · b ≤ 4 · � (1 − a) · b (1 − 2 · a) · (2 · b − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (13) Multiplying both sides by both denominators, we get the following equivalent inequality: (1 − 2 · a) · (2 · b − 1) ≤ 4 · (1 − a) · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (14) If we open parentheses, this inequality takes the equivalent form 2 · b − 4 · a · b − 1 + 2 · a ≤ 4 · b − 4 · a · b, (15) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', by adding 4 · a · b − 2 · b to both sides, the form −1 + 2 · a ≤ 2 · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (16) We are considering the case when a ≤ b – since, as we have mentioned earlier, it is sufficient to only consider this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the equivalent inequality (12) is also true and hence, for the case when ρ > 0, we indeed have d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' To complete the proof, it is now sufficient to consider the case when ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, if one of the values a and b is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 and another one is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5, then the differences 1 − 2 · a and 1 − 2 · b have different signs, so the right-hand side of the expression (5) for d is larger than 1 and thus, non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, it is sufficient to consider the cases when: either both a and b are larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 or both a and b are smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us consider these two cases one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us first consider the case when a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 and b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, a + b − 1 > 0, so the inequality (4) takes the form a · b + ρ · � a · (1 − a) · b · (1 − b) ≥ a + b − 1, (17) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, that |ρ| · � a · (1 − a) · b · (1 − b) ≤ a · b − a − b + 1 = (1 − a) · (1 − b), (18) or that |ρ| ≤ (1 − a) · (1 − b) � a · (1 − a) · b · (1 − b) = � (1 − a) · (1 − b) √ a · b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (19) In this case, the condition d ≥ 0 that the value (5) is non-negative takes the form 1 − |ρ| · (2 · a − 1) · (2 · b − 1) 4 · � a · (1 − a) · b · (1 − b) , (20) 6 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, |ρ| · (2 · a − 1) · (2 · b − 1) 4 · � a · (1 − a) · b · (1 − b) ≤ 1 (21) and |ρ| ≤ 4 · � a · (1 − a) · b · (1 − b) (2 · a − 1) · (2 · b − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (22) Similarly to the case when ρ > 0, to check that all values |ρ| satisfying the inequality (19) also satisfies the inequality (22), it is sufficient to check that the largest possible value |ρ| satisfying the inequality (19) satisfies the inequality (22), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', that � (1 − a) · (1 − b) √ a · b ≤ 4 · � a · (1 − a) · b · (1 − b) (2 · a − 1) · (2 · b − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (23) If we divide both sides by � (1 − a) · (1 − b), we get the following equivalent inequality 1 √ a · b ≤ 4 · √ a · b (2 · a − 1) · (2 · b − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (24) Multiplying both sides by both denominators, we get the following equivalent inequality (2 · a − 1) · (2 · b − 1) ≤ 4 · a · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (25) Opening parentheses, we get 4 · a · b − 2 · a − 2 · b + 1 ≤ 4 · a · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (26) Adding 2 · a + 2 · b − 4 · a · b to both sides, we get an equivalent inequality 1 ≤ 2 · a + 2 · b, (27) which is true since we consider the case when a + b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, in this case, we indeed have d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us now consider the case when a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5 and b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, a + b − 1 < 0, so the inequality (4) takes the form a · b + ρ · � a · (1 − a) · b · (1 − b) ≥ 0, (28) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, that |ρ| · � a · (1 − a) · b · (1 − b) ≤ a · b, (29) or that |ρ| ≤ a · b � a · (1 − a) · b · (1 − b) = √ a · b � (1 − a) · (1 − b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (30) 7 In this case, the condition d ≥ 0 that the value (5) is non-negative takes the form 1 − |ρ| · (1 − 2 · a) · (1 − 2 · b) 4 · � a · (1 − a) · b · (1 − b) , (31) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, |ρ| · (1 − 2 · a) · (1 − 2 · b) 4 · � a · (1 − a) · b · (1 − b) ≤ 1 (32) and |ρ| ≤ 4 · � a · (1 − a) · b · (1 − b) (1 − 2 · a) · (1 − 2 · b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (33) Similarly to the cases when ρ > 0 and when a + b > 1, to check that all values |ρ| satisfying the inequality (30) also satisfies the inequality (33), it is sufficient to check that the largest possible value |ρ| satisfying the inequality (30) satisfies the inequality (33), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', that √ a · b � (1 − a) · (1 − b) ≤ 4 · � a · (1 − a) · b · (1 − b) (1 − 2 · a) · (1 − 2 · b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (34) If we divide both sides by √ a · b, we get the following equivalent inequality 1 � (1 − a) · (1 − b) ≤ 4 · � (1 − a) · (1 − b) (1 − 2 · a) · (1 − 2 · b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (35) Multiplying both sides by both denominators, we get the following equivalent inequality (1 − 2 · a) · (1 − 2 · b) ≤ 4 · (1 − a) · (1 − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (36) Opening parentheses, we get 1 − 2 · a − 2 · b + 4 · a · b ≤ 4 − 4 · a − 4 · b + 4 · a · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (37) Adding 4 · a + 4 · b − 4 · a · b − 1 to both sides, we get an equivalent inequality 2 · a + 2 · b ≤ 3, (38) which is true since we consider the case when a + b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, in this case, we indeed have d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In all cases when have d ≥ 0, thus, the and-operation fρ(a, b) is indeed a copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, for boxes in which all four vertices belong to the area described by the expression (1), the inequality (4a) is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us now consider the boxes in which two vertices belong to the boundary between two areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' First, we will consider the case when ρ > 0 and then, we will consider the case when ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us first consider the case when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For this case, let us first describe the boundaries between the areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us analyze which of the three areas listed in formula (4) are possible in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' When ρ > 0, we have a · b + ρ · � a · (1 − a) · b · (1 − b) ≥ a · b, and since it is known that we always have a · b ≥ max(a + b − 1, 0), we have a · b + ρ · � a · (1 − a) · b · (1 − b) ≥ max(a + b − 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, for ρ > 0, we cannot have the first of the three cases described by the formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, we only have two areas: the area where the and-operation is described by the formula (1), and the area where the and-operation is described by the formula min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us describe the two possible areas and the boundary between these two areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The first area is characterized by the inequality a · b + ρ · � a · (1 − a) · b · (1 − b) ≤ min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (39) Similarly to the previous part of the proof, without losing generality, we can consider the case when a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, the inequality (39) describing the first area takes the following form: a · b + ρ · � a · (1 − a) · b · (1 − b) ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (40) If we subtract a · b from both sides of this inequality, we get the following equivalent inequality: ρ · � a · (1 − a) · b · (1 − b) ≤ a · (1 − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (41) Both sides of this inequality are non-negative, so we can get an equivalent inequality is we square both sides: ρ2 · a · (1 − a) · b · (1 − b) ≤ a2 · (1 − b)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (42) The cases when a or b are equal to 0 or 1 can be obtained by taking a limit from the cases when both a and b are located insyed the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For such values, we can divide both side of the inequality by positive numbers a2, b, and 1 − b, and get the following equivalent inequality: ρ2 · 1 − a a ≤ 1 − b b , (43) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, ρ2 · 1 − a a ≤ 1 b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (44) 9 By adding 1 to both sides of this inequality, we get a + ρ2 · (1 − a) a ≤ 1 b , (45) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, that b ≤ a a + ρ2 · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (46) This inequality describes the first area, in which the and-operation is described by the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the boundary between the two areas is described by the equality b = a a + ρ2 · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (47) Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' One can see that for a = 0 we get b = 0, for a = 1, we get b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us prove that for all a, the corresponding boundary value b is greater than or equal to a – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', that for all the points (a, b) on this boundary, we have a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Indeed, for the expression (47), the desired inequality a ≤ b takes the form a ≤ a a + ρ2 · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (48) If we divide both sides by a and multiply both sides by the denominator of the right-hand side, we get the following equivalent inequality a + ρ2 · (1 − a) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (49) If we move all the terms to the right-hand side, we get an equivalent inequality 0 ≤ 1 − a − ρ2 · (1 − a) = (1 − ρ2) · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (50) This inequality is always true, since ρ2 ≤ 1 and a ≤ 1, so indeed, for all boundary points, we have a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us prove that the boundary describes b as an increasing function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' By applying, to the equality (47) that describes the boundary, the same transformations that show the equivalent of inequalities (43) and (46), we can conclude that the equality (47) is equivalent to ρ2 · 1 − a a = 1 − b b , (51) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', to ρ2 · �1 a − 1 � = 1 b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (52) 10 The left-hand side is decreasing with respect to a, the right-hand side is a decreasing function of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, as a increases, the left-hand side decreases, thus the right-hand side also decreases and hence, the value b increases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For ρ = 1 the condition (46) describing the first area takes the form b ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Since we have a ≤ b, this means that this condition is only satisfies for a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For these values, the expression (4a) is equal to a · a + � a · (1 − a) · a · (1 − a) = a2 + a · (1 − a) = a2 + a − a2 = a = min(a, b), (53) which means that our and-operation is always equal to min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The expression min(a, b) is known to be a copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, we only need to prove the fact that our and-operation is a copula for the case when ρ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This is the case we will consider from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us prove that for ρ < 1, the only boundary points for which a = b are points for which a = b = 0 and a = b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Indeed, as we have mentioned, the points (0, 0) and (1, 1) are boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us prove, by contradiction, that there are no other boundary points for which a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Indeed, when a = b, the equality (52) that describes the boundary takes the form: ρ2 · �1 a − 1 � = 1 a − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (54) Dividing both sides of this equality by the non-zero right-hand side, we get ρ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This contradicts to the fact that we are considering the case when ρ < 1 and thus, ρ2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This contradiction shows that other boundary points with a = b are not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The boundary consists of a curved line that is separate from the line a = b – except for the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, if we limit ourselves to a sub-box [ε, 1−ε]×[ε, 1−ε] for some small ε > 0, the boundary line is separated from the line a = b – there is the smallest distance δ > 0 between points of these two lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, if we have a box that includes both points with a ≤ b and with a ≥ b, we can divide this box into sub-boxes of linear size < δ/2 and thus, make sure that every sub-box that contains boundary points with a ≤ b cannot contain any points with a = b – and therefore, only contains points with a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, due to additivity, it is sufficient to prove the inequality (4a) for boxes for which: two vertices lie on the boundary, and we have a ≤ b for all the points from this sub-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This will allow us to prove the inequality (4a) for all sub-boxes of the square [ε, 1−ε]×[ε, 1−ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' We can do it for any ε and thus, in the limit, get the desired inequality for all sub-boxes of the original square [0, 1] × [0, 1] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, suppose that we have a box for which: 11 two vertices lie on the boundary, and we have a ≤ b for all the points from this box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Since the boundary describes the increasing function of a, the corresponding box has the form � � � � � So, in the corresponding box: the two vertices (a, b) and (a, b) are on the boundary, the vertex (a, b) is in the first area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', for this point, we have the expres- sion (1), and the vertex (a, b) is in the second area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', here C(a, b) = min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The desired inequality (4a) has the form C(a, b) − C(a, b) ≤ C(a, b) − C(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (55) The points (a, b) and (a, b) are both in the second area for which C(a, b) = min(a, b) – to be more precise, the second of these points is in the boundary, which means it also satisfies the condition C(a, b) = min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For all the points from the box, a ≤ b, so we have C(a, b) − C(a, b) = min(a, b) − min(a, b) = a − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (56) On the other hand, for the difference in the left-hand side of the formula (55), we have C(a, b) − C(a, b) = � a a ∂C ∂a da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (57) So, if we prove that the partial derivative ∂C/∂a is always smaller or equal than 1, we would indeed conclude that C(a, b) − C(a, b) = � a a 1 da = a − a, (58) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', exactly, the desired inequality (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For the points (a, b) and (a, b) – and the points from the interval connecting these two points – the expression C(a, b) is described by the formula (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the partial derivative of C(a, b) with respect to a is described by the formula (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the inequality ∂C ∂a (a, b) ≤ 1, (59) 12 takes the form b + ρ · 1 − 2 · a 2 · � a · (1 − a) � b · (1 − b) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (60) Subtracting b from both sides of (60), we get an equivalent inequality ρ · 1 − 2 · a 2 · � a · (1 − a) � b · (1 − b) ≤ 1 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (61) To separate the variables, we can divide both sides by � b · (1 − b), then we get an equivalent inequality ρ · 1 − 2 · a 2 · � a · (1 − a) ≤ � 1 − b b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (62) By taking the square root of both sides of the inequality (46), we conclude that: ρ · � 1 − a a ≤ � 1 − b b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (63) Thus, if we prove that the left-hand side of the inequality (62) is smaller than or equal to the left-hand side of the inequality (63), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', that ρ · 1 − 2 · a 2 · � a · (1 − a) ≤ ρ · � 1 − a a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (64) this will prove the inequality (62) and thus, the desired upper bound (60) on the partial derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' We can simplify the inequality (64) by dividing both sides by ρ and multiplying both sides by 2 · � a · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Then, we get an equivalent inequality 1 − 2 · a ≤ 2 · (1 − a) = 2 − 2 · a, (65) which is equivalent to 1 ≤ 2 and is, thus, always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, (55) holds, so the inequality (4a) is true for all the boxes in which two vertices are located on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This completes the proof of the Proposition for the case when ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us now consider the case when ρ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' For this case, let us first describe the boundaries between the areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us analyze which of the three areas listed in formula (4) are possible in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' When ρ < 0, we have a · b + ρ · � a · (1 − a) · b · (1 − b) ≤ a · b, and since it is known that we always have a · b ≤ min(a, b), we have a · b + ρ · � a · (1 − a) · b · (1 − b) ≤ min(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, for ρ < 0, we cannot have the third of the three cases described by the formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, we only have two areas: 13 the area where the and-operation is described by the formula (1), and the area where the and-operation is described by the formula max(a + b − 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us describe the two possible areas and the boundary between these two areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The first area is characterized by the inequality C(a, b) ≥ max(a + b − 1, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, by two inequalities a · b − |ρ| · � a · (1 − a) · b · (1 − b) ≥ 0 (66) and a · b − |ρ| · � a · (1 − a) · b · (1 − b) ≥ a + b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (67) Let us consider these two inequalities one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The inequality (66) is equivalent to: a · b ≥ |ρ| · � a · (1 − a) · b · (1 − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (68) To separate the variables, let us divide both sides of this inequality by a · � b · (1 − b), then we get an equivalent inequality � b 1 − b ≥ |ρ| · � 1 − a a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (69) Both sides of this inequality are non-negative, thus if we square both sides, we get an equivalent inequality b 1 − b ≥ ρ2 · 1 − a a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (70) Reversing both sides, we get an equivalent inequality 1 − b b ≤ a ρ2 · (1 − a), (71) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, 1 b − 1 ≤ a ρ2 · (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (72) By adding 1 to both sides, we get 1 b ≤ ρ2 · (1 − a) + a ρ2 · (1 − a) , (73) 14 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, b ≥ ρ2 · (1 − a) ρ2 · (1 − a) + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (74) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The inequality (67) is equivalent to a · b − a − b + 1 ≥ |ρ| · � a · (1 − a) · b · (1 − b), (75) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', (1 − a) · (1 − b) ≥ |ρ| · � a · (1 − a) · b · (1 − b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (76) To separate the variables, let us divide both sides by (1 − a) · � b · (1 − b), then we get an equivalent inequality � 1 − b b ≥ |ρ| · � a 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (77) Both sides of this inequality are non-negative, thus if we square both sides, we get an equivalent inequality 1 − b b ≥ ρ2 · a 1 − a, (78) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, 1 b − 1 ≥ ρ2 · a 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (79) By adding 1 to both sides, we get 1 b ≥ ρ2 · a + (1 − a) 1 − a , (80) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', equivalently, b ≤ 1 − a ρ2 · a + (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (81) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' By combining the inequalities (74) and (81), we get the following de- scription of the area in which the and-operation is described by the formula (1): ρ2 · (1 − a) ρ2 · (1 − a) + a ≤ b ≤ 1 − a ρ2 · a + (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (82) Thus, the boundary between the two areas consists of the following two curves: b = ρ2 · (1 − a) ρ2 · (1 − a) + a (83) and b = 1 − a ρ2 · a + (1 − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (84) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us prove that: 15 the curve (83) lies in the area where a + b ≤ 1, and the curve (84) lies in the area where a + b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us first prove that for each value b described by the formula (83), we have a + b ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' We need to prove the inequality a + ρ2 · (1 − a) ρ2 · (1 − a) + a ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (85) Subtracting a from both sides, we get an equivalent inequality ρ2 · (1 − a) ρ2 · (1 − a) + a ≤ 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (86) Dividing both sides by 1 − a and multiplying both sides by the denominator of the left-hand side, we get the following equivalent inequality: ρ2 ≤ ρ2 · (1 − a) + a = ρ2 + (1 − ρ2) · a, (87) which is, of course, always true, since ρ2 ≤ 1 and a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The statement is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us now prove that for each value b described by the formula (84), we have a + b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' We need to prove the inequality a + 1 − a ρ2 · a + (1 − a) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (88) Subtracting a from both sides, we get an equivalent inequality 1 − a ρ2 · a + (1 − a) ≥ 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (89) Dividing both sides by 1 − a and multiplying both sides by the denominator of the left-hand side, we get the following equivalent inequality: 1 ≥ ρ2 ≤ a + (1 − a) = 1 − (1 − ρ2) · a, (90) which is, of course, always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The statement is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Similarly to Part 6 of this proof, it is sufficient to prove the inequality (4a) for boxes in which two vertices are on the boundary and for which: either we have a + b ≤ 1 for all the points from the box, or we have a + b ≥ 1 for all the points from the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 16 Let us consider the two parts of the boundary one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us first consider the case when we have a + b ≤ 1 for all the points from the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, the corresponding part of the boundary is described by the formula (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' By reformulating this expression in the equivalent form b = 1 1 + a ρ2 · (1 − a) = 1 1 + 1 ρ2 · 1 1 a − 1 , (91) we can see that b is a decreasing function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the corresponding box has the form ❅ ❅ ❅ ❅ ❅ So, in the corresponding box: the two vertices (a, b) and (a, b) are on the boundary, the vertex (a, b) is in the first area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', for this point, we have the expres- sion (1), and the vertex (a, b) is in the second area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', here C(a, b) = max(a+b−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' So, for three vertices, we have C(a, b) = max(a + b − 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Since for all the points from the box, we have a + b ≤ 1, this means that for three vertices, we have C(a, b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, the inequality (4a) is clearly true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Let us now consider the case when we have a + b ≥ 1 for all the points from the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' In this case, the corresponding part of the boundary is described by the formula (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' By reformulating this expression in the equivalent form b = 1 1 + ρ2 · a 1 − a = 1 1 + ρ2 · 1 1 a − 1 , (91) we can see that b is also a decreasing function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the corresponding box has the same form as in the case a + b ≤ 1: ❅ ❅ ❅ ❅ ❅ 17 So, in the corresponding box: the two vertices (a, b) and (a, b) are on the boundary, the vertex (a, b) is in the first area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', for this point, we have the expres- sion (1), and the vertex (a, b) is in the second area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', here C(a, b) = max(a+b−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Similarly to Part 6 of the proof, we can show that the desired inequality (4a) is satisfied if we the corresponding partial derivatives is smaller than or equal to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', if ∂C ∂a = b − |ρ| · 1 − 2 · a 2 · � a · (1 − a) � b · (1 − b) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (92) Subtracting b from both sides, we get an equivalent inequality −|ρ| · 1 − 2 · a 2 · � a · (1 − a) � b · (1 − b) ≤ 1 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (93) We can separate the variable if we divide both sides by � b · (1 − b), then we get an equivalent inequality −|ρ| · 1 − 2 · a 2 · � a · (1 − a) ≤ � 1 − b b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (94) We know a lower bound on the expression in the right-hand side – it is provided by the inequality (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, to prove the inequality (94), it is sufficient to prove that the left-hand side of the formula (94) is smaller than or equal to this lower bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', that −|ρ| · 1 − 2 · a 2 · � a · (1 − a) ≤ |ρ| · � a 1 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' (95) Let us prove this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Dividing both sides of (95) by |ρ| and multiplying both sides by 2· � a · (1 − a), we get an equivalent inequality −(1−2·a) ≤ 2·a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=', 2 · a − 1 ≤ 2 · a, which is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, the inequality (94) holds, hence the inequality (92) also holds, and therefore, in this case, the inequality (4a) that describes a copula is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' We have considered all possible cases, and in all these cases, we have shown that the inequality (4a) – that defines a copula – is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Thus, our and-operation is indeed a copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' The proposition is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Acknowledgments This research was partly funded by the EPSRC and ESRC CDT in Risk and Uncertainty (EP/L015927/1), established within the Institute for Risk and 18 Uncertainty at the University of Liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 - EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Com- mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Neither the European Union nor the European Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' was supported in part by the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes), and by the AT&T Fellowship in Information Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' He was also supported by the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' 075-02-2020-1478, and by a grant from the Hungarian National Research, Development and Innovation Office (NRDI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Durante and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE5T4oBgHgl3EQfPw5R/content/2301.05507v1.pdf'} +page_content=' Jaworski, “A new characterization of bivariate copulas”, 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a/79AzT4oBgHgl3EQfE_pK/content/tmp_files/2301.01002v1.pdf.txt b/79AzT4oBgHgl3EQfE_pK/content/tmp_files/2301.01002v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..596f9add4fd836d1cb7348a66bbfcb060169818e --- /dev/null +++ b/79AzT4oBgHgl3EQfE_pK/content/tmp_files/2301.01002v1.pdf.txt @@ -0,0 +1,1552 @@ +1 +Monolayer-to-mesoscale modulation of the +optical properties in 2D CrI3 mapped by +hyperspectral microscopy +Marta Galbiati1*†, Fernando Ramiro-Manzano1,2*†, José Joaquín Pérez Grau1, Fernando Cantos- +Prieto1, Jaume Meseguer-Sanchez1, Ivona Kosic1, Filippo Mione1, Ana Pallarés Vilar1, Andrés +Cantarero1, David Soriano3,4, and Efrén Navarro-Moratalla1* +1 Instituto de Ciencia Molecular, Universitat de València, Calle Catedrático José Beltrán Martínez 2, 46980, Paterna, Spain. +2 Instituto de Tecnología Química, Universitat Politècnica de València - Consejo Superior de Investigaciones Científicas (UPV-CSIC), Avd. de los +Naranjos s/n, 46022, Valencia, Spain. +3 Information Engineering Department, University of Pisa, Via Caruso 16, 56122, Pisa, Italy. +4 Departamento de Física Aplicada, Universidad de Alicante, 03690, San Vicente del Raspeig, Alicante, Spain. +† M.G. and F. R.-M. contributed equally to this work. * e-mail:ferraman@fis.upv.es, marta.galbiati@uv.es, efren.navarro@uv.es +Magnetic 2D materials hold promise to change the +miniaturization paradigm of unidirectional photonic +components. However, the integration of these materials +in devices hinges on the accurate determination of the +optical properties down to the monolayer limit, which is +still missing. By using hyperspectral wide-field imaging +we reveal a non-monotonic thickness dependence +of the complex optical dielectric function in the +archetypal magnetic 2D material CrI3 extending across +different length scales: onsetting at the mesoscale, +peaking at the nanoscale and decreasing again down +to the single layer. These results portray a modification +of the electronic properties of the material and align +with the layer-dependent magnetism in CrI3, shedding +light into the long-standing structural conundrum in +this material. The unique modulation of the complex +dielectric function from the monolayer up to more than +100 layers will be instrumental for understanding and +manipulating the magneto-optical effects of magnetic +2D materials. +The dependence of the physical properties of van der +Waals materials with the number of layers has been +the flagship of two-dimensional (2D) materials since +their discovery. In general, this layer dependence gains +importance upon approaching the single layer limit, +where the strict confinement of electrons in a 2D lattice +imposes dramatic changes in the electronic structure of +the crystal, enabling the realization of new electronic +states and quantum correlated phases of matter. This +has opened the door for the realization of massless Dirac +fermions in graphene and superconducting twisted +bilayer +graphene1,2, +direct-gap +photoluminescence3,4 +and valley polarization in single-layer transition metal +dichalcogenides5,6, or Ising-like superconductivity in +few-layer metallic transition in metal dichalcogenides7, +to name just a few prominent examples. Beyond the +single layer limit, the properties of van der Waals +materials change gradually until reaching the bulk +properties generally within the nanoscale thickness. A +much more unusual case features a continuous change +of the material properties at much larger thicknesses. +One of the few examples is the effect of layer number +on the photoluminescence of hexagonal boron nitride8. +However, this phenomenon arises from a variation of +the bandgap and activation energies of impurities in the +system and is not a consequence of the layer-dependence +of the intrinsic electronic properties. Another remarkable +case is chromium triiodide (CrI3), one of the first layered +materials to exhibit a non-zero net magnetization down +to the monolayer9, accompanied by a transition from +layered antiferromagnetic in the few-layer regime to +ferromagnetic in the bulk, with a crossover thickness to +the bulk that is still unclear but that is unambiguously +located in the mesoscale10-12. Seminal studies attribute this +change to differences in the stacking between the layers13 +being the ferromagnetic and layered antiferromagnetic +states related to the rhombohedral and monoclinic +stacking, respectively. This points at a profound layer- +dependent electronic effect at the mesoscale, setting an +imperious need to study the evolution of the electronic +properties as a function of the layer count in order to +understand the underlying mechanisms giving rise to the +magnetism in CrI3. +An insightful and non-destructive way to study the +layer-dependent evolution of the electronic properties +of a layered material is to determine the complex +dielectric function via light-matter interaction. Although +optical ellipsometry is usually employed to extract these +parameters from thin films, the small sample footprints +of most mechanically exfoliated 2D materials hinder +the direct application of this technique. One strategy to +overcome the spatial resolution limitation is to use an +optical microscope equipped with a broad-band white +light source coupled to a spectrometer, where a continuous +adjustment of the illumination wavelength in constrained +areas of the sample allows for an accurate extraction of +the dielectric function in the visible range14. However, the +sampling speed and data statistics of point spectroscopy +are generally low. Wide-field hyperspectral microscopy +on the other hand has been successfully employed for the +high-throughput layer-dependent characterization of bare + +2 +2D materials and heterostructures in air15-18. Unfortunately, +the magnetic 2D materials are very sensitive to the +presence of humidity and oxygen and therefore require an +appropriate isolation from ambient conditions and rapid +characterization to preclude the effect of degradation, a +combination of requirements that until now has not been +met by the established techniques. By developing a simple +encapsulation technique compatible with wide-field +hyperspectral imaging, we herein circumvent the low- +throughput of deterministic encapsulation and achieve +high sampling frequency and data reliability in the +characterization of the layer-dependent optical properties +of CrI3. The use of a monochrome camera with a high +linear dynamic range permits a single-shot acquisition of +the light intensity coming from tens of bare CrI3 crystals +with different layer number, providing simultaneous +access to large statistics while minimizing the acquisition +times. By modeling the spectral information obtained as +a function of the layer number, we identify at least two +crossover points of the complex optical dielectric function +of CrI3 in the nanoscale and the mesoscale thickness +regime. The crossover in the mesoscale portrays a +sizeable modification of the electronic properties of the +material, which could underpin the structural preference +for the monoclinic phase and provide an explanation for +the layer-dependence of the magnetic properties of the +material. Our theoretical results, based on first-principles +calculations, provide support to the electronic origin of +the evolution of the dielectric function. +CrI3 flakes from the single layer to the hundreds-of- +layers thickness range were obtained by mechanical +exfoliation of bulk crystals on standard microscopy glass +slides and encapsulated with a coverslip sealed using a +thermoplastic material. Spatially resolved hyperspectral +maps of different regions of the sample were then acquired +by combining a sequence of optical images recorded +under different monochromatic illumination spanning +the full visible range (more details in the Supplementary +Information SI2). +By analyzing the hyperspectral optical transmission data +in selected regions of the CrI3 crystals of homogeneous +thickness, examined by atomic force microscopy (AFM) +for their correct estimation (see SI2 for details), we +obtained spectral traces of the material for different +layer numbers. Figure 1 shows the transmittance spectra +of flakes with a thickness ranging from 1 layer (L) to 164 +L. Two absorption features are visible at about 1.95 eV +and 2.7 eV, corresponding to the ligand-to-metal charge +transfer absorption peaks reported for bulk CrI3 and more +recently for CrI3 exfoliated samples19-21. Remarkably, both +transmittance dips present a non-monotonic trend with +the number of layers, i.e. a red shift when flake thickness +increases from 1 L to ~ 13 L and a blue shift above ~ 50 L, +thus suggesting a change in CrI3 optical properties with +flake thickness. It is also interesting to point out that these +features remain almost unchanged for crystals thicker +than ~ 100 L, hinting to a smooth saturation towards bulk +values for layers thickness located at the mesoscopic scale. +After +considering +different +approaches +for +the +evaluation of the complex optical dielectric function (ε(ω) += ε1 - iε2) from optical data (see SI3), we chose to focus on +transmission measurements and large statistics analysis +to achieve high data reliability. We simultaneously +analysed transmittance spectra acquired over flakes with +different thicknesses, including a large number of samples +with ample statistics (thousands of pixels each) in every +dataset, in order to limit the interdependence between ε1 +and ε2 from a single experiment and further constrain the +range of possible solutions of CrI3 ε(ω). +To extract the complex dielectric function from the +transmittance data, we consider our experimental system +as a basic Fabry-Pérot cavity formed by a stack of 3 layers +and we assume that the dielectric function of CrI3 follows +a modified Lorentzian oscillator model22: +This model is composed by two oscillators (n = 1,2), +where ε∞ is the permitivity for infinite optical frequencies, +ωn, ωp, and ω represent the resonant, plasma and photon +frequency, respectively, and γn and Γn are damping related +parameters. Figure 2 shows the real and imaginary parts +of ε(ω) and their evolution with flake thickness calculated +by simultaneous fitting of all the transmittance spectra +shown in Figure 1 (see SI3 for details). The dielectric +function of CrI3 monolayer is found to be significantly +different from the rest of the layers. This is in line with the +transmittance spectra experimentally measured in the +Figure 1 | Visible range transmittance spectra of CrI3 crystals with +different layer number. The data displayed was extracted from +selected areas of a wide-field hyperspectral image that are found to be +atomically flat by AFM. Red curves correspond to thin layers (from 1L to +13L), blue curves to layers ranging from 14L to ~100L and green curves +correspond to bulk (up to 164L). +~ +~ +~ +~ +Number of layers (L) +1 +13 +100 +164 +˜ε(ω) = ˜ε∞ + +2 +∑ +n=1 +ω2 +p + 2iωΓn +ω2n − ω2 + 2iωγn + +3 +monolayer, where absorption features are significantly +shifted compared to the bilayer. Besides the discontinuity +found for the monolayer, we find two additional critical +points at about 13 L (~8.9 nm) and 100 L (~68.7 nm), where +ε2 intensity increases and peaks red shift (< 13 L), then +decreases and peaks blue shift (> 13L), until the dielectric +function starts to asymptotically saturate toward the bulk +value above ~100 L. This delimits three different thickness +domains: (i) few layer, (ii) multilayer and (iii) bulk. These +changes are also clearly illustrated in Figure 3 which +shows the evolution with layer thickness of ε1 and ε2 at 2.66 +eV extracted from the fitting process (Figure 3a), and the +evolution of the experimental absorption feature minima +of the transmittance spectra (Figure 3b). Hyperspectral +analysis also gives the possibility to spatially resolve the +optical properties of the 2D material. As such, Figure 3c +Figure 2 | Evolution of the dielectric functions in CrI3 as a function of the layer number. The calculated real (ε1) and imaginary (ε2) parts of +are shown in panels (a) and (b), respectively. (c) and (d) have been also plotted as a function of the excitation energy for crystals of different +thicknesses (these are line cuts of the image plots shown in (a) and (b)). +Figure 3 | Identification of different thickness regimes in CrI3. The few-layer, multilayer and bulk thickness regimes are depicted according to the +layer dependence of ε1 and ε2 at 2.66 eV (a) and the low (bottom) and high (top) energy transmittance minima (b). Markers display experimental data +while the dotted lines are guides to the eye. (c) Spatially resolved wide field image of the high energy transmittance dip position calculated from +hyperspectral images of CrI3 flakes with different thicknesses. The energy crossover of dip position when increasing the number of layers is clearly +visible, highlighting the different thickness regimes. +0 +0 +100 +150 +1.95 +2.05 +2.15 +2.65 +2.69 +2.73 +2.77 +5 +Number of layers (L) +Number of layers (L) +Energy (eV) +ε1, ε2 (2.66 eV) +ε1 +ε2 +0 +0 +100 +150 +0 +1 +2 +3 +4 +5 +5 +6 +7 +8 +a +b +c +d +a +b +c +2.75 +67 L +14 L +3 L +70 L +20 L +8 L +30 L +53 L +2 L +57 L +13 L +44 L +2.73 +2.71 +2.69 +2.67 +2.65 +Energy (eV) + +5 +10 +b +34567 +a +Dielectric function - , +Dielectric function - 2 +160 +160 +140 +140 +120 +120 +100 +100 +(7) s +layers +Number of layers ( +80 +80 +60 +60 +Number of +40 +40 +20 +20 +87654321 +８７６５4３２1 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +Energy (eV) +Energy (eV) +101 +102 +101 +102 +1 +Number of layers (L) +d +Number of layers (L) +c +8 +12 +10 +6 +8 +13 +c2 +4 +6 +1L +4 +2 +1L +2 +0 +0 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +Energy (eV) +Energy (eV)4 +displays the spatially resolved wide field image of the +higher-energy transmittance dip of CrI3 flakes for different +thicknesses (see SI4 for calculation details) where the dip +in energy of the crossover from a few layers to bulk is also +clearly visible. These experimental observations hence +point at profound changes in the optical properties of CrI3 +in these three different thickness domains. +The evolution of the dielectric function with the +layer number has been previously reported in other 2D +materials such as transition metal dichalcogenides23, +In2Se3 +24, +PdSe2 +25 +and +a +non-monotonic +behaviour, +consistent with our work, was observed in MoS2 at the +nanoscale26. However, in all these cases this behaviour +has been reported only in the nanoscale while this is the +first time that a modulation of the dielectric function has +been experimentally observed at the mesoscale range of +thickness, allowing for a continuous modulation of the +optical properties from 1 to more than 100 layers. +To gain further insight into the possible origins of this +mesoscopic transition, we carried out first-principles +calculations on few-layer CrI3, up to 10 L (see SI5 for +details). In the real and imaginary parts of the computed +dielectric function (Figure 4a-b), we identify 2 peaks +located at 2.1 eV and 2.8 eV, in good agreement with the +experimental results. A look at the band structure and +DOS of monolayer CrI3 (left panel in Figure 4c) confirms +that these peaks can be ascribed to metal-to-ligand charge +transfer processes between the p-orbitals of the ligands, +localized in the last occupied bands, and the empty dx2-y2 and +dz2 orbitals of the chromium atoms localized in the eg set of +conduction bands. In the right panel of Figure 4c, we plot +the evolution of the peak at 2.1 eV with increasing number +of layers. We observe a strong red shift from 2.1 eV to 1.9 +eV in agreement with experimental data and indicating a +clear connection between the number of layers and the +electronic structure in few-layer CrI3. The calculations +also show a tendency towards saturation of the value of the +complex dielectric function when increasing the thickness +from 1 L to 10 L, which confirms the electronic origin of +the layer-dependence. The discrepancies in the absolute +value of the complex dielectric functions extracted from +the calculations and the experimental ones may originate +from subtle differences of the inter-layer distance at +different temperatures and layer numbers, which have a +strong effect on the polarizability of the wave-functions +in the direction perpendicular to the layers (see SI5 for +more details). This effect also suggests that small changes +in the crystal structure could eventually modify the +intensity of the complex dielectric function. Indeed, our +calculations indicate that at 0 K the interlayer distance in +bulk layered antiferromagnetic CrI3 increases up to 0.2 Å +compared to few layer samples (see Table S2) which may +lead to a reduction of the polarizability and the dielectric +function. This result hence highlights the dipolar and +long-range nature of the van der Waals interactions which +can be one of the possible explanations for the observed +mesoscopic crossover of the complex dielectric function +experimentally observed in the different thickness +regimes. +The saturation of the value of the complex dielectric +function at the mesoscale is well aligned with the critical +crystal thickness at which the low-temperature magnetic +properties of CrI3 change from antiferromagnetic to +ferromagnetic. Although this critical thickness has not +yet been unambiguously determined, most works report +on the layered antiferromagnetic state persisting in +crystals up to 50 nm27,28. Both the magnetic and the optical +crossover thickness to the bulk regime being in the same +range of thickness is a strong indication of a connection +between both phenomena. Our findings, supported by the +theoretical calculations, point at an electronic origin of +these effects, with an evolution of the electronic structure +with the layer number onsetting below 100 L (68.7 nm) +as revealed by the non-monotonic modulation of the +dielectric function as the crystal is thinned down. These +results will contribute to shed light into the open structural +conundrum of the layer-dependent phase diagram of CrI3. +Our results demonstrate that hyperspectral transmission +microscopy is instrumental to study the layer-dependent +evolution of the electronic properties of air-sensitive +Figure 4 | Theory calculations of the layer-dependent electronic and optical properties of CrI3. (a-b) Evolution of ε1 and ε2 of CrI3 with increasing +number of layers calculated in the z-direction perpendicular to the layers. (c) Projected band structure and DOS of monolayer CrI3. Filled and empty +circles represent I p-orbitals and Cr d-orbitals respectively. The size of the circles is the weight of the orbital wavefunction in each band. Red and +blue stand for spin up and down respectively. The yellow arrow shows the most probable transition responsible for the low-energy peak in the +dielectric function. In (d), we show the evolution of the low-energy peak in the imaginary complex dielectric function with increasing number of +layers. +a +c +I +Cr +Spin up +Spin down +eg +b +d +I +Cr +Spin up +Spin down +eg +E - EF (eV) +ε2 +ε1 +Spin ↑ +Spin ↓ +0 +0 +1 +2 +3 +4 +5 +6 +0.5 +Κ +-1.5 +-1 +1 +DOS (eV-1) +ω (eV) +ω (eV) +Number of layers (L) +ω (eV) +10 +1.8 +2.8 +2 +2.2 2.4 2.6 +0 +-10 +0 +5 +10 +Κ +-0.5 +Γ +1.9 +1.95 +2 +2.05 +2.1 +1.8 +2.8 +2 +2.2 2.4 2.6 +0 +1 +2 +3 +4 +5 +6 + +6 +10L +8 +5 +6 +L +4 +L +4 +3 +L +2 +L +3 +3 +2 +1 +0 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +w (eV)0,5 +0 +eV) +L +i +-0,5 +00000 +8:8888 +K +r +K +10 +0 +-10 +DOS (ev-6 +10 L +8 L +5 +6 L +4L +4 +3 L +2 L +1 L +2 +3 +3 +2 +1 +O +1.8 +2 +2.2 +2.4 +2.6 +2.8 +w (eV)2,1 +2,05 +(eV) +3 +2 +1,95 +1,9 +0 +5 +10 +Number of layers (L)5 +layered materials through the determination of its +complex dielectric function. The possibility to obtain in a +single acquisition the light intensity coming from tens of +bare CrI3 crystals with different layer number guarantees +high throughput statistics and allows for a robust and +simultaneous determination of both the real and imaginary +layer-dependent components of the dielectric function. +This approach reveals at least two crossover points of the +electronic properties of CrI3, depicting changes of the +trends of the complex dielectric function both in the nano- +and the mesoscale thickness regimes. The significantly +wide span of the continuous layer number modulation +of the optical and electronic structure, covering more +than 100 layers, will be pivotal to enable the fine tuning +of the optical properties of magnetic 2D materials for the +development of new magnetic 2D-based devices, such as +unidirectional miniaturized photonic components. +References +1 +K. S. Novoselov et al. Two-Dimensional Gas of Massless Dirac +Fermions in Graphene. 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Science 360 , 1214 (2018). +Acknowledgments +The project that gave rise to these results received +the financial support of a fellowship from “la Caixa” +Foundation (ID 100010434, fellowship codes LCF/BQ/ +PR21/11840011 and LCF/BQ/DI22/11940022) and the grant +PID2020-118938GA-100 from the Spanish Ministerio de +Ciencia e Innovación (MICINN). ENM acknowledges the +European Research Council (ERC) under the Horizon +2020 research and innovation program (ERC StG, grant +agreement No. 803092) and to the MICINN for financial +support from the Ramon y Cajal program (Grant No. +RYC2018-024736-I). FCP also acknowledges the MICINN +for the FPU program (Grant No. FPU17/01587). This work +was also supported by the Spanish Unidad de Excelencia +“María de Maeztu” (CEX2019-000919-M) and is part of the +Advanced Materials programme supported by MICIN +with funding from European Union NextGenerationEU +(PRTR-C17.I1) and by Generalitat Valenciana. + +6 +Methods +Crystal growth and isolation of few-layer crystals: High quality CrI3 crystals were grown by chemical vapor transport and thoroughly +characterized by X-ray diffraction (XRD), Energy-dispersive X-ray spectroscopy (EDAX) and Raman spectroscopy to verify their +pristine quality (see Supporting Information SI1 for details). Atomically thin layers were obtained by mechanical exfoliation of +bulk crystals using a scotch tape technique and deposited over a transparent glass slide to perform transmittance measurements. +Selected flakes were first selected by optical microscopy screening and successively characterized by atomic force microscopy to +accurately determine their thickness. The whole process and characterization were carried out inside an Ar-filled glove box where +the O2 and H2O levels were maintained below 0.5 ppm to avoid material degradation. +Optical transmission experiment: Once the thicknesses of all flakes were determined, the sample was encapsulated by gluing +on the glass slide substrate a coverslip with a thermoplastic material and exposed to the air avoiding degradation. Hyperspectral +transmittance images were acquired under monochromatic light ranging from 430 nm to 720 nm in an upright transmittance +microscope equipped with a high-resolution CMOS monochromatic camera (Hamamatsu Orca Fusion) and a homemade grating +monochromator for the incidence light, for accurate quantitative transmittance measurements. +Extraction of the optical dielectric function: The transmittance spectra have been fitted simultaneously by employing a multilayer +model whose complex coefficients of reflectance and transmittance are described by Fresnel equations. Their complex parameters +(refractive index and extinction coefficient) have been calculated from the optical dielectric function (Equation 1) assuming a +relative permeability of µr=1 (at room temperature). To fit the model simultaneously with all the experimental data (Figure 1), we +have considered a spline for modelling the layer dependence of the parameters of Equation 1. In particular, a faithful data fit could +be reached employing two or three cubic polynomials (C1 continuity) excluding or including the monolayer, respectively. +Density functional theory: The electronic structure calculations and geometry relaxations have been carried out using Quantum- +Espresso ab initio package. Each structure has been relaxed using the Grimme-D3 van der Waals correction until all the forces acting +on atoms were lower than 10-3 Ry/bohr. For the relaxation, we have used projector augmented waves (PAW) pseudopotentials within +the generalized gradient approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation potential, with a 4x4x1 +k-point grid. The dielectric functions were obtained using the epsilon.x code, which is part of the Quantum-Espresso package. The +calculations were carried out at the independent particle (IP) level and non-local contributions from pseudopotentials are neglected. +For the optical properties, the electronic structure calculations are carried out using the DFT+U+J scheme with U = 4.1 and J = 0.6 eV +norm-conserving GGA-PBE pseudopotentials from the PseudoDojo library, with energy cutoffs 80 and 320 Ry for the wavefunctions +and charge density, respectively. In our case, the dielectric functions converged for a 8x8x1 k-point grid. + +7 +Supporting information +Monolayer-to-mesoscale modulation of the +optical properties in 2D CrI3 mapped by +hyperspectral microscopy +Marta Galbiati1*†, Fernando Ramiro-Manzano1,2*†, José Joaquín Pérez Grau1, Fernando Cantos- +Prieto1, Jaume Meseguer-Sanchez1, Ivona Kosic1, Filippo Mione1, Ana Pallarés Vilar1, Andrés +Cantarero1, David Soriano3,4, and Efrén Navarro-Moratalla1* +1 Instituto de Ciencia Molecular, Universitat de València, Calle Catedrático José Beltrán Martínez 2, 46980, Paterna, Spain. +2 Instituto de Tecnología Química, Universitat Politècnica de València - Consejo Superior de Investigaciones Científicas (UPV-CSIC), Avd. de los +Naranjos s/n, 46022, Valencia, Spain. +3 Information Engineering Department, University of Pisa, Via Caruso 16, 56122, Pisa, Italy. +4 Departamento de Física Aplicada, Universidad de Alicante, 03690, San Vicente del Raspeig, Alicante, Spain. +† M.G. and F. R.-M. contributed equally to this work. * e-mail:ferraman@fis.upv.es, marta.galbiati@uv.es, efren.navarro@uv.es +Contents +1. +Materials characterization and sample preparation +8 +2. +Wide-field hyperspectral setup for air-sensitive materials +11 +3. +Optical model +13 +4. +Spatially resolved optical properties +16 +5. +Dependence of the complex dielectric function with layer number +and interlayer distance +17 + +8 +1. Materials characterization and sample preparation +1.1. Bulk CrI3 crystal characterization +Powder X-ray diffraction (PXRD): +The crystal structure of CrI3 was characterized by powder X-ray diffraction (PXRD). Figure S1 shows the +XRD of the of CrI3 single crystals taken employing PANalytical Empyrean X-ray diffractometer. X-Rays +diffraction analysis was performed on sample of single crystalline CrI3 by loading the material into a +capillary and sealing it inside the glove box. The powder pattern of sample is consistent with the monoclinic +AlCl3-type structure (C2/m) reported for CrI3. In the XRD spectrum, no evidence of any other phases was +detected indicating that the product is of high purity. +Energy Dispersive Analysis of X-ray (EDAX): +The chemical composition of as-grown CrI3 single crystals was determined by Energy Dispersive Analysis +of X-ray (EDAX) technique using Hitachi S4800 for confirming stoichiometry. The obtained spectrum is +shown in Figure S2. The atomic and weight % obtained using EDAX is tabulated in Table S1. The data +clearly states that the as-grown CVT single crystals have chemical composition of CrI3 and they are free +from any impurity. +Figure S1 | XRD pattern and unit cell refinement of a representative CrI3 single crystal. The continuous black line shows +the experimental pattern, the blue line shows the Rietveld refined total intensity and the red trace depicts the difference. +The green marks indicate the refined reflections. A selection of the most prominent reflections are labeled using their +corresponding Miller indices. +Figure S2 | EDAX spectrum of the as-grown CrI3 single crystals. + +10 +kev9 +Raman spectroscopy: +CrI3 bulk flakes have been characterized by Raman spectroscopy after their mechanical exfoliation +to verify their pristine quality after encapsulation with the glass cover (see section 1.2 below). Raman +characterization has been performed using a Horiba LabRAM HR Evolution Raman spectrometer under a +532 nm excitation laser. Results in Figure S3 show peaks at frequencies of 78, 100−110, 128, and 234 cm-1 in +agreement to that previously reported. +1.2. Preparation of samples for transmission experiment +For our transparent substrates we use standard microscope glass slides thoroughly rinsed to obtain very +clean and low roughness surfaces. Glass slides were soaked in a freshly prepared solution of NH4OH : H2O2 +(1 : 1) and sonicated for 8 min. Next, they were rinsed with Milli-Q water, sonicated 5 min in Milli-Q water +and dried under a nitrogen stream. Clean slides were introduced inside the glove box and heated above +100ºC for at least 30 min before deposition of the 2D material, in order to remove any possible H2O traces +from the substrate surface. +Atomically thin CrI3 layers were obtained by mechanical exfoliation of bulk CrI3 crystal, grown by +chemical vapor transport, using a scotch tape technique and deposited over the glass slide. An optical +microscope was then used to screen the sample and select regions of interest with abundant flakes. +Selected layers were characterized by atomic force microscopy to determine their thickness. Remarkably, +the surface of the glass slides was observed to be perfectly clean with a low roughness (RMS ~ 0.23 nm) +which allows accurate flakes thickness measurement down to the monolayer. +The whole exfoliation and characterization processes were carried out inside an Ar-filled glove box where +O2 and H2O levels were maintained below 0.5 ppm to avoid material degradation. Once flakes characterized, +sample was sealed with a thin glass cover slide (100 µm thick) to prevent CrI3 layers degradation when +exposed to the air. The samples were finally extracted from the glove box to proceed with the wide field +hyperspectral experiment. +Table S1 | EDAX spectrum of the as-grown CrI3 single crystals. +Figure S3 | Raman spectrum of bulk CrI3 flake. The flake under observation was well within the bulk thickness regime +described in the main text. +Element +Weight % +Atomic % +Cr +11.73 +24.49 +I +88.27 +75.51 + +7000 +6000 +Intensity (arb. units) +5000 +4000 +3000 +2000 +1000 +0 +50 +100 +150 +200 +250 +300 +Raman shift (cm-1)10 +2. Wide-field hyperspectral setup for air-sensitive materials +A wide field hyperspectral setup has been implemented by modifying a standard optical transmittance +microscope as schematically shown in Figure S4. White light coming from a high intensity LED phosphor +light source enters a homemade monochromator, which allows selecting the excitation wavelength with +a bandwidth of ~1 nm. Monochromatic light is then connected to the optical microscope through an +optical fiber and is used to irradiate the transparent samples in transmittance mode. For the hyperspectral +analysis, transmission images of a region of interest of the sample are acquired sweeping the wavelength +from 430 nm (2.9 eV) to 720 nm (1.7 eV) with a step of 1 nm. To achieve this, the optical microscope is +equipped with a monochrome CMOS camera (Hamamatsu Orca Fusion), conveniently chosen for its low +noise and high linear range, which allows precise quantitative analysis of the images even for thin layers +and in the full visible range. Data acquisition is performed in complete dark conditions, in order to avoid +any source of external light, which could result in reduced data reproducibility. With an exposure time of +about 300 ms per image, a complete hyperspectral image takes less than 10 minutes. +Figure S5 shows transmission images of CrI3 atomically thin crystals down to the monolayer under +different wavelengths, with a contrast that ranges from almost transparent to very dark for increasing +thickness. Figure S5 also demonstrates the high throughput of this imaging technique, which permits +White light +source +CMOS monochrome +camera +Monochromator +Figure S4 | Schematic of the wide-field hyperspectral microscope setup. Solid straight lines depict selected light ray traces. +The drawing is not to scale. +Figure S5 | Few-layer CrI3 crystals deposited on standard glass slides. (a-d) Optical transmission micrographs of a +representative sample, showing different layer number plateaus illuminated with white, 630 nm, 532 nm and 460 nm light +respectively. Scale bars in the optical micrographs are 10 µm long. (e) Atomic force microscopy topographic image of single +layer CrI3 located inside the dashed box drawn in panel (d), with the corresponding probability density distribution shown +in panel (f), which allows to estimate a flake thickness of 1.15 ± 0.001 nm, compatible with a monolayer within experimental +uncertainty. Probability density has been calculated in the region marked with a dotted line in panel (e). + +2 μm11 +exploring the layer-dependent optical properties in the visible range from the single layer to the hundreds- +of-layers range with ample statistics coming from just a single frame of hyperspectral images. To accurately +determine the number of layers of each flake we employed atomic force microscopy (AFM). As shown in +Figure S5e,f, the topographic image of a CrI3 single layer on a glass slide depicts a thickness of ~1.1 nm +which is in agreement, within the experimental uncertainty, with the interlayer space extracted from the +lattice parameter of the bulk structure [1], and demonstrates the atomically-flat quality of the micrometer- +size samples obtained down to the single layer limit. It is also worth noticing that the root-mean-square +(RMS) roughness measured over the commercial glass slide is only ~ 0.23 nm, comparable to that obtained +on standard Si/SiO2 wafers, hence making commercial glass slides a convenient transparent substrate +even for the investigation of the thinnest layers. +The collected hyperspectral data are arranged in a 3D matrix where each image pixel (x and y spatial +coordinates) is associated to all the swept wavelengths. To reduce signal to noise ratio in data analysis +we select a region with homogeneous thickness on the analyzed flake (blue mask) and a second +neighboring region on the substrate (green mask). Typical flake/substrate regions are displayed in Figure +S6a. A dark background frame is then subtracted and the pixels intensities are averaged for each region +(Iflake and Isubstrate). Quantitative transmittance is calculated as T(E) = Iflake(E)/ Isubstrate(E). Figure S6c shows +examples of typical transmittance spectra for different CrI3 flake thicknesses. Good reproducibility of the +transmittance spectra for different flakes with equal thickness was verified in order to ensure a reliable +layer-dependent analysis. Reflection data showed to be strongly affected by the choice of regions and by +spurious reflections, not complying with our data reproducibility requirement and resulting in a high +dispersion of spectral traces for different flakes with equal thickness. Reflection data was consequently +excluded from our analysis. +Figure S6 | Transmission data processing and representative results. (a) Optical transmission image of CrI3 flakes +deposited over a glass slide. In order to analyze transmittance spectra a zone of the flake with the same thickness is +selected over the image (area labeled with an f), and a zone over the substrate just nearby the flake (area labeled with an +s). Transmission intensities are then averaged over the selected zones (If and Is) for each image acquired in a wavelength +range of 430 - 720 nm. Finally, transmittance is calculated as T = If / Is for each swept wavelength (energy). (b) Schematic +of the simplified system where monochromatic light is coming from the back of the sample in transmittance mode and is +transmitted through the glass slide and the CrI3 layers before being collected by the camera. (c) Examples of transmittance +spectra collected for different CrI3 layers thickness. To ensure data reliability good reproducibility of transmittance spectra +over different flakes with the same thickness has been verified as it can be observed in this graph. +a +b +c +Energy (eV) +Transmittance +a +b +c +2 L +3 L +4 L +5 L + +a) +Crl3: n2 +Glass slide: n10.95 +0.9 +0.85 +0.8 +0.75 +1.8 +2 +2.2 +2.4 +2.6 +2.812 +3. Optical model +In previous works, the reflectance of 2D materials deposited on solid substrates has been modeled using the +Fresnel equations [1–4]. In this regard, the real (ε1) and imaginary (ε2) part of the optical dielectric function +of individual flakes of the materials are derived from a single reflectance experiment using a Kramers- +Kronig (KK)-constrained analysis [5]. This approach has been used, for instance, to find the dielectric +function of TMDC monolayers deposited on fused silica substrates modeling ε(ω) as a superposition of +Lorentzian oscillators, whose complex equations fulfill the KK relations.[3] Similarly, KK relations have +been used to calculate the phase of the amplitude reflection coefficient θ to extract the refractive index +and extinction coefficient of bulk CrI3 in the visible range [1], or in the terahertz to near-infrared region +[6]. However, this approach has the important limitation that, for an accurate determination of ε(ω), KK +relations require a wide energy range (ideally from 0 to ∞ while optical contrast experiments are usually +limited to the small range of visible light. To address this issue, a possible solution has been to make use +of literature data to extrapolate high and low energy limits. While we initially applied this same approach, +we realized that for our spectral range, the extracted phase (θ) showed a remarkable dependence with the +reference data. As a result, it was difficult to reach consistent solutions over different spectra. +An alternative technique to determine the optical dielectric function is to acquire both reflectance and +transmittance spectra for solving a system of two independent equations. In this way, both ε1 and ε2 can +be unequivocally obtained. At this aim, we attempted to simultaneously analyze the reflectance spectra of +CrI3 over two different substrates: SiO2/Si and a glass slide, successively trying to strengthen our analysis by +additionally relating them to transmittance measurements performed on the same CrI3/glass slide sample. +However, this extended analysis led us to the following conclusions: firstly, reflection spectra introduce +a high experimental error, which hampered finding a solution for the computed ε(ω) of the 2D material. +While SiO2/Si substrate interference strongly masked the small absorption peaks of CrI3 layers, external +reflections due to optical elements, very difficult to get rid of, were observed on the transparent glass +substrate, resulting in a poor reproducibility of reflectance data and subsequent inconsistent analysis. +Secondly, we verified that ε2 dominates both the transmittance (see Figure S8) and reflectance spectra [3], +leading to ambiguity of the solution of the equation due to the high experimental uncertainty. +We thus focused on transmission measurements as a source of high data reliability. To limit the +interdependence between ε1 and ε2 from a single experiment we simultaneously analyzed transmittance +spectra acquired over flakes with different thicknesses, including a large number of samples with ample +statistics (thousands of pixels each) in every dataset. This approach allows to further constrain the range +of possible solutions of CrI3 ε(ω), hence increasing the consistency of our analysis. +For the transmittance analysis we describe our system as a simple Fabry-Perot cavity composed of three +layers as shown in Figure S6b: (1) glass slide / (2) CrI3 / (3) air. Other contributions such as the presence +of a glass cover or the rest of optical elements cancel each other in the transmittance model. The energy +dependent complex refractive index for each layer is written as: +n1(ω) = nglass(ω) - ikglass(ω); n2(ω) = n2D(ω) - ik2D(ω); n3(ω) = 1. +To calculate the complex refractive index of CrI3 (n2(ω)), we assume CrI3 dielectric function to follow a +modified Lorentzian oscillatory model: +composed by two oscillators (n = 1, 2), where ωn, ωp, ω represent the resonant, plasma and scan frequencies +respectively, ε∞ is assumed to be 1, and γn and Γn correspond to damping related parameters. We then +extract n2 from n2(ω) ≈ √ε(ω). +Figure S7 shows examples of transmittance spectra (black lines) and fit (red lines) performed with our +model on CrI3 samples with different thickness. A very good agreement between experimental data and +theoretical fit can be observed. +Figure S8 shows maps of solutions of the model around the energy values of the transmittance dips +(1.95 and 2.66 eV). These maps were generated by plotting the difference, in absolute value, between +the experimental transmittance signal and different theoretical solutions. These model solutions are +simulated by scanning different combinations of ε1 and ε2. The zeros of this subtraction (black/dark +region) correspond to possible solutions. Given that these values form quasi horizontal lines, the solution +is dominated by a small and limited range of ε2 values in contrast to a larger dispersion of ε1. Indeed, +~ +~ +~ +~ +~ +~ +~ +~ +˜ε(ω) = ˜ε∞ + +2 +∑ +n=1 +ω2 +p + 2iωΓn +ω2n − ω2 + 2iωγn + +13 +a certain deviation of ε1 results in a slight perturbation of the highly reliable ε2 value. In any case, the +solutions of the maps show a small curvature due to the multilayer nature of the system. Consequently, +this shape changes with the sample thickness. Importantly, in our experiments the layer complexity was +streamlined to the bare minimum, i.e. consisting of a single layer substrate holding the sample, and as a +result, the Figure S8 shows simple and almost flat solution curves. The white marker in each map indicates +the solution captured by the fitting procedure. Note that the fitting procedure not only considers a single +spectral energy (as shown in Figure S8), but it considers the full spectroscopic experimental data range +(Figure S7) for obtaining the solution Note that our final fitting procedure not only considers a single +spectral energy (as shown in Figure S8), but it considers the full spectroscopic experimental data range +(Figure S7) for obtaining a complete solution as a function of the layer number. +To further increase consistency of our data analysis, we perform a simultaneous fit using the previously +described model with > 40 transmittance spectra of CrI3 layers with thickness ranging from 1 L to 164 L. +Although each spectrum represents a different scenario, they are interconnected by shared parameters +with a defined tendency. We rely on the fits performed individually on each spectrum for the different +layer thicknesses in order to gain a first idea on the fitting parameters evolution with the number of layers +(Figure S9). Then, we fit parameters tendency with respect to layer thicknesses with a spline consisting +in 3 cubic polynomials defining two C1 intersection points (or knots) at about 1L and 46 L. Our model +allows to fits accurately to all transmittance spectra while interconnecting fitting parameters allows to +highly constrain the number of possible solutions, hence increasing the accuracy of our analysis (Figure +2). It is important to point out that it was not possible to attain a good fit of all transmittance spectra +without defining the spline intersection points (at about 1L and 46L). This, again, supports the idea of +three thickness regimes with different optical properties: the so-called thin, multilayer and bulk regions +as discussed in the main text. +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.95 +0.96 +0.97 +0.98 +0.99 +Transmission +1L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +Transmission +31L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.8 +0.85 +0.9 +0.95 +Transmission +4L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.2 +0.3 +0.4 +Transmission +78L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +Transmission +6L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.2 +0.3 +0.4 +Transmission +131L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.5 +0.6 +0.7 +0.8 +Transmission +14L +1.8 +2 +2.2 2.4 2.6 2.8 +Energy (eV) +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Transmission +164L +Transmittance +Transmittance +Transmittance +Transmittance +Transmittance +Transmittance +Transmittance +Transmittance +Figure S7 | Examples of transmittance spectra experimentally measured over CrI3 layers of different thickness (black +curves) and their model fit (red curves). The results plotted were obtained by using the oscillatory model for different +number of layers. + +14 +1L +4L +6L +14L +31L +71L +131L +164L +1L +4L +6L +14L +31L +71L +131L +164L +ENERGY 2.66 eV +ENERGY 1.95 eV +Figure S8 | Map of solutions of the model computed at 1.95 eV and 2.66 eV. Black regions correspond to the possible +solutions (zeros) as a function of ε1 and ε2. White dots represent individual solutions extracted from the fitting process. + +4 +2 +0 +05 +105 +105 +1010 +8 +6 +20.2 +0.4 +0.64 +2 +0 +05 +1010 +8 +6 +20.2 0.4 0.6 0.84 +2 +0 +05 +1010 +8 +6 +210 +8 +6 +20.2 0.4 0.6 0.84 +2 +0 +05 +1010 +8 +6 +20.02 +0.06 +0.10.2 0.4 0.6 0.84 +2 +0 +05 +1010 +8 +6 +20.010.020.034 +2 +0 +05 +1010 +8 +6 +20.05 +0.1 +0.154 +2 +0 +05 +1010 +8 +6 +210 +8 +6 +20.05 +0.1 +0.154 +2 +0 +05 +1010 +8 +6 +20.01 +0.020.1 +0.24 +2 +0 +05 +1010 +8 +6 +20.1 +0.2 +0.34 +2 +0 +05 +1010 +8 +6 +20.2 +0.44 +2 +0 +05 +1010 +8 +6 +20.2 +0.44 +2 +0 +05 +1010 +8 +6 +20.1 +0.3 +0.54 +2 +0 +05 +1010 +8 +6 +20.2 +0.4 +0.64 +2 +0 +04 +2 +0 +05 +1010 +8 +6 +20.2 0.4 0.6 0.84 +2 +0 +015 +Figure S9 | Map of the complex optical dielectric function of CrI3 composed by independently fitting the transmittance +spectra of samples with different thicknesses. The calculated real (ε1) and imaginary (ε2) parts of ε(ω) are shown in panels +(a) and (b), respectively. The computed ε1 (ℏω) (c) and ε2 (ℏω) (d) have been also plotted as a function of the excitation energy +ℏω for different CrI3 crystals of different thicknesses. The red traces correspond to thin layers (from 1L to 13L), the blue traces +to layers ranging from 14L to ~100L and the green traces to bulk (up to 164L). This figure displays independent fits of specific +thickness spectra in comparison to Figure 2 that shows a simultaneous fit to the complete layer-dependent dataset. A good +agreement between the two figures can be observed. +~ + +0 +5 +10 +b +2 +4 +6 +a +Dielectric function - ε, +Dielectric function - , +160 +160 +140 +140 +120 +120 +100 +100 +(L) +Number of layers (L) +80 +80 +Number of layers +60 +40 +20 +20 +８7654321 +８7６54３２1 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +Energy (eV) +Energy (eV) +101 +102 +101 +102 +d +c +Number of layers (L) +Number of layers (L) +8 +12 +1 L +10 +6 +8 +13 +1 L +4 +6 +4 +2 +2 +0 +0 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +Energy (eV) +Energy (eV)16 +4. Spatially resolved optical properties +Hyperspectral analysis allows to spatially resolve the optical properties of a 2D material. In Figure S10a +we show the optical transmission image of different CrI3 flakes with their respective thickness measured +by AFM, while Figure S10b displays the spatially resolved intensity of the high energy transmittance dip +of CrI3 (at 2.66 eV). As for the peak position map of Figure 3 of the manuscript, this has been obtained +by calculating pixel by pixel the corresponding transmittance spectrum and fitting its high energy dip +position (Figure 3c) and transmittance intensity (Figure S10b), then ascribing these values to each pixel. +Background pixels were automatically set to the minimum value. The color scale can also be compared to +the values extracted from the analyzed transmittance spectra as a function of the number of layers (Figure +S10c) showing a good agreement. +The spatial resolution of the optical properties of a 2D material results to be a very intuitive and powerful +method to gain a fast insight on the properties of many different sample thicknesses at a same time. By +tracking the peak transmittance intensity, as shown in Figure S10b, it is possible to gain a large contrast +on the thinnest layers. Once transmittance intensity has been associated to the respective layers thickness +characterized by AFM, this becomes a very powerful tool to quickly and reliably gain information on +all flakes thickness of a whole image with a high spatial resolution. This process could be used for the +automatic analysis of flakes thicknesses with clear advantages in terms of time saving (data images +acquisition takes about 10 minutes for a whole zone) and non-invasive way, very relevant especially for air +sensitive 2D materials. +Figure S10 | Spatially resolved layer dependence of the optical properties of CrI3. (a) Optical transmission image of CrI3 +flakes deposited on a glass slide. The number of layers of each section of the sample has been characterized by AFM and +labeled on each flake in the image. (b) Spatially resolved map of zone shown in (a) displaying the transmittance value at the to +display intensity at the high energy transmittance dip for each flake. This has been obtained by extracting the transmittance +spectrum at each pixel, fitting the transmittance in the spectral vicinity of the high energy dip and obtaining the energy at the +local minimum, and ascribing this value to each pixel. Background pixels where automatically set to 0. (c) Evolution with the +number of layers of the minimum transmittance intensity at the high energy dip, obtained by relating analyzed transmission +images to results obtained by AFM. Dots are experimental data while dotted line is guide to the eye. Once transmittance +intensity dependence with layer thickness is calibrated, this technique allows to optically analyze thin layers thickness with a +high throughput. This method is particularly convenient as it is fast and non-invasive. +b +c +1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +50 +100 +150 +High energy dip transmittance intensity +Number of layers (L) + +67L +70L +42L +14L +53L +30L +3L +8L +70L +44L +2L +20L +13L +20L +aioy +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +017 +5. Dependence of the complex dielectric function with layer number +and interlayer distance +We carried out first-principles calculations on few-layer CrI3, up to 10 L using the DFT+U+J scheme with U += 4.1 and J = 0.6 eV, as previously established for this material [7], based on the constrained random-phase +approximation (cRPA) [8]. For all cases, we consider an ideal monoclinic stacking between adjacent layers +and antiferromagnetic coupling. +Figure S11 shows the theoretically computed evolution of the real and imaginary parts of the dielectric +function of CrI3 with increasing number of layers for the full energy range calculated from 0 eV to 4 eV. +In Figure S12 we show the evolution of the intensity of the e2 peak at 1.5 eV for bilayer CrI3 with the inter- +layer distance. The other peaks reported a similar trend. The increase in the inter-layer distance decreases +the polarizability of the wave-functions in the direction perpendicular to the layers, also decreasing the +magnitude of the complex dielectric function. These results clearly indicate that small changes in the +crystal structure and stacking could eventually modify the intensity of the complex dielectric function. +Considering that in layered bulk crystals the interlayer distance increases due to stronger van der Waals +interactions and modifications in the crystal structure, our theoretical results give hints on the crossover +and the different thickness regimes observed experimentally. +a +b +a +b +Figure S11 | Evolution of the real and imaginary parts of the dielectric function of CrI3 with increasing number of layers. +This calculation was performed in the z-direction perpendicular to the layers. +Figure S12 | Dependence of the intensity of the 1.5 eV peak in the imaginary dielectric function with the inter-layer +distance. +Table S2 | Evolution of the Cr-Cr interlayer distance with the number of layers based on first principles calculations. +Number of layers +2 +4 +6 +8 +10 +Bulk +Average Cr-Cr interlayer distance (Å) +6.654 +6.640 +6.644 +6.649 +6.681 +6.845 +a +a +a +b + +18 +References +[1] +B. Huang et al., Layer-Dependent Ferromagnetism in a van Der Waals Crystal down to the Monolayer +Limit. Nature 546, 270 (2017). +[2] +P. Blake et al. Making Graphene Visible. Appl. Phys. Lett. 91, 063124 (2007). +[3] +Y. Li et al. Measurement of the Optical Dielectric Function of Monolayer Transition-Metal +Dichalcogenides: MoS2, MoSe2, WS2, and WSe2. Phys. Rev. B CCondens. Mater. Phys. 90, 205422 (2014). +[4] +A. Castellanos-Gomez et al. Fast and Reliable Identification of Atomically Thin Layers of TaSe2 +Crystals. Nano Res. 6, 191 (2013). +[5] +H. Ibach and H. Lüth, Solid-State Physics, 3rd ed. (Springer, Berlin, Germany, 2003). +[6] +L. Tomarchio et al. Low Energy Electrodynamics of CrI3 Layered Ferromagnet. Sci. Rep. 11, 23405 +(2021). +[7] +D. Soriano et al. Environmental Screening and Ligand-Field Effects to Magnetism in CrI3 Monolayer. +Npj Comput. Mater. 7, (2021). +[8] +F. Aryasetiawan et al. Frequency-Dependent Local Interactions and Low-Energy Effective Models +from Electronic Structure Calculations. Phys. Rev. B Condens. Mater. Phys. 70, (2004). + diff --git a/79AzT4oBgHgl3EQfE_pK/content/tmp_files/load_file.txt b/79AzT4oBgHgl3EQfE_pK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..793f32afb7f340eb2656cde9aabd960b24c886f3 --- /dev/null +++ b/79AzT4oBgHgl3EQfE_pK/content/tmp_files/load_file.txt @@ -0,0 +1,830 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf,len=829 +page_content='1 Monolayer-to-mesoscale modulation of the optical properties in 2D CrI3 mapped by hyperspectral microscopy Marta Galbiati1*†, Fernando Ramiro-Manzano1,2*†, José Joaquín Pérez Grau1, Fernando Cantos- Prieto1, Jaume Meseguer-Sanchez1, Ivona Kosic1, Filippo Mione1, Ana Pallarés Vilar1, Andrés Cantarero1, David Soriano3,4, and Efrén Navarro-Moratalla1* 1 Instituto de Ciencia Molecular, Universitat de València, Calle Catedrático José Beltrán Martínez 2, 46980, Paterna, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 2 Instituto de Tecnología Química, Universitat Politècnica de València - Consejo Superior de Investigaciones Científicas (UPV-CSIC), Avd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' de los Naranjos s/n, 46022, Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 3 Information Engineering Department, University of Pisa, Via Caruso 16, 56122, Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 4 Departamento de Física Aplicada, Universidad de Alicante, 03690, San Vicente del Raspeig, Alicante, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' † M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' * e-mail:ferraman@fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='upv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es, marta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='galbiati@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es, efren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='navarro@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es Magnetic 2D materials hold promise to change the miniaturization paradigm of unidirectional photonic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, the integration of these materials in devices hinges on the accurate determination of the optical properties down to the monolayer limit, which is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' By using hyperspectral wide-field imaging we reveal a non-monotonic thickness dependence of the complex optical dielectric function in the archetypal magnetic 2D material CrI3 extending across different length scales: onsetting at the mesoscale, peaking at the nanoscale and decreasing again down to the single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These results portray a modification of the electronic properties of the material and align with the layer-dependent magnetism in CrI3, shedding light into the long-standing structural conundrum in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The unique modulation of the complex dielectric function from the monolayer up to more than 100 layers will be instrumental for understanding and manipulating the magneto-optical effects of magnetic 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The dependence of the physical properties of van der Waals materials with the number of layers has been the flagship of two-dimensional (2D) materials since their discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In general, this layer dependence gains importance upon approaching the single layer limit, where the strict confinement of electrons in a 2D lattice imposes dramatic changes in the electronic structure of the crystal, enabling the realization of new electronic states and quantum correlated phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This has opened the door for the realization of massless Dirac fermions in graphene and superconducting twisted bilayer graphene1,2, direct-gap photoluminescence3,4 and valley polarization in single-layer transition metal dichalcogenides5,6, or Ising-like superconductivity in few-layer metallic transition in metal dichalcogenides7, to name just a few prominent examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Beyond the single layer limit, the properties of van der Waals materials change gradually until reaching the bulk properties generally within the nanoscale thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A much more unusual case features a continuous change of the material properties at much larger thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' One of the few examples is the effect of layer number on the photoluminescence of hexagonal boron nitride8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, this phenomenon arises from a variation of the bandgap and activation energies of impurities in the system and is not a consequence of the layer-dependence of the intrinsic electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Another remarkable case is chromium triiodide (CrI3), one of the first layered materials to exhibit a non-zero net magnetization down to the monolayer9, accompanied by a transition from layered antiferromagnetic in the few-layer regime to ferromagnetic in the bulk, with a crossover thickness to the bulk that is still unclear but that is unambiguously located in the mesoscale10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Seminal studies attribute this change to differences in the stacking between the layers13 being the ferromagnetic and layered antiferromagnetic states related to the rhombohedral and monoclinic stacking, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This points at a profound layer- dependent electronic effect at the mesoscale, setting an imperious need to study the evolution of the electronic properties as a function of the layer count in order to understand the underlying mechanisms giving rise to the magnetism in CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' An insightful and non-destructive way to study the layer-dependent evolution of the electronic properties of a layered material is to determine the complex dielectric function via light-matter interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Although optical ellipsometry is usually employed to extract these parameters from thin films, the small sample footprints of most mechanically exfoliated 2D materials hinder the direct application of this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' One strategy to overcome the spatial resolution limitation is to use an optical microscope equipped with a broad-band white light source coupled to a spectrometer, where a continuous adjustment of the illumination wavelength in constrained areas of the sample allows for an accurate extraction of the dielectric function in the visible range14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, the sampling speed and data statistics of point spectroscopy are generally low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Wide-field hyperspectral microscopy on the other hand has been successfully employed for the high-throughput layer-dependent characterization of bare 2 2D materials and heterostructures in air15-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Unfortunately, the magnetic 2D materials are very sensitive to the presence of humidity and oxygen and therefore require an appropriate isolation from ambient conditions and rapid characterization to preclude the effect of degradation, a combination of requirements that until now has not been met by the established techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' By developing a simple encapsulation technique compatible with wide-field hyperspectral imaging, we herein circumvent the low- throughput of deterministic encapsulation and achieve high sampling frequency and data reliability in the characterization of the layer-dependent optical properties of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The use of a monochrome camera with a high linear dynamic range permits a single-shot acquisition of the light intensity coming from tens of bare CrI3 crystals with different layer number, providing simultaneous access to large statistics while minimizing the acquisition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' By modeling the spectral information obtained as a function of the layer number, we identify at least two crossover points of the complex optical dielectric function of CrI3 in the nanoscale and the mesoscale thickness regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The crossover in the mesoscale portrays a sizeable modification of the electronic properties of the material, which could underpin the structural preference for the monoclinic phase and provide an explanation for the layer-dependence of the magnetic properties of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Our theoretical results, based on first-principles calculations, provide support to the electronic origin of the evolution of the dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' CrI3 flakes from the single layer to the hundreds-of- layers thickness range were obtained by mechanical exfoliation of bulk crystals on standard microscopy glass slides and encapsulated with a coverslip sealed using a thermoplastic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Spatially resolved hyperspectral maps of different regions of the sample were then acquired by combining a sequence of optical images recorded under different monochromatic illumination spanning the full visible range (more details in the Supplementary Information SI2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' By analyzing the hyperspectral optical transmission data in selected regions of the CrI3 crystals of homogeneous thickness, examined by atomic force microscopy (AFM) for their correct estimation (see SI2 for details), we obtained spectral traces of the material for different layer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure 1 shows the transmittance spectra of flakes with a thickness ranging from 1 layer (L) to 164 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Two absorption features are visible at about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 eV, corresponding to the ligand-to-metal charge transfer absorption peaks reported for bulk CrI3 and more recently for CrI3 exfoliated samples19-21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Remarkably, both transmittance dips present a non-monotonic trend with the number of layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' a red shift when flake thickness increases from 1 L to ~ 13 L and a blue shift above ~ 50 L, thus suggesting a change in CrI3 optical properties with flake thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' It is also interesting to point out that these features remain almost unchanged for crystals thicker than ~ 100 L, hinting to a smooth saturation towards bulk values for layers thickness located at the mesoscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' After considering different approaches for the evaluation of the complex optical dielectric function (ε(ω) = ε1 - iε2) from optical data (see SI3), we chose to focus on transmission measurements and large statistics analysis to achieve high data reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' We simultaneously analysed transmittance spectra acquired over flakes with different thicknesses, including a large number of samples with ample statistics (thousands of pixels each) in every dataset, in order to limit the interdependence between ε1 and ε2 from a single experiment and further constrain the range of possible solutions of CrI3 ε(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To extract the complex dielectric function from the transmittance data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' we consider our experimental system as a basic Fabry-Pérot cavity formed by a stack of 3 layers and we assume that the dielectric function of CrI3 follows a modified Lorentzian oscillator model22: This model is composed by two oscillators (n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' where ε∞ is the permitivity for infinite optical frequencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' ωn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' ωp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' and ω represent the resonant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' plasma and photon frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' and γn and Γn are damping related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure 2 shows the real and imaginary parts of ε(ω) and their evolution with flake thickness calculated by simultaneous fitting of all the transmittance spectra shown in Figure 1 (see SI3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The dielectric function of CrI3 monolayer is found to be significantly different from the rest of the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This is in line with the transmittance spectra experimentally measured in the Figure 1 | Visible range transmittance spectra of CrI3 crystals with different layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The data displayed was extracted from selected areas of a wide-field hyperspectral image that are found to be atomically flat by AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Red curves correspond to thin layers (from 1L to 13L), blue curves to layers ranging from 14L to ~100L and green curves correspond to bulk (up to 164L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' ~ ~ ~ ~ Number of layers (L) 1 13 100 164 ˜ε(ω) = ˜ε∞ + 2 ∑ n=1 ω2 p + 2iωΓn ω2n − ω2 + 2iωγn 3 monolayer, where absorption features are significantly shifted compared to the bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Besides the discontinuity found for the monolayer, we find two additional critical points at about 13 L (~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 nm) and 100 L (~68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 nm), where ε2 intensity increases and peaks red shift (< 13 L), then decreases and peaks blue shift (> 13L), until the dielectric function starts to asymptotically saturate toward the bulk value above ~100 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This delimits three different thickness domains: (i) few layer, (ii) multilayer and (iii) bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These changes are also clearly illustrated in Figure 3 which shows the evolution with layer thickness of ε1 and ε2 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV extracted from the fitting process (Figure 3a), and the evolution of the experimental absorption feature minima of the transmittance spectra (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Hyperspectral analysis also gives the possibility to spatially resolve the optical properties of the 2D material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' As such, Figure 3c Figure 2 | Evolution of the dielectric functions in CrI3 as a function of the layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The calculated real (ε1) and imaginary (ε2) parts of are shown in panels (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (c) and (d) have been also plotted as a function of the excitation energy for crystals of different thicknesses (these are line cuts of the image plots shown in (a) and (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure 3 | Identification of different thickness regimes in CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The few-layer, multilayer and bulk thickness regimes are depicted according to the layer dependence of ε1 and ε2 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV (a) and the low (bottom) and high (top) energy transmittance minima (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Markers display experimental data while the dotted lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (c) Spatially resolved wide field image of the high energy transmittance dip position calculated from hyperspectral images of CrI3 flakes with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The energy crossover of dip position when increasing the number of layers is clearly visible, highlighting the different thickness regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 0 0 100 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='77 5 Number of layers (L) Number of layers (L) Energy (eV) ε1, ε2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV) ε1 ε2 0 0 100 150 0 1 2 3 4 5 5 6 7 8 a b c d a b c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='75 67 L 14 L 3 L 70 L 20 L 8 L 30 L 53 L 2 L 57 L 13 L 44 L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='65 Energy (eV) 5 10 b 34567 a Dielectric function - , Dielectric function - 2 160 160 140 140 120 120 100 100 (7) s layers Number of layers ( 80 80 60 60 Number of 40 40 20 20 87654321 ８７６５4３２1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) Energy (eV) 101 102 101 102 1 Number of layers (L) d Number of layers (L) c 8 12 10 6 8 13 c2 4 6 1L 4 2 1L 2 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) Energy (eV)4 displays the spatially resolved wide field image of the higher-energy transmittance dip of CrI3 flakes for different thicknesses (see SI4 for calculation details) where the dip in energy of the crossover from a few layers to bulk is also clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These experimental observations hence point at profound changes in the optical properties of CrI3 in these three different thickness domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The evolution of the dielectric function with the layer number has been previously reported in other 2D materials such as transition metal dichalcogenides23, In2Se3 24, PdSe2 25 and a non-monotonic behaviour, consistent with our work, was observed in MoS2 at the nanoscale26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, in all these cases this behaviour has been reported only in the nanoscale while this is the first time that a modulation of the dielectric function has been experimentally observed at the mesoscale range of thickness, allowing for a continuous modulation of the optical properties from 1 to more than 100 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To gain further insight into the possible origins of this mesoscopic transition, we carried out first-principles calculations on few-layer CrI3, up to 10 L (see SI5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In the real and imaginary parts of the computed dielectric function (Figure 4a-b), we identify 2 peaks located at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 eV, in good agreement with the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A look at the band structure and DOS of monolayer CrI3 (left panel in Figure 4c) confirms that these peaks can be ascribed to metal-to-ligand charge transfer processes between the p-orbitals of the ligands, localized in the last occupied bands, and the empty dx2-y2 and dz2 orbitals of the chromium atoms localized in the eg set of conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In the right panel of Figure 4c, we plot the evolution of the peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 eV with increasing number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' We observe a strong red shift from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 eV to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 eV in agreement with experimental data and indicating a clear connection between the number of layers and the electronic structure in few-layer CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The calculations also show a tendency towards saturation of the value of the complex dielectric function when increasing the thickness from 1 L to 10 L, which confirms the electronic origin of the layer-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The discrepancies in the absolute value of the complex dielectric functions extracted from the calculations and the experimental ones may originate from subtle differences of the inter-layer distance at different temperatures and layer numbers, which have a strong effect on the polarizability of the wave-functions in the direction perpendicular to the layers (see SI5 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This effect also suggests that small changes in the crystal structure could eventually modify the intensity of the complex dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Indeed, our calculations indicate that at 0 K the interlayer distance in bulk layered antiferromagnetic CrI3 increases up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 Å compared to few layer samples (see Table S2) which may lead to a reduction of the polarizability and the dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This result hence highlights the dipolar and long-range nature of the van der Waals interactions which can be one of the possible explanations for the observed mesoscopic crossover of the complex dielectric function experimentally observed in the different thickness regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The saturation of the value of the complex dielectric function at the mesoscale is well aligned with the critical crystal thickness at which the low-temperature magnetic properties of CrI3 change from antiferromagnetic to ferromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Although this critical thickness has not yet been unambiguously determined, most works report on the layered antiferromagnetic state persisting in crystals up to 50 nm27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Both the magnetic and the optical crossover thickness to the bulk regime being in the same range of thickness is a strong indication of a connection between both phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Our findings, supported by the theoretical calculations, point at an electronic origin of these effects, with an evolution of the electronic structure with the layer number onsetting below 100 L (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 nm) as revealed by the non-monotonic modulation of the dielectric function as the crystal is thinned down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These results will contribute to shed light into the open structural conundrum of the layer-dependent phase diagram of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Our results demonstrate that hyperspectral transmission microscopy is instrumental to study the layer-dependent evolution of the electronic properties of air-sensitive Figure 4 | Theory calculations of the layer-dependent electronic and optical properties of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (a-b) Evolution of ε1 and ε2 of CrI3 with increasing number of layers calculated in the z-direction perpendicular to the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (c) Projected band structure and DOS of monolayer CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Filled and empty circles represent I p-orbitals and Cr d-orbitals respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The size of the circles is the weight of the orbital wavefunction in each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Red and blue stand for spin up and down respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The yellow arrow shows the most probable transition responsible for the low-energy peak in the dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In (d), we show the evolution of the low-energy peak in the imaginary complex dielectric function with increasing number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' a c I Cr Spin up Spin down eg b d I Cr Spin up Spin down eg E - EF (eV) ε2 ε1 Spin ↑ Spin ↓ 0 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 Κ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 1 1 DOS (eV-1) ω (eV) ω (eV) Number of layers (L) ω (eV) 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0 10 0 5 10 Κ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 Γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0 1 2 3 4 5 6 6 10L 8 5 6 L 4 L 4 3 L 2 L 3 3 2 1 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 w (eV)0,5 0 eV) L i 0,5 00000 8:8888 K r K 10 0 10 DOS (ev-6 10 L 8 L 5 6 L 4L 4 3 L 2 L 1 L 2 3 3 2 1 O 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 w (eV)2,1 2,05 (eV) 3 2 1,95 1,9 0 5 10 Number of layers (L)5 layered materials through the determination of its complex dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The possibility to obtain in a single acquisition the light intensity coming from tens of bare CrI3 crystals with different layer number guarantees high throughput statistics and allows for a robust and simultaneous determination of both the real and imaginary layer-dependent components of the dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This approach reveals at least two crossover points of the electronic properties of CrI3, depicting changes of the trends of the complex dielectric function both in the nano- and the mesoscale thickness regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The significantly wide span of the continuous layer number modulation of the optical and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Giant Tunneling Magnetoresistance in Spin-Filter van Der Waals Heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Science 360 , 1214 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Acknowledgments The project that gave rise to these results received the financial support of a fellowship from “la Caixa” Foundation (ID 100010434, fellowship codes LCF/BQ/ PR21/11840011 and LCF/BQ/DI22/11940022) and the grant PID2020-118938GA-100 from the Spanish Ministerio de Ciencia e Innovación (MICINN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' ENM acknowledges the European Research Council (ERC) under the Horizon 2020 research and innovation program (ERC StG, grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 803092) and to the MICINN for financial support from the Ramon y Cajal program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' RYC2018-024736-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' FCP also acknowledges the MICINN for the FPU program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' FPU17/01587).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This work was also supported by the Spanish Unidad de Excelencia “María de Maeztu” (CEX2019-000919-M) and is part of the Advanced Materials programme supported by MICIN with funding from European Union NextGenerationEU (PRTR-C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='I1) and by Generalitat Valenciana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 6 Methods Crystal growth and isolation of few-layer crystals: High quality CrI3 crystals were grown by chemical vapor transport and thoroughly characterized by X-ray diffraction (XRD), Energy-dispersive X-ray spectroscopy (EDAX) and Raman spectroscopy to verify their pristine quality (see Supporting Information SI1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Atomically thin layers were obtained by mechanical exfoliation of bulk crystals using a scotch tape technique and deposited over a transparent glass slide to perform transmittance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Selected flakes were first selected by optical microscopy screening and successively characterized by atomic force microscopy to accurately determine their thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The whole process and characterization were carried out inside an Ar-filled glove box where the O2 and H2O levels were maintained below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 ppm to avoid material degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Optical transmission experiment: Once the thicknesses of all flakes were determined, the sample was encapsulated by gluing on the glass slide substrate a coverslip with a thermoplastic material and exposed to the air avoiding degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Hyperspectral transmittance images were acquired under monochromatic light ranging from 430 nm to 720 nm in an upright transmittance microscope equipped with a high-resolution CMOS monochromatic camera (Hamamatsu Orca Fusion) and a homemade grating monochromator for the incidence light, for accurate quantitative transmittance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Extraction of the optical dielectric function: The transmittance spectra have been fitted simultaneously by employing a multilayer model whose complex coefficients of reflectance and transmittance are described by Fresnel equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Their complex parameters (refractive index and extinction coefficient) have been calculated from the optical dielectric function (Equation 1) assuming a relative permeability of µr=1 (at room temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To fit the model simultaneously with all the experimental data (Figure 1), we have considered a spline for modelling the layer dependence of the parameters of Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In particular, a faithful data fit could be reached employing two or three cubic polynomials (C1 continuity) excluding or including the monolayer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Density functional theory: The electronic structure calculations and geometry relaxations have been carried out using Quantum- Espresso ab initio package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Each structure has been relaxed using the Grimme-D3 van der Waals correction until all the forces acting on atoms were lower than 10-3 Ry/bohr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' For the relaxation, we have used projector augmented waves (PAW) pseudopotentials within the generalized gradient approximation (GGA) and the Perdew-Burke-Ernzerhof (PBE) exchange-correlation potential, with a 4x4x1 k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The dielectric functions were obtained using the epsilon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='x code, which is part of the Quantum-Espresso package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The calculations were carried out at the independent particle (IP) level and non-local contributions from pseudopotentials are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' For the optical properties, the electronic structure calculations are carried out using the DFT+U+J scheme with U = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 eV norm-conserving GGA-PBE pseudopotentials from the PseudoDojo library, with energy cutoffs 80 and 320 Ry for the wavefunctions and charge density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In our case, the dielectric functions converged for a 8x8x1 k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 7 Supporting information Monolayer-to-mesoscale modulation of the optical properties in 2D CrI3 mapped by hyperspectral microscopy Marta Galbiati1*†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Fernando Ramiro-Manzano1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2*†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' José Joaquín Pérez Grau1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Fernando Cantos- Prieto1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Jaume Meseguer-Sanchez1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Ivona Kosic1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Filippo Mione1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Ana Pallarés Vilar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Andrés Cantarero1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' David Soriano3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' and Efrén Navarro-Moratalla1* 1 Instituto de Ciencia Molecular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Universitat de València,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Calle Catedrático José Beltrán Martínez 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 46980,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Paterna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 2 Instituto de Tecnología Química, Universitat Politècnica de València - Consejo Superior de Investigaciones Científicas (UPV-CSIC), Avd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' de los Naranjos s/n, 46022, Valencia, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 3 Information Engineering Department, University of Pisa, Via Caruso 16, 56122, Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 4 Departamento de Física Aplicada, Universidad de Alicante, 03690, San Vicente del Raspeig, Alicante, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' † M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' * e-mail:ferraman@fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='upv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es, marta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='galbiati@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es, efren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='navarro@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='es Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Materials characterization and sample preparation 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Wide-field hyperspectral setup for air-sensitive materials 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Optical model 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Spatially resolved optical properties 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Dependence of the complex dielectric function with layer number and interlayer distance 17 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Materials characterization and sample preparation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Bulk CrI3 crystal characterization Powder X-ray diffraction (PXRD): The crystal structure of CrI3 was characterized by powder X-ray diffraction (PXRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S1 shows the XRD of the of CrI3 single crystals taken employing PANalytical Empyrean X-ray diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' X-Rays diffraction analysis was performed on sample of single crystalline CrI3 by loading the material into a capillary and sealing it inside the glove box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The powder pattern of sample is consistent with the monoclinic AlCl3-type structure (C2/m) reported for CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In the XRD spectrum, no evidence of any other phases was detected indicating that the product is of high purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Energy Dispersive Analysis of X-ray (EDAX): The chemical composition of as-grown CrI3 single crystals was determined by Energy Dispersive Analysis of X-ray (EDAX) technique using Hitachi S4800 for confirming stoichiometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The obtained spectrum is shown in Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The atomic and weight % obtained using EDAX is tabulated in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The data clearly states that the as-grown CVT single crystals have chemical composition of CrI3 and they are free from any impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S1 | XRD pattern and unit cell refinement of a representative CrI3 single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The continuous black line shows the experimental pattern, the blue line shows the Rietveld refined total intensity and the red trace depicts the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The green marks indicate the refined reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A selection of the most prominent reflections are labeled using their corresponding Miller indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S2 | EDAX spectrum of the as-grown CrI3 single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 10 kev9 Raman spectroscopy: CrI3 bulk flakes have been characterized by Raman spectroscopy after their mechanical exfoliation to verify their pristine quality after encapsulation with the glass cover (see section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Raman characterization has been performed using a Horiba LabRAM HR Evolution Raman spectrometer under a 532 nm excitation laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Results in Figure S3 show peaks at frequencies of 78, 100−110, 128, and 234 cm-1 in agreement to that previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Preparation of samples for transmission experiment For our transparent substrates we use standard microscope glass slides thoroughly rinsed to obtain very clean and low roughness surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Glass slides were soaked in a freshly prepared solution of NH4OH : H2O2 (1 : 1) and sonicated for 8 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Next, they were rinsed with Milli-Q water, sonicated 5 min in Milli-Q water and dried under a nitrogen stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Clean slides were introduced inside the glove box and heated above 100ºC for at least 30 min before deposition of the 2D material, in order to remove any possible H2O traces from the substrate surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Atomically thin CrI3 layers were obtained by mechanical exfoliation of bulk CrI3 crystal, grown by chemical vapor transport, using a scotch tape technique and deposited over the glass slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' An optical microscope was then used to screen the sample and select regions of interest with abundant flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Selected layers were characterized by atomic force microscopy to determine their thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Remarkably, the surface of the glass slides was observed to be perfectly clean with a low roughness (RMS ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='23 nm) which allows accurate flakes thickness measurement down to the monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The whole exfoliation and characterization processes were carried out inside an Ar-filled glove box where O2 and H2O levels were maintained below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 ppm to avoid material degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Once flakes characterized, sample was sealed with a thin glass cover slide (100 µm thick) to prevent CrI3 layers degradation when exposed to the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The samples were finally extracted from the glove box to proceed with the wide field hyperspectral experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Table S1 | EDAX spectrum of the as-grown CrI3 single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S3 | Raman spectrum of bulk CrI3 flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The flake under observation was well within the bulk thickness regime described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Element Weight % Atomic % Cr 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='73 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='49 I 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='27 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='51 7000 6000 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' units) 5000 4000 3000 2000 1000 0 50 100 150 200 250 300 Raman shift (cm-1)10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Wide-field hyperspectral setup for air-sensitive materials A wide field hyperspectral setup has been implemented by modifying a standard optical transmittance microscope as schematically shown in Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' White light coming from a high intensity LED phosphor light source enters a homemade monochromator, which allows selecting the excitation wavelength with a bandwidth of ~1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Monochromatic light is then connected to the optical microscope through an optical fiber and is used to irradiate the transparent samples in transmittance mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' For the hyperspectral analysis, transmission images of a region of interest of the sample are acquired sweeping the wavelength from 430 nm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 eV) to 720 nm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 eV) with a step of 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To achieve this, the optical microscope is equipped with a monochrome CMOS camera (Hamamatsu Orca Fusion), conveniently chosen for its low noise and high linear range, which allows precise quantitative analysis of the images even for thin layers and in the full visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Data acquisition is performed in complete dark conditions, in order to avoid any source of external light, which could result in reduced data reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' With an exposure time of about 300 ms per image, a complete hyperspectral image takes less than 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S5 shows transmission images of CrI3 atomically thin crystals down to the monolayer under different wavelengths, with a contrast that ranges from almost transparent to very dark for increasing thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S5 also demonstrates the high throughput of this imaging technique, which permits White light source CMOS monochrome camera Monochromator Figure S4 | Schematic of the wide-field hyperspectral microscope setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Solid straight lines depict selected light ray traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The drawing is not to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S5 | Few-layer CrI3 crystals deposited on standard glass slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (a-d) Optical transmission micrographs of a representative sample, showing different layer number plateaus illuminated with white, 630 nm, 532 nm and 460 nm light respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Scale bars in the optical micrographs are 10 µm long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (e) Atomic force microscopy topographic image of single layer CrI3 located inside the dashed box drawn in panel (d), with the corresponding probability density distribution shown in panel (f), which allows to estimate a flake thickness of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='001 nm, compatible with a monolayer within experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Probability density has been calculated in the region marked with a dotted line in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 2 μm11 exploring the layer-dependent optical properties in the visible range from the single layer to the hundreds- of-layers range with ample statistics coming from just a single frame of hyperspectral images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To accurately determine the number of layers of each flake we employed atomic force microscopy (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' As shown in Figure S5e,f, the topographic image of a CrI3 single layer on a glass slide depicts a thickness of ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 nm which is in agreement, within the experimental uncertainty, with the interlayer space extracted from the lattice parameter of the bulk structure [1], and demonstrates the atomically-flat quality of the micrometer- size samples obtained down to the single layer limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' It is also worth noticing that the root-mean-square (RMS) roughness measured over the commercial glass slide is only ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='23 nm, comparable to that obtained on standard Si/SiO2 wafers, hence making commercial glass slides a convenient transparent substrate even for the investigation of the thinnest layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The collected hyperspectral data are arranged in a 3D matrix where each image pixel (x and y spatial coordinates) is associated to all the swept wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To reduce signal to noise ratio in data analysis we select a region with homogeneous thickness on the analyzed flake (blue mask) and a second neighboring region on the substrate (green mask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Typical flake/substrate regions are displayed in Figure S6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A dark background frame is then subtracted and the pixels intensities are averaged for each region (Iflake and Isubstrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Quantitative transmittance is calculated as T(E) = Iflake(E)/ Isubstrate(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S6c shows examples of typical transmittance spectra for different CrI3 flake thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Good reproducibility of the transmittance spectra for different flakes with equal thickness was verified in order to ensure a reliable layer-dependent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Reflection data showed to be strongly affected by the choice of regions and by spurious reflections, not complying with our data reproducibility requirement and resulting in a high dispersion of spectral traces for different flakes with equal thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Reflection data was consequently excluded from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S6 | Transmission data processing and representative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (a) Optical transmission image of CrI3 flakes deposited over a glass slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In order to analyze transmittance spectra a zone of the flake with the same thickness is selected over the image (area labeled with an f), and a zone over the substrate just nearby the flake (area labeled with an s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Transmission intensities are then averaged over the selected zones (If and Is) for each image acquired in a wavelength range of 430 - 720 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Finally, transmittance is calculated as T = If / Is for each swept wavelength (energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (b) Schematic of the simplified system where monochromatic light is coming from the back of the sample in transmittance mode and is transmitted through the glass slide and the CrI3 layers before being collected by the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (c) Examples of transmittance spectra collected for different CrI3 layers thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To ensure data reliability good reproducibility of transmittance spectra over different flakes with the same thickness has been verified as it can be observed in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' a b c Energy (eV) Transmittance a b c 2 L 3 L 4 L 5 L a) Crl3: n2 Glass slide: n10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='812 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Optical model In previous works, the reflectance of 2D materials deposited on solid substrates has been modeled using the Fresnel equations [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In this regard, the real (ε1) and imaginary (ε2) part of the optical dielectric function of individual flakes of the materials are derived from a single reflectance experiment using a Kramers- Kronig (KK)-constrained analysis [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This approach has been used, for instance, to find the dielectric function of TMDC monolayers deposited on fused silica substrates modeling ε(ω) as a superposition of Lorentzian oscillators, whose complex equations fulfill the KK relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' [3] Similarly, KK relations have been used to calculate the phase of the amplitude reflection coefficient θ to extract the refractive index and extinction coefficient of bulk CrI3 in the visible range [1], or in the terahertz to near-infrared region [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, this approach has the important limitation that, for an accurate determination of ε(ω), KK relations require a wide energy range (ideally from 0 to ∞ while optical contrast experiments are usually limited to the small range of visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To address this issue, a possible solution has been to make use of literature data to extrapolate high and low energy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' While we initially applied this same approach, we realized that for our spectral range, the extracted phase (θ) showed a remarkable dependence with the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' As a result, it was difficult to reach consistent solutions over different spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' An alternative technique to determine the optical dielectric function is to acquire both reflectance and transmittance spectra for solving a system of two independent equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In this way, both ε1 and ε2 can be unequivocally obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' At this aim, we attempted to simultaneously analyze the reflectance spectra of CrI3 over two different substrates: SiO2/Si and a glass slide, successively trying to strengthen our analysis by additionally relating them to transmittance measurements performed on the same CrI3/glass slide sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' However, this extended analysis led us to the following conclusions: firstly, reflection spectra introduce a high experimental error, which hampered finding a solution for the computed ε(ω) of the 2D material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' While SiO2/Si substrate interference strongly masked the small absorption peaks of CrI3 layers, external reflections due to optical elements, very difficult to get rid of, were observed on the transparent glass substrate, resulting in a poor reproducibility of reflectance data and subsequent inconsistent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Secondly, we verified that ε2 dominates both the transmittance (see Figure S8) and reflectance spectra [3], leading to ambiguity of the solution of the equation due to the high experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' We thus focused on transmission measurements as a source of high data reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To limit the interdependence between ε1 and ε2 from a single experiment we simultaneously analyzed transmittance spectra acquired over flakes with different thicknesses, including a large number of samples with ample statistics (thousands of pixels each) in every dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This approach allows to further constrain the range of possible solutions of CrI3 ε(ω), hence increasing the consistency of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' For the transmittance analysis we describe our system as a simple Fabry-Perot cavity composed of three layers as shown in Figure S6b: (1) glass slide / (2) CrI3 / (3) air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Other contributions such as the presence of a glass cover or the rest of optical elements cancel each other in the transmittance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The energy dependent complex refractive index for each layer is written as: n1(ω) = nglass(ω) - ikglass(ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' n2(ω) = n2D(ω) - ik2D(ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' n3(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To calculate the complex refractive index of CrI3 (n2(ω)), we assume CrI3 dielectric function to follow a modified Lorentzian oscillatory model: composed by two oscillators (n = 1, 2), where ωn, ωp, ω represent the resonant, plasma and scan frequencies respectively, ε∞ is assumed to be 1, and γn and Γn correspond to damping related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' We then extract n2 from n2(ω) ≈ √ε(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S7 shows examples of transmittance spectra (black lines) and fit (red lines) performed with our model on CrI3 samples with different thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A very good agreement between experimental data and theoretical fit can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S8 shows maps of solutions of the model around the energy values of the transmittance dips (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These maps were generated by plotting the difference, in absolute value, between the experimental transmittance signal and different theoretical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These model solutions are simulated by scanning different combinations of ε1 and ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The zeros of this subtraction (black/dark region) correspond to possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Given that these values form quasi horizontal lines, the solution is dominated by a small and limited range of ε2 values in contrast to a larger dispersion of ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Indeed, ~ ~ ~ ~ ~ ~ ~ ~ ˜ε(ω) = ˜ε∞ + 2 ∑ n=1 ω2 p + 2iωΓn ω2n − ω2 + 2iωγn 13 a certain deviation of ε1 results in a slight perturbation of the highly reliable ε2 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In any case, the solutions of the maps show a small curvature due to the multilayer nature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Consequently, this shape changes with the sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Importantly, in our experiments the layer complexity was streamlined to the bare minimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' consisting of a single layer substrate holding the sample, and as a result, the Figure S8 shows simple and almost flat solution curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The white marker in each map indicates the solution captured by the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Note that the fitting procedure not only considers a single spectral energy (as shown in Figure S8), but it considers the full spectroscopic experimental data range (Figure S7) for obtaining the solution Note that our final fitting procedure not only considers a single spectral energy (as shown in Figure S8), but it considers the full spectroscopic experimental data range (Figure S7) for obtaining a complete solution as a function of the layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' To further increase consistency of our data analysis, we perform a simultaneous fit using the previously described model with > 40 transmittance spectra of CrI3 layers with thickness ranging from 1 L to 164 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Although each spectrum represents a different scenario, they are interconnected by shared parameters with a defined tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' We rely on the fits performed individually on each spectrum for the different layer thicknesses in order to gain a first idea on the fitting parameters evolution with the number of layers (Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Then, we fit parameters tendency with respect to layer thicknesses with a spline consisting in 3 cubic polynomials defining two C1 intersection points (or knots) at about 1L and 46 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Our model allows to fits accurately to all transmittance spectra while interconnecting fitting parameters allows to highly constrain the number of possible solutions, hence increasing the accuracy of our analysis (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' It is important to point out that it was not possible to attain a good fit of all transmittance spectra without defining the spline intersection points (at about 1L and 46L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This, again, supports the idea of three thickness regimes with different optical properties: the so-called thin, multilayer and bulk regions as discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='99 Transmission 1L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='65 Transmission 31L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 Transmission 4L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 Transmission 78L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 Transmission 6L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 Transmission 131L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Transmission 14L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='3 Transmission 164L Transmittance Transmittance Transmittance Transmittance Transmittance Transmittance Transmittance Transmittance Figure S7 | Examples of transmittance spectra experimentally measured over CrI3 layers of different thickness (black curves) and their model fit (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The results plotted were obtained by using the oscillatory model for different number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 14 1L 4L 6L 14L 31L 71L 131L 164L 1L 4L 6L 14L 31L 71L 131L 164L ENERGY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV ENERGY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 eV Figure S8 | Map of solutions of the model computed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='95 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Black regions correspond to the possible solutions (zeros) as a function of ε1 and ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' White dots represent individual solutions extracted from the fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 4 2 0 05 105 105 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='64 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='34 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='44 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='44 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='54 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='64 2 0 04 2 0 05 1010 8 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='84 2 0 015 Figure S9 | Map of the complex optical dielectric function of CrI3 composed by independently fitting the transmittance spectra of samples with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The calculated real (ε1) and imaginary (ε2) parts of ε(ω) are shown in panels (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The computed ε1 (ℏω) (c) and ε2 (ℏω) (d) have been also plotted as a function of the excitation energy ℏω for different CrI3 crystals of different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The red traces correspond to thin layers (from 1L to 13L), the blue traces to layers ranging from 14L to ~100L and the green traces to bulk (up to 164L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This figure displays independent fits of specific thickness spectra in comparison to Figure 2 that shows a simultaneous fit to the complete layer-dependent dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' A good agreement between the two figures can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' ~ 0 5 10 b 2 4 6 a Dielectric function - ε, Dielectric function - , 160 160 140 140 120 120 100 100 (L) Number of layers (L) 80 80 Number of layers 60 40 20 20 ８7654321 ８7６54３２1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) Energy (eV) 101 102 101 102 d c Number of layers (L) Number of layers (L) 8 12 1 L 10 6 8 13 1 L 4 6 4 2 2 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='8 Energy (eV) Energy (eV)16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Spatially resolved optical properties Hyperspectral analysis allows to spatially resolve the optical properties of a 2D material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In Figure S10a we show the optical transmission image of different CrI3 flakes with their respective thickness measured by AFM, while Figure S10b displays the spatially resolved intensity of the high energy transmittance dip of CrI3 (at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='66 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' As for the peak position map of Figure 3 of the manuscript, this has been obtained by calculating pixel by pixel the corresponding transmittance spectrum and fitting its high energy dip position (Figure 3c) and transmittance intensity (Figure S10b), then ascribing these values to each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Background pixels were automatically set to the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The color scale can also be compared to the values extracted from the analyzed transmittance spectra as a function of the number of layers (Figure S10c) showing a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The spatial resolution of the optical properties of a 2D material results to be a very intuitive and powerful method to gain a fast insight on the properties of many different sample thicknesses at a same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' By tracking the peak transmittance intensity, as shown in Figure S10b, it is possible to gain a large contrast on the thinnest layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Once transmittance intensity has been associated to the respective layers thickness characterized by AFM, this becomes a very powerful tool to quickly and reliably gain information on all flakes thickness of a whole image with a high spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This process could be used for the automatic analysis of flakes thicknesses with clear advantages in terms of time saving (data images acquisition takes about 10 minutes for a whole zone) and non-invasive way, very relevant especially for air sensitive 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S10 | Spatially resolved layer dependence of the optical properties of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (a) Optical transmission image of CrI3 flakes deposited on a glass slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The number of layers of each section of the sample has been characterized by AFM and labeled on each flake in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (b) Spatially resolved map of zone shown in (a) displaying the transmittance value at the to display intensity at the high energy transmittance dip for each flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This has been obtained by extracting the transmittance spectrum at each pixel, fitting the transmittance in the spectral vicinity of the high energy dip and obtaining the energy at the local minimum, and ascribing this value to each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Background pixels where automatically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' (c) Evolution with the number of layers of the minimum transmittance intensity at the high energy dip, obtained by relating analyzed transmission images to results obtained by AFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Dots are experimental data while dotted line is guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Once transmittance intensity dependence with layer thickness is calibrated, this technique allows to optically analyze thin layers thickness with a high throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This method is particularly convenient as it is fast and non-invasive.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 0 0 50 100 150 High energy dip transmittance intensity Number of layers (L) 67L 70L 42L 14L 53L 30L 3L 8L 70L 44L 2L 20L 13L 20L aioy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='9 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 017 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Dependence of the complex dielectric function with layer number and interlayer distance We carried out first-principles calculations on few-layer CrI3, up to 10 L using the DFT+U+J scheme with U = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='1 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='6 eV, as previously established for this material [7], based on the constrained random-phase approximation (cRPA) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' For all cases, we consider an ideal monoclinic stacking between adjacent layers and antiferromagnetic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S11 shows the theoretically computed evolution of the real and imaginary parts of the dielectric function of CrI3 with increasing number of layers for the full energy range calculated from 0 eV to 4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' In Figure S12 we show the evolution of the intensity of the e2 peak at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 eV for bilayer CrI3 with the inter- layer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The other peaks reported a similar trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' The increase in the inter-layer distance decreases the polarizability of the wave-functions in the direction perpendicular to the layers, also decreasing the magnitude of the complex dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' These results clearly indicate that small changes in the crystal structure and stacking could eventually modify the intensity of the complex dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Considering that in layered bulk crystals the interlayer distance increases due to stronger van der Waals interactions and modifications in the crystal structure, our theoretical results give hints on the crossover and the different thickness regimes observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' a b a b Figure S11 | Evolution of the real and imaginary parts of the dielectric function of CrI3 with increasing number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' This calculation was performed in the z-direction perpendicular to the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Figure S12 | Dependence of the intensity of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='5 eV peak in the imaginary dielectric function with the inter-layer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Table S2 | Evolution of the Cr-Cr interlayer distance with the number of layers based on first principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Number of layers 2 4 6 8 10 Bulk Average Cr-Cr interlayer distance (Å) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='654 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='640 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='644 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='649 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='681 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content='845 a a a b 18 References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=', Layer-Dependent Ferromagnetism in a van Der Waals Crystal down to the Monolayer Limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Nature 546, 270 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' [2] P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} +page_content=' 70, (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfE_pK/content/2301.01002v1.pdf'} diff --git a/7tE0T4oBgHgl3EQffQCV/content/tmp_files/2301.02402v1.pdf.txt b/7tE0T4oBgHgl3EQffQCV/content/tmp_files/2301.02402v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4942526360ceda11c44f887878b0437613b0b8a --- /dev/null +++ b/7tE0T4oBgHgl3EQffQCV/content/tmp_files/2301.02402v1.pdf.txt @@ -0,0 +1,2130 @@ +Hawkeye: Hectometer-range Subcentimeter Localization +for Large-scale mmWave Backscatter +Kang Min Bae†, Hankyeol Moon†, Sung-Min Sohn§, Song Min Kim¶ +Korea Advanced Institute of Science and Technology (KAIST) , §Arizona State University +{bkm2259, moonkyul1, songmin}@kaist.ac.kr, smsohn@asu.edu +ABSTRACT +Accurate localization of a large number of objects over a +wide area is one of the keys to the pervasive interaction +with the Internet of Things. This paper presents Hawkeye, +a new mmWave backscatter that, for the first time, offers +over (i) hundred-scale simultaneous 3D localization at (ii) +subcentimeter accuracy for over an (iii) hectometer distance. +Hawkeye generally applies to indoors and outdoors as well +as under mobility. Hawkeye tag’s Van Atta Array design +with retro-reflectivity in both elevation and azimuth planes +offers 3D localization and effectively suppresses the multi- +path. Hawkeye localization algorithm is a lightweight signal +processing compatible with the commodity FMCW radar. It +uniquely leverages the interplay between the tag signal and +clutter, and leverages the spetral leakage for fine-grained po- +sitioning. Prototype evaluations in corridor, lecture room, +and soccer field reveal 6.7 mm median accuracy at 160 m +range, and simultaneously localizes 100 tags in only 33.2 ms. +Hawkeye is reliable under temperature change with signifi- +cant oscillator frequency offset. +1. +INTRODUCTION +Precise interaction with a large number of objects +spread over a region has long been a vision for the +IoT, where accurate localization is one of the essen- +tial features for an immersive experience. Backscatter +possess great potential towards this goal, with the low- +cost and ultra low-power tags that can be massively de- +ployed over a large area with the minimum deployment +cost and maintenance efforts. Localization of large-scale +(e.g., hundreds to thousands) tags with subcentimeter +accuracy installed over an area (e.g., hectometer-range) +would offer benefits to a wide range of applications in- +cluding asset tracking, inventory management, ware- +house automation, smart factories, virtual/augmented +reality, and structural health monitoring. +To this end, backscatter (including RFID) localiza- +tion has been extensively studied in sub-6GHz bands. +However, their accuracy, scalability, and range are fun- +damentally throttled by the hard bandwidth constraint +†Co-primary Student Authors. +¶Song Min Kim is the corresponding author. +Figure 1: Hawkeye is tested under large-scale (left), over +a long range (center), and on mobile objects (right). +(e.g., tens of KHz for UHF RFID). This limits their +performance to tens of cm accuracy [4, 8, 37, 45, 65], +restricts the deployment scenarios by requiring fixed +movement trajectories [42, 50, 60, 62], or requires de- +ploying dense reference tags with prior knowledge [5, +13, 58]. A recent line of research, RFind [33] and Tur- +boTrack [32], resolve the bandwidth issue by enabling +RFID to emulate the wide bandwidth of 220 MHz that +extends beyond the ISM band, to achieve subcentime- +ter accuracy. +However, the range is bounded to sev- +eral meters to remain compliant to the FCC regulations, +limiting the usage to a room-scale and requiring a cus- +tomized reader. The latest work of millimetro [55] takes +a fundamental approach of exploring the rich, 250 MHz +bandwidth in the 24 GHz mmWave band, by utilizing +FMCW radar and backscatter tag to reach over 100 m +range. However, the median accuracy of millimetro is +limited to 15 cm, which is essentially the accuracy of 24 +GHz FMCW radar. +This paper presents Hawkeye (Figure 1), a mmWave +backscatter localization with the empirical performance +of (i) 6.7 mm median accuracy (ii) at 160 m range +(@1D), (iii) simultaneously localizes 100 tags in only +33.2 ms (scales up to 1024 tags in theory), and (iv) +uses an affordable commodity radar (∼200 USD [17]). +Hawkeye blends a new backscatter tag for efficient sig- +nal delivery and lightweight radar-side signal processing +for accurate and rapid localization. Hawkeye backscat- +ter tag is a planar Van Atta Array (VAA) combined +1 +arXiv:2301.02402v1 [eess.SP] 6 Jan 2023 + +Large Scale +Long Range +Sub-cm +(100 Tag) +(160m) +&Mobile +1153Systems +Accuracy @ 5 m +Range +Bandwidth +Simultaneous Localization +Fixed Trajectory +Hawkeye +2.5 mm +180 m +250 MHz +100 Tags (1024 in theory) +No +Millimetro [55] +78 mm +180 m +250 MHz +6 Tags (106 in theory) +No +RFind [33] +3.4 mm +6 m +220 MHz +No +No +TurboTrack [32] +5.1 mm +10 m +100 MHz +2 Tags +No +Tagoram [60] +10 mm +12 m +6 MHz +No +Yes +Table 1: Comparison with the state-of-the-arts +with a power-efficient low-loss FSK modulator using hy- +brid coupler. +The tag retro-reflects in both azimuth +(90◦ FoV) and elevation (140◦ FoV) to enable 3D local- +ization and effectively suppresses multipath. The de- +sign is robust against oscillator frequency offset, with +only 4 mm localization error across low (9.45◦C), high +(38.43◦C), and room (23.7◦C) temperatures. This is an +essential property for practical subcentimeter localiza- +tion under disparate deployment scenarios, using low- +end tags. Furthermore, one-shot interrogation localizes +up to 1024 tags (evaluated with 100) simultaneously, +supporting large-scale rapid localization. Table 1 sum- +marizes the comparison to the state-of-the-art backscat- +ter localization systems, showcasing that Hawkeye is +uniquely positioned to achieve high scalability, long range, +and precision at the same time. +Hawkeye exploits the interplay between Hawkeye back- +scatter FSK signal and the chirp-based Frequency Mod- +ulated Continuous Wave (FMCW) radar to improve the +localization performance by over ×60 over using the +FMCW alone. Hawkeye tag is tuned to demonstrate S11 +of -10 dB throughout the entire 250 MHz bandwidth, +where FSK modulation is performed by the combina- +tion of reflective network and low-loss 90◦ hybrid cou- +pler co-optimized for efficient VAA reflection. The use +of the VAA, along with the severe signal attenuation of +the mmWave, naturally suppresses the multipath intef- +erence. To the best of our knowledge, Hawkeye tag is +the first planar VAA mmWave backscatter design with +modulation capability. +The radar-side subcentimeter +localization is designed as a lightweight post-processing +on top of FMCW demodulation, without requiring any +change to the commodity FMCW radar. Specifically, +Hawkeye localization algorithm is built on the recent +technique of HD-FMCW [11] that isolates the tag FSK +signal from the clutter noise. The key insight of Hawk- +eye is to leverage the relationship between the tag sig- +nal and the clutter, and the spectral leakage signature +embedded in the tag signal, from which the precise loca- +tion can be extracted. Hawkeye supports single radar or +multilateration positioning, where 1D-3D localization +was evaluated throughout indoors (corridors and lec- +ture rooms), outdoors (soccer field), NLOS, and varying +temperatures to demonstrate practicality. +Hawkeye is an accurate, long-range, and large-scale lo- +calization for mmWave backscatter, essentially enabling +tracking many objects spread over an area, ranging from +everyday spaces like homes and offices, to industrial sec- +tors such as inventories and warehouses. +Hawkeye is +kept economic with low-cost tags and compatibility to +affordable commodity radar. +To sum up, we believe +Hawkeye takes a solid step towards bringing pervasive +tag deployment and localization to practice. Our con- +tribution is three-fold: +• We design Hawkeye, a mmWave backscatter-based +subcentimeter localization that works over hecto- +meter-range and simultaneously localizes over a +thousand tags. +• To the best of our knowledge, we design the first +planar VAA mmWave backscatter with modula- +tion capability. +• We prototype Hawkeye tags on Rogers RO4003C +substrate for antenna with planar VAA structure, +VXCO-based control board on the PCB. Hundred +tags were produced for large-scale simultaneous 3D +localization. Readers were implemented on com- +modity 24GHz radars [1, 17]. +• We will release Hawkeye’s source code and HFSS +tag design file upon acceptance, for facilitating +community’s future works. +2. +BACKGROUND +This section provides the technical background for +Hawkeye, followed by the design overview. +FMCW Radar. An FMCW radar leverages chirp, whose +frequency linearly increases with time. Objects in the +radar’s vicinity reflect the transmitted chirp, which re- +turns to the radar with a round-trip propagation delay. +FMCW radar mixes the transmitted chirp with the re- +ceived chirp (with propagation delay) to produce an IF +signal. The range is measured by performing FFT on +the IF signal, where each reflected object is represented +as a signal with range frequency fr proportional to the +propagation delay. +Planar Van Atta Array. Planar Van Atta array (VAA) +passively reflects back the signal to the direction of ar- +rival, achieving retro-reflectivity in both azimuth and +elevation planes. It is a simple antenna array structure +where centrosymmetric pairs of the antenna in 2D plane +are interconnected by transmission lines (TLs) with a +length difference equal to λg (i.e., guided wavelength, +which is the wavelength of EM wave in the dielectric). +2 + +Figure 2: Planar +Van Atta array +The centrosymmetrically intercon- +nected antenna pair flips the inci- +dent signal’s phase sequence, which +directs the signal to the source. The +phase induced by the TLs does not +affect the radiation direction, be- +cause it is applied equally to all +lines. As an illustrative example in +Figure 2, consider a 2×2 planar VAA, where the signal +comes in the azimuth plane with the phase sequence of +[−2ϕ, −ϕ] at both [A,B] and [B’,A’]. After the propaga- +tion in TL, the phase sequence is inverted and produces +a reflected signal with a phase sequence of [−ϕ, −2ϕ] at +both [A,B] and [B’,A’]. This makes the reflected wave +back to the incident angle achieving retro-reflectivity in +the azimuth plane. Retro-reflectivity in the elevation +plane is also achieved in the same manner, by flipping +the phase sequence at [A,B’] and [B, A’]. +Figure 3: Hawkeye tag with key geometrical parameters +optimized for the 24 GHz band. Centrosymmetric an- +tenna pairs are interconnected by TLs. All dimensions +are in mm. +(a) +(b) +Figure 4: Comparison of measured normalized mono- +static RCS of Hawkeye tag and equal-sized flat plate +with the same substrate (a) in azimuth plane and (b) +in elevation plane. In each cases, we achieve over 17.2 +dB and 23.9 dB beamforming gain over the flat plate. +3. +HAWKEYE TAG DESIGN +Figure 3 presents Hawkeye tag prototype, which uniq- +uely blends a 4×4 planar Van Atta array with FSK +modulation which serves as a basis to long-range sub- +centimeter 3D localization with effective multipath sup- +pression. In the following, we present detailed structure +and design choices. +3.1 +Retro-reflectivity via Planar VAA +As depicted in Figure 4, Hawkeye VAA achieves retro- +reflectivity with an average beamforming gain of 17.2 +dB with FoV (10dB beamwidth) of 90◦ and 140◦ in az- +imuth and elevation, respectively. +To meet the VAA +condition, centrosymmetric antenna pairs are intercon- +nected via TLs, whose lengths are based on the guided +wavelength of λg = 7.56 mm, derived from the 24.125 +GHz (i.e., the center frequency of 24GHz band). Specif- +ically, the length differences of TLs are multiples of λg. +To limit the phase misalignment within the 250MHz +bandwidth, the difference between the maximum and +the minimum TL lengths is capped to 9λg. This trans- +lates to the maximum phase misalignment of 16.9◦, or +equivalently, beamforming power loss of only 5×10−2dB. +(a) S11 +(b) Smith chart +Figure 5: Measured antenna performance of Hawkeye +tag, with (a) S11 and (b) Smith chart. +(a) +(b) +Figure 6: (a) Magnified view of Hawkeye tag with key +geometrical parameters. All dimensions are in mm. (b) +Equivalent circuit of one pair of antenna, equipped with +a 90◦ hybrid coupler and a reflective network. +Hawkeye tag adopts inset-fed rectangular microstrip +patch antenna with the patch, edge notch depth (i.e., +the antenna feeding point [38]), and TL dimensions op- +timized to 50 Ω impedance matching at 24GHz, through +HFSS parametric sweep. +Figure 5 demonstrates the +antenna performance measured via VNA; Figure 5(a) +depicts the antenna response (S11) of −10 dB through- +out the 24 GHz band, and Figure 5(b) shows the cor- +responding impedance matching results. The parame- +ters are shown in Figure 6(a) on Rogers RO4003C sub- +strate (dielectric constant ϵr = 3.55, dissipation factor +tanδ = 2.7 × 10−3), where the top layer holds Hawkeye +tag circuit while bottom layer is ground plate. Lastly, +4×4 antenna elements are chosen to balance between +the FoV of the retro-reflectivity and the tag size – i.e., +3 + +- Reflected wave +- Incident wave +A +-29 +B +B +A' +-2Φ +11-2966 +L118.856 +5111.341 +L103.826 +96.311 +66 +5 +L52.280 +15 +Biaspin +L59.781 +L67.281 +74.7810 +Normalized RCS (dB) +-20 +Avg 17.2dB +-40 +-60 +Hawkeye +-Flat Plate +-90 +-60 +-30 +0 +30 +60 +90 +Azimuth (degrees)0 +Normalized RCS (dB) +-20 +.40 +Avg 23.9dB +-60 +Hawkeye +-Flat Plate +-90 +-60 +-30 +0 +30 +60 +90 +Elevation (degrees0 +-10 +0 -20 +-30 +-40 +23 +23.5 +24 +24.5 +25 +Frequency (GHz)j50 +j25 +j100 +j250 +j10 +U +0 +8 +10 +2550 +100250 +j250 +-j10 +-j100 +-j25 +-j504.2 +4 +RF choke +Inset-fed +1.9 +2 +Patch Antenna +3.2 +0.175 +1.6 +11.1 +Reflective +1.6 +2.64 +Network +1.1 +3.355 +1.275 +0.45 +Via +Hybrid Coupler +PIN +13 ++0.8 +0.45 +DiodeVAA +90° Hybrid Coupler +Reflective Network +Low Bias : 10000Q +00o° Bias +Low Bias +High Bias : 62 +Z1 +000 +① +? +OH +High Bias +Z2 +Z2 +High Bias +4 +PiN diode +Z1 +Low Bias +Low Bias : 0.023pF +- +High Bias : OpFthe maximum beam inclination1 increased beyond 4×4 +becomes marginal, relative to the exponentially increas- +ing size. +3.2 +Low-loss Modulator for Planar VAA +An efficient modulator directly affects the SNR of +the tag signal, which determines the tag’s detection ro- +bustness. Given Hawkeye operating throughout the 250 +MHz bandwidth in the 24 GHz, this section discusses +how it effectively performs FSK modulation without af- +fecting the retro-reflectivity. Figure 6(b) presents the +equivalent circuit representation of an antenna pair, cor- +responding to Figure 6(a). Hawkeye tag consists of 8 +such antenna pairs, each equipped with a 90◦ hybrid +coupler and a reflective network for FSK modulation +via periodic 180◦ phase shifts. As shown in Figure 6(a), +both the coupler and the reflective networks have mi- +crostrip architecture with a couple of PIN diodes. This +structure enables Hawkeye to maximize SNR by (i) low- +loss characteristic of the 90◦ hybrid coupler in combina- +tion with symmetric reflective networks, and (ii) keep- +ing the retro-reflectivity intact whilst FSK modulation +via carefully designed bias and compensation of the cou- +pling effects, avoiding leakage or distortion of the signal. +Our design collectively brings low-loss FSK modulation, +laying a solid foundation for hectometer-range support, +in combination with the unique localization algorithm +in Section 4. +Low-loss Phase Shifting using Hybrid Coupler. Hawk- +eye performs energy-efficient FSK via (i) 180◦ phase +shifts with (ii) Hybrid Coupler, minimizing the backscat- +ter reflection loss at the Hawkeye tag. As in Figure 6(b), +the phase is 180◦ flipped depending on the length of the +signal path at the reflection network, controlled by the +ON (high bias) and OFF (low bias) states of the PIN +diode. This retains the incident signal power for low- +loss modulation (cf. many state-of-the-art backscatters +modulate via SPDT switches, to toggle between absorp- +tion and reflection states [11, 39, 55] – essentially sac- +rificing the half of the interrogation signal power (dur- +ing absorption)). For retro-reflectivity, the modulated +interrogation signal flows into the other side of the an- +tenna pair. This is achieved with low-loss via a hybrid +coupler and a reflective network in Figure 6(b), perform- +ing as the impedance matched TL (details in the later +part of the section). Alternative low-loss modulator is +the switched line phase shifter that switches between +two TLs with different lengths. However, this design +requires 2× diodes than the hybrid coupler [24] to in- +duce higher power consumption and cost. +While the +switched line phase shifter has advantage in the form- +factor, this advantage is insignificant in mmWave. +1Max beam inclination for antenna array is 90◦−47.83◦× +(λ/Md)1/2 [19], where M, λ, and d are # of antenna ele- +ments, wavelength, and antenna spacing, respectively. +Figure 7: The measured response from the fabricated +modulator unit throughout the 24GHz band. (a) Loss +for low and high biases, (b) phase shift between low and +high biases. +To understand the operation of Hawkeye phase shifter, +let us refer to Figure 6(b) where the incident interro- +gation signal is received at the upper antenna (port +1⃝) and flows into the lower antenna (port 4⃝). +To +maintain the retro-reflectivity alongside FSK modula- +tion, the hybrid coupler performs low-loss signal trans- +fer from the port 1⃝ to port 4⃝, with the reflective net- +work in between. For this, the hybrid coupler acts as an +impedance-matched TL with low insertion loss [24], by +the following operation: The incident signal is divided +into four paths by the coupler, each flowing into reflec- +tive networks and bouncing off back to the coupler – i.e., +path1: +1⃝ → 2⃝ → 1⃝, path2: +1⃝ → 3⃝ → 1⃝, path3: +1⃝ → 2⃝ → 4⃝, and path4: +1⃝ → 3⃝ → 4⃝. Hybrid +coupler with four λg/4 TLs yields 180◦ phase difference +between path1 and path2, and the same phase between +path3 and path4 [49]. By keeping the signal amplitudes +in the four paths identical, path1 and path2 cancel each +other at port 1⃝. Therefore, no signal is radiated on +the upper antenna, thereby minimizing the reflection +(i.e., S11=0). On the other hand, path3 and path4 are +constructively added at port 4⃝ for maximized radia- +tion on the lower antenna (i.e., S41 = 1). In summary, +the incident signal is maximally delivered from the up- +per to the lower antenna, or equivalently, the insertion +loss is minimized when the signal amplitudes in the four +paths are the same. The path amplitude ratios are com- +puted as path1/path2 = Z2 +2 and path3/path4 = Z2/Z2 +when Z1 = Z2/ +� +Z2 +2 + 1 [49]. For the ratios of 1, we +get Z1 = 1/ +√ +2 and Z2 = 1, that is, 35.3Ω and 50Ω, +respectively. VNA measurement in Figure 7(a) shows +the low-loss of under 1.17 dB, where the modulator was +isolated from the tag (DUT in the figure) for precise +measurement. +Incorporating the Modulator and Planar VAA. For pre- +cise 180◦ phase shifts throughout the entire 250 MHz in +the 24 GHz we leverage an equivalent circuit in Fig- +ure 6(b) with the PIN diode MADP-000907-14020 rep- +resented in corresponding R, L, and C as per the datash- +eet [35]. Also, to maintain VAA retro-reflectivity while +combining with the FSK modulator, RF leakage through +the bias line (driving the PIN diodes) should be avoided +4 + +1.2 +210 +(a) +(q) +179.8° +Max. Loss=1.17dB + SSOT +180 +177° +High Blas + Measured +DUT +0.9 +Low Bias +150 +--Ideal shift +24 +24.25 +24 +24.25 +Frequency (GHz) +Fregquency (GHz)– the leakage would distort the phase and corrupt the +retro-reflectivity. Hawkeye adopts λg/4 open stub (i.e., +an open-ended microstrip line [15]) as an RF choke to +prevent RF leakage, which is represented as an induc- +tor in the equivalent circuit. The parameter values are +found through an extensive HFSS parameter sweep sim- +ulation, including the TL lengths and gaps between TLs +to compensate for the coupling effect between the mod- +ulator and the TLs. The parameter values are shown in +Figures 3 and 6(a). As in Figure 7(b), Hawkeye modu- +lator yields an accurate FSK with the maximum error +of 3◦ (i.e., [177◦, 179.8◦]), indicating only 3 × 10−3dB +power loss throughout the entire 250 MHz in the 24 GHz +band. The RF signal was effectively isolated by greater +than 30dB by the RF choke to achieve FSK modulated +retro-reflectivity. +Figure 8: +Delay profiles in an (a) office, and a (b) +hallway. +Environment delay profile demonstrates the +multipath-rich indoor scenario with the delay spanning +over 230us. Hawkeye effectively suppresses the multi- +path (i.e., delay) via retro-reflectivity. +3.3 +Indoor Multipath Suppression +Retro-reflectivity of Hawkeye effectively suppresses the +multipath, enabling robust indoor localization. Figure 8 +presents the delay profile measured from two multipath- +rich indoor settings of an office and a hallway. In both +scenarios Hawkeye significantly reduces the delay spread +and limits the multipath signal power to be 20dB or +more below the LOS signal. +This is because, under +retro-reflectivity, the received NLOS (i.e., multipaths) +signals are strictly limited to the direction to which the +signals are sent (instead of all directions without retro- +reflectivity). Thus, higher the retro-reflectivity, less the +delay spread. The multipath suppressed by Hawkeye tag +in Figure 8 induces only 8.8 mm (office) and 2 mm (hall- +way) error in Hawkeye localization (Section 4), enabling +subcentimeter indoor positioning as demonstrated in +Section 5.3. +(a) FMCW +(b) HD-FMCW [11] +Figure 9: IF of FMCW and HD-FMCW with tag FSK +at 40 Hz. HD-FMCW isolates tag signal from clutter. +4. +HAWKEYE LOCALIZATION +In this section, we describe how Hawkeye achieves +subcentimeter localization at hectometer-range, using +the commodity radar and Hawkeye tag. Essentially, it +is an extremely accurate ranging design that can be +extended to 2D/3D localization with mutliple (multi- +lateration) or single radar. +At a high level, Hawkeye +leverages the spectral leakage signature of a tag signal +to extract super-resolution range frequency. Our design +leverages a recent technique of HD-FMCW presented +in OmniScatter [11]. We first provide a brief primer on +HD-FMCW, followed by the subcentimeter localization +supporting mobile tags and large-scale simultaneous lo- +calization. +HD-FMCW Primer. A recent technique of HD-FMCW, +with a light add on signal processing on commodity +FMCW radars, effectively isolates the FSK signal from +the clutter noise in the frequency domain. Compared to +the original FMCW which uses a single-chirp symbol, +HD-FMCW leverages multiple (periodic) chirp symbol, +s(t) = c(t) ∗ �N +n=1 δ(t − nT), where c(t) is a chirp with +duration T, N is the number of chirp repetitions, and +∗ is the convolution. This interrogation signal, when +reflected from the clutter (i.e., clutter noise), simply +becomes s(t − ∆t) where ∆t is the round trip propa- +gation delay between the radar and the clutter. Since +it is simply a time-shifted version of the interrogation +signal, it maintains the period of T. This is therefore +represented as peaks on the multiples of 1 +T Hz frequency +bins in the IF signal, where all other bins in between are +left zero2. Note that, this applies to clutter noise from +all sources – i.e., all noises are concentrated on the same +set of frequency bins, leaving other bins zero. On the +contrary, the interrogation signal reflected off the tag +(i.e., the tag signal) is not only time-shifted, but also +modulated by the tag FSK. Specifically, the tag signal +is +s(t − ∆t) +� +�� +� +Interrogation signal +(period T ) +· ej2πfmt +� �� � +FSK +(period +1 +fm ) +(1) +where fm is the modulation frequency of the FSK. This +yields a new period, other than T (i.e., the least com- +mon multiple of T and +1 +fm ). Therefore, in IF, tag signal +and clutter noise are placed in different frequency bins +– isolating the tag signal from all environmental clutter +noise. The separation of tag and clutter is demonstrated +at Figure 9, where HD-FMCW (Figure 9(b)) isolates the +tag signal unlike the original FMCW (Figure 9(a)). +2From the Fourier’s Theorem [7], an arbitrary signal with +period T seconds in the time domain is represented as the +multiples of +1 +T Hz in the frequency domain. +5 + + Hawkeye Delay Profile +Environment Delay Profile (w/o Hawkeye +Amplitude +(a) +(b) +0.5 +0.5 +Normalized +0 +0 +0 +50 +150 +200 +50 +150 +200 +100 +0 +100 +Delay (ns) +Delay (ns)I Clutter +Tag Signal +30 +20 +10 +124.8 +125 +125.2 +125.4 +125.6 +Frequency (kHz)1000 +Amplitude (mV) +Clutter +Tag Signal +30 +20 +10 +124.8 +125 +125.2 +125.4 +125.6 +Frequency (kHz)Figure 10: Hawkeye one-shot localization process. The fm is determined in real time (a) to be detached from IF, +where IFFT is applied to (b) the isolated clean signal with only fr (including the spectral leakage). At (c) time +domain of the signal, (d) a subset with duration T is zero-padded, where (e) FFT is performed to reveal the envelope +sinc function. The frequency of the maximum peak amplitude is chosen as fr. +Figure 11: The time-frequency representation of tag sig- +nal at HD-FMCW. The ∆t is the round trip propaga- +tion delay between radar and tag. +4.1 +Subcentimeter-accuracy Localization +Hawkeye localization uniquely leverages the relation- +ship between the isolated tag signal and the clutter noise +in the HD-FMCW, and the spectral leakage signature +embedded in the tag signal. Figure 11 illustrates the re- +flected tag signal in relation to the interrogation signal, +where the offset between the two signals is the combi- +nation of fr and fm. The range frequency fr is from +the tag-radar propagation delay ∆t whereas fm is the +FSK modulation frequency. In other words, fr indicates +the true position of the tag and therefore, accurate lo- +calization translates to the problem of finding fr. This +is achieved in two steps: (i) Removing the effect of fm +from the relationship between the tag signal and clutter +noise, and (ii) precise estimation of fr from the spectral +leakage signature, which we discuss in the following. +Eliminating Tag Modulation fm. The first step to achiev- +ing accurate estimate of the range frequency fr is to +eliminate the effect of fm in the IF signal. +We note +that, in practice, fm is not known in advance; It needs +to be detected in real time due to the instability of +the oscillator speed. +For instance, a crystal oscilla- +tor can have 500 ppm variance under different environ- +ments [40]. This indicates 25 Hz error at 50 kHz, which +can translate to a vast amount of over 24.6 cm localiza- +tion error. Hawkeye is inherently robust to the oscillator +variance, as it detects the accurate fm on the fly, with- +out any prior knowledge on the modulation speeds or +the environment under which the tags are installed. +In order to precisely identify fm, we begin from the +fact that fr originates from the propagation delay be- +tween the radar and tag, or equivalently, s(t − ∆t) ( +Eq. 1) that has the period T. +On the contrary, fm +stems from the FSK signal of ej2πfmt with period +1 +fm . +As a result, the IF signal is represented as peaks at +frequencies with +1 +T Hz interval (from s(t − ∆t)), with +offset of fm (from ej2πfmt) as depicted in Figure 10(a). +Thus, the fm can be precisely determined in real time, +simply from the offset from +n +T Hz, (n ∈ N) frequency +bins. We note that those frequency bins hold the clut- +ter noise; Therefore, to remove fm, the clutter noise are +first nullified to zero, and then the tag signal is shifted +to the nullified frequency bins. Figure 10(b) depicts the +resulting clean signal with only fr, free of contamina- +tion from fm and clutter noise. This is a key technique +for subcentimeter localization unaffected from the tag +clock offset prevalent in practice. +Extracting Super-resolution Range Frequency fr. Hawk- +eye subcentimeter localization is achieved by accurately +identifying the fr (indicating tag-radar distance) with +boosted frequency resolution. This deviates from the +conventional HD-FMCW, where the fr is represented +with frequency resolution of 1 +T Hz – i.e., identical to the +original FMCW frequency resolution. To do so, Hawk- +eye exploits the spectral leakage in the discrete Fourier +Transform (DFT), where the DFT of a signal with the +period of T and frequency of fr is represented as peaks +at the multiples of 1 +T Hz whose envelope follows the sinc +function centered at fr – i.e., Tsinc(πT(f − fr)) [46]. +Therefore, accurately deriving fr becomes the problem +of fine-grained identification of the envelope sinc func- +tion from which the center frequency can be pinpointed. +To achieve this, Hawkeye zero-pads a subset with dura- +tion T to increase the period to Tpad (≫ T) in the time +domain, as depicted in Figures 10(c),(d). Figure 10(e) +demonstrates the zero-padding result, where the peaks +in the frequency domain are densified to precisely re- +veal the envelope sinc function. In our experiment we +set Tpad = 128T to keep the computation overhead low +while achieving the subcentimeter localization accuracy. +Given the dense peaks, the center frequency fr is simply +found as the frequency with the maximum peak ampli- +tude. +6 + +Zero-Padding +fm Elimination +IFFT +FFT +26 m +Estimated fr. +20 +20 +Clutter +Zero-padded +Tag +- Tag w/o fm +Signal +frm +5 +-Tag w/o fm +Period Tpad +10 +Hz +Hz +T +50 + 50 +Period T +５ +50 +Zero-Padding +0 +0 +0 1 +6 +5.2 +5.4 +5.6 +5.8 +5.8 +2 +6 +2 +4 +5 +5 +5.2 +5.4 +5.6 +5 +5.2 +5.4 +5.6 +5.8 +Time (ms) +Frequency (kHz) +Frequency (kHz) +Time (ms) +×10 +×10 +Frequency (kHz) +(a) +(b) +(c) +(d) +(e)Interrogation Signal +Frequency +Delayed Signal (t) +Tag Signal +At +TimeAn extensive experiment reveals Hawkeye median range +error of 2.5 mm, over ×60 improvement compared to +the original FMCW with 15 cm median error. +With +the super-resolution fr acquisition mechanism, Hawkeye +achieves subcentimeter localization up to 160 m out- +doors, and 80 m indoors. We note that entire Hawkeye +localization algorithm has the computation complexity +of O(N log N) (for FFT/IFFT) where N is the number +of samples – retaining the complexity of the original +FMCW that mandatorily runs FFT. +Figure 12: IF signal of a mobile tag. (a) The tag sig- +nal has offset of fm + fd from clutter noise, where (b) +varying fr incurs frequency dispersion. +4.2 +Mobile Tags +On the contrary to the static tags, a mobile tag in- +duces Doppler frequency fd and time-varying range fre- +quency fr(t). +Figure 12(a) illustrates the IF of the +mobile tag (↔ Figure 10(a) for static tag), where fd +is added on top of fm. The fd is effortlessly removed +together with fm by simply following the fm elimina- +tion mechanism discussed in the previous section. On +the other hand, time-varying range frequency, fr(t), +causes frequency dispersion of the peak as shown in +Figure 12(b). Hawkeye tracks fr(t) with subcentimeter +accuracy, through fine-grained temporal analysis. For +mobile localization, we begin by distinguishing the mov- +ing tags from the static ones via frequency dispersion, +proportional to the tag velocity3. For subcentimeter lo- +calization, we define mobile tags as those with > 1 cm +movement within a symbol, revealed to be the frequency +dispersion of ≥ 1.4 Hz according to our empirical study. +Then, the movement is tracked by the following. +Extracting Time-varying Range Frequency fr(t). Lo- +calization of mobile tags essentially follows the same +design principles as Section 4.1, where mobile tag signal +fm is eliminated to run IFFT (Figures 10(a)-(c)), recon- +structing the range frequency in the time domain (i.e., +fr(t)). Subsequently, each subset with duration T of +fr(t) can be zero-padded to reveal the precise location +at the corresponding time (Figures 10(d),(e)). Hawk- +eye mobile localization can be configured for balance +between time granularity and computation overhead, +by choosing the location update interval. For instance, +zero-padding can be applied on fr(t) with a 520 T inter- +val to provide 60 localization updates per second (under +3This is known as the dispersion factor [22] in radar con- +text +T = 32 µs). Our evaluations show 2.6 mm median error +for a humanoid robot with 17 cm/s speed, providing evi- +dence for mobile tag localization Hawkeye. We note that +the minor modification of Hawkeye sustains O(N log N) +computation complexity for mobile localization. +(a) IF for 5 tags +(b) Localization +Figure 13: Five tags (a) IF from a single interrogation +and (b) its localization results. +4.3 +Large-scale One-shot Localization +The lightweight localization of Hawkeye can be di- +rectly extended to large-scale, for simultaneous local- +ization of mobile and static Hawkeye tags with a single +interrogation. +For instance, localizing 100 tags takes +less than 33.2 ms end-to-end (3.2 ms interrogation + +30 ms processing time) on a mediocre desktop PC (i7- +8700, 32 GB RAM). We verify simultaneous localization +of 100 tags in Section 5.6, where each tag is identified +according to its unique modulating frequencies. The lo- +calization runs iteratively for each tag, to eliminate tag +modulation before extracting accurate range frequency. +Figure 13(a) illustrates simultaneous localization of 5 +mobile and static tags, where the tags with 200, 500, 890 +Hz modulation are mobile and 350, 770 Hz modulation +are static. Figure 13(b) depicts successful localization of +each tag, where individual tag signals are distinguished +according to the modulation frequency. We note that +Hawkeye supports up to 1024 tags under 32.8 ms in- +terrogation signal (T = 32 µs and N = 1025), which +translates to 30.5 Hz interval between each tag IDs. +The ample ID space tolerates over 500 ppm frequency +offset in low-end crystal oscillators, demonstrating the +scalability of Hawkeye in practice. The large scalability, +in combination with long-range localization and retro- +reflective tag, offers a wide-area coverage. +4.4 +Radar Setup +Multilateration. Hawkeye is capable of supporting seam- +less 2D/3D tag localization, where multiple Hawkeye +radars concurrently interrogate Hawkeye tags for multi- +lateration. Essentially, concurrent interrogation is made +possible by Hawkeye plannar VAA tag which retro-reflects +interrogation signal back to the source radar, efficiently +avoiding tag signal interference amongst radars. Hence, +Hawkeye radars can be set up for multilateration with- +out the need for access control. Furthermore, Hawkeye +radars can be time-synchronized to support 2D/3D lo- +calization for mobile tags, utilizing the Network Time +7 + + Clutter +150 + Tag +(m +ude +100 +mplitu +Frequency +50 +Dispersion +A +0 +4 +6 +8 +Frequency (kHz) +(q) +(a)0.6 +- Clutter +- Tag 1 (mobile) +(m +0.5 +- Tag 2 (static) +0.4 +e + Tag 3 (mobile) +Amplitud +Tag 4 (static) +0.3 + Tag 5 (mobile) +0.2 +0.1 +A +0 +4 +5 +6 +7 +8 +Frequency (kHz)3.558 mo +0.6 +(mV) +Tag +1 +Tag +2 +0.5 +Tag +3 +Amplitude +0.4 +Tag +4 +0.3 +Tag +5 +3.203 m +3.081 m0 +0.2 +0.1 +0 +4 +5 +6 +7 +8 +Frequency (kHz)Protocol [43]. +The protocol provides sub-millisecond +accuracy in local area networks, where a millisecond er- +ror translates to 3.6 mm localization error for a typical +human running speed of 13 kmph, sustaining subcen- +timeter accuracy. +Single Radar Localization. Single radar 2D/3D localiza- +tion can be achieved by utilizing the AoA of the MIMO +radar. Compared to multilateration, single radar local- +ization trades off accuracy for lower deployment cost +(less number of radars). +Localization error from the +AoA inaccuracy is amplified over distance. For instance, +an AoA error of 5◦ causes 8.7 m localization error at +100 m (100 m × 0.087 rad). Single radar localization is +demonstrated in Section 5.4. +Figure 14: Hawkeye tag evaluation setup. We utilize a +separate control board to provide the control signal to +tag. The tag is mounted on a linear stage with 0.01 mm +resolution. +5. +EVALUATION +This section presents the implementation details and +evaluation results of Hawkeye. +5.1 +Implementation +Hawkeye radar is implemented on Eval-TinyRad (Ana- +log Devices) [1] commodity 24GHz radar, where the op- +eration of Hawkeye localization is verified. The radars +provide the IF data to PC, where it is collected to per- +form Hawkeye localization. To deliver the control sig- +nal to Hawkeye tag, a fabricated control board with +VCXO (i.e., Voltage Controlled Crystal Oscillator) is +used with a small form factor of 26.67 mm×38.354 mm, +as depicted in Figure 14. +The board uses Skyworks +515NDAM 134200BAG [53] oscillator for an accurate +fm generation with 20 ppm variance. To control the +frequency of the VCXO, a variable resistor is combined +with a coin cell battery, where a LDO voltage regula- +tor (Toshiba TAR5SB33 [56]) is utilized to stabilize the +voltage. Arduino Uno is also implemented as a control +board for large-scale localization, where it provides a +wider range of fm (4 MHz bandwidth, 0−4 MHz) com- +pared to the VCXO (25.2 Hz bandwidth, 134.1748 − +134.2252 kHz). +Tag Power Consumption. Hawkeye tag is composed of +16 PIN diodes (Macom MADP-000907-14020), with a +separated control board for operation. The tag power +consumption is highly variant according to the control +signal voltage, as the diodes consume more power at +higher control voltage. Figure 15 provides our evalua- +Figure 15: The power consumption (excluding the con- +trol board power) is plotted versus the SNR at 2.4 m +tag-radar distance with 6.45 dBm transmit power. The +power consumption ranges from 6.4 µW to 7.68 mW. +(a) +(b) +Figure 16: For the ground truth of localization, we uti- +lize a (a) laser distance meter with 1 mm resolution +mounted on a tripod, then (b) mount the radar on the +tripod for evaluation. +tion on the power consumption vs. SNR at 2.4 m tag- +radar distance with 6.45 dBm interrogation signal. The +power consumption is calculated utilizing the diode IV +data [34]. The results demonstrate the control signal +voltage of 0.9 V (i.e., tag operating at 6.4 µW power) +is sufficient for the operation of Hawkeye tag at 2.4 m +distance with over 20 dB SNR. Meanwhile, the control +board power consumption is analyzed by simulating an +IC using the Libero SoC SmartPower [41]. The simula- +tion consists of a ring oscillator and a modulator circuit, +where the power consumption results in 2 µW. Thus, +Hawkeye tag can run with 8.4 µW power, which can be +easily operated by energy harvesting [63], or with a coin +cell battery of 1000 mAh for 40.7 years. +5.2 +Evaluation settings +By default, we conduct evaluations using Analog De- +vices Eval-Tinyrad as Hawkeye radar, with the specific +radar parameters set as follows: bandwidth 250 MHz +(24 GHz to 24.25 GHz), transmit power 8 dBm, IF +sampling frequency 1 MHz, and 8192 samples per chirp +with 2048 chirps for interrogation signal. We use a sin- +gle transmit antenna and a single receive antenna for +omni-directional Hawkeye operation, unless otherwise +mentioned. We note that the PCB fabrication and sol- +dering error may result in separate distance offset per +tag, which we calibrate by measuring the distance offset +of a tag with known distance. In all evaluations, Hawk- +eye tag is mounted on a acrylic plate to avoid unneces- +sary electric coupling. The Arduino Uno is utilized for +our control board, which supplies 1.3 V control voltage +to Hawkeye tag. +8 + +Tag +Hawkeye Tag +吧 +66 mm +Control Board +Ctrl. +26.67 mm +品 +Sig. +Linear +Stage +38.354 mm +66 mm40 +1.4 +Voltage ( +SNR (dB) +35 +1.25 +1.1 +SNR +Ctrl. Sig +25 +0.95 +Voltage +0.8 +20 +101 +102 +103 +104 +Power Consumption (uW)Tag +LaserDistance MeterTag +Radar(a) The tag deployment at Hawkeye 1D localization ex- +periment, performed over 180 m at every 20 m. +(b) Box plot of Hawkeye 1D localization result. +(c) Detection rate of Hawkeye. +Hawkeye tags +achieve stable localization up to 160 m. +Figure 17: The 1D localization evaluation setup and +performance. +Ground Truth. In order to obtain precise ground truth +of localization, laser distance meter with 1 mm resolu- +tion up to 200 meters range (Leica DISTO D510 [27]) +is utilized at radar side, as depicted in Figure 16. The +tag-radar distance is first measured on the laser distance +meter mounted on a tripod, where subsequently Hawk- +eye radar is mounted on the same tripod for evaluation. +At tag side, a linear stage of 0.01 mm resolution up to +150 mm range (Soar STMX1020-D [54]) is utilized as +shown in Figure 14. Hawkeye tag is mounted on the +linear stage, where the linear stage relocates the tag +for evaluation. The ground truth error caused by the +laser distance meter does not exceed 1 mm. For mobile +tag evaluation, we utilize OptiTrack PrimeX [47] with +0.2 mm 3D accuracy for ground truth. +5.3 +1D Localization +To verify Hawkeye’s subcentimeter accuracy at hec- +tometer range, a 1D localization experiment is conducted +at a soccer field, where the measurement is conducted +up to 180 meters in a straight line. +Hawkeye tag is +located at every 20 meters as shown in Figure 17(a), +where a total of 100 experimental trials are performed +at 20 different locations within each 20 m position. Fig- +ure 17(b) demonstrates Hawkeye 1D localization perfor- +mance, where the edge of the box indicates the 75th and +25th percentile error, while the whiskers indicate the +90th and 10th percentile error. Data outside the 90th +and 10th percentile error is considered as outliers, which +are marked as red dots outside the whiskers. Hawkeye +achieves subcentimeter accuracy with 90th percentile er- +ror up to 100 m, where the error is 8.9 mm. Further- +more, the subcentimeter accuracy is sustained up to +160 m with 50th percentile (i.e., median) error, where +the median error is 6.7 mm at 160 m, demonstrating +successful hectometer range subcentimeter localization. +For control signal, Arduino Uno produces 150 kHz fm. +Figure 17(c) shows the detection rate of Hawkeye, where +we achieve 100 % detection rate up to 140 m, which is +decreased to 96 % and 54 % at 160 m and 180 m. The +subcentimeter accuracy at hectometer range proves the +robustness of Hawkeye tag, in combination with the ef- +ficiency of Hawkeye localization. +Figure 18: +(a) Indoor hallway experiment setup of +Hawkeye, where the tag is placed at every 20 m from +the radar. (b) Box plot of hallway experiment results. +Indoor 1D Localization. We further evaluate 1D local- +ization at multipath rich hallway to verify Hawkeye op- +eration in indoors. As depicted in Figure 18(a), the hall- +way experiment is conducted up to 80 m in a straight +line. The tag is located at every 20 m, where 50 ex- +perimental trials are conducted at 10 different locations +within each 20 m positions. Figure 18(b) plots the box +plot of the indoor 1D localization. The median error +stays below 4 mm throughout 80 meters, validating +Hawkeye’s subcentimeter localization even in indoors. +The detection rate stayed at 100 % up to the 80 m +range. +This results totally prove the multipath sup- +pression of Hawkeye tag. +Figure 19: Experiment setup for testing the effect of +blockage in the indoor environment (a) without block- +age (LOS) (b) with cardboard (c) with plywood. +LOS +Cardboard +Plywood +Median (mm) +2.2 +3.4 +5.7 +90th percentile (mm) +5.9 +5.7 +11.9 +Table 2: 1D localization accuracy under blockages. +Robustness Against Blockage and Temperature Change. +Localization for IoT applications may face various envi- +ronment changes during practical use. In order to sub- +stantiate Hawkeye operation for pervasive deployment, +we analyze the 1D localization performance impact of +9 + +Tag Locations +Radar +口 +口 +口 +口 +口 +180m +160m +140 m +120m +100m +80 m +60 m +40m +20m(mm +40 +goth percentile +Median +30 +sub-cm error +sub-cm error +Error +T +20 +： +Range +10 ++ +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Distance(m)100 +96 +Detection rate +50 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Distance (m)Error (mm) +: +15 +Tag Locations +10 +Radar +Range +5 +FTL +60 m +80 m +20 m +40 m +40 +60 +20 +80 +(b) +(a) +Distance (m)(a) +(b) +(c) +Radar +3 +Plywood +Cardboard +TagFigure 20: Experiment setup for testing the effect of +temperature at indoor environment with (a) ice pack +(b) hand warmer. +(c) Hawkeye effectively eliminates +the effect of the instability of oscillator. +two common error sources – blockages and tempera- +ture variance. +The experiments are conducted in an +indoor concert hall, where the tag-radar distance is set +to 2.2 m. +For each experiment, total 40 experimen- +tal trials are conducted at 4 different locations, with +Arduino Uno as the control board. +Figure 19 shows +the setup for the blockage experiments, where card- +board and plywood of 3.25 mm and 5 mm thickness +is utilized as blockage materials. The experiment re- +sults are summarized at Table 2, where the cardboard +and plywood blockage increased the median localization +error by 1.2 mm and 3.5 mm each. The results show +that Hawkeye sustains subcentimeter accuracy even un- +der NLOS, demonstrating our robustness to the block- +ages. Localization performance under varying temper- +ature (i.e., varying fm due to the instability of oscil- +lator) is also evaluated, to show Hawkeye’s ability to +eliminate the effect of fm for precise localization. As +shown in Figure 20, the Arduino oscillator temperature +is set to 9.54◦C, 23.7◦C and 38.43◦C, for evaluation. +At each temperature, the fm varied from 150.901 kHz +to 150.949 kHz, showing high frequency instability of +the control board. To control the temperature of the +oscillator, the control board was either surrounded by +ice pack or hand warmers, as shown in Figures 20(a) +and (b). Figure 20(c) compares the localization result +with and without Hawkeye fm elimination. +Without +the fm elimination, the error induced by the tempera- +ture at 9.54◦C and 38.43◦C is 58.9 mm and 176.8 mm +each. +Contrarily, with Hawkeye’s fm elimination ap- +plied, the error induced by the temperature change at +9.54◦C and 38.43◦C stays under 4 mm. Altogether, the +results prove the performance of Hawkeye at diverse en- +vironments. +5.4 +2D Localization +We evaluate hectometer range 2D localization at soc- +cer field with track, utilizing two radars and a single tag +for multilateration. As demonstrated at Figure 21(a), +the two radars are located at 100 m distance from the +tag, where the radar interrogates the tag with 30◦ inci- +dence angle. The distance between the radars is 100 m, +Figure 21: (a) The experimental setup of 2D localiza- +tion experiment. The tag and two radars form an equi- +lateral triangle with a side of 100 m. (b) The CDF of +Hawkeye 2D localization error. +and the tag modulation fm is 150 kHz. +The exper- +iment consists of total 400 experimental trials, where +the tag localization is conducted at 20 positions on the +linear stage moving towards positive y direction. Fig- +ure 21(b) plots the CDF of 2D localization error, where +Hawkeye achieves subcentimeter median error of 8.7 mm +and 90th percentile accuracy of 15.9 mm in 2D. Median +error along x and y dimensions are 6.2 mm and 4 mm, +while the 90th percentile error is 13.6 mm and 9.8 mm +each. +The 2D localization, which inherently requires +retro-reflectivity, is made possible by Hawkeye tag’s high +retro-reflectivity. The subcentimeter 2D localization at +hectometer range demonstrates the capability of Hawk- +eye at practical applications. +Figure 22: The CDF of Hawkeye 2D localization error +with single radar. +Single Radar 2D Localization. +Hawkeye 2D localiza- +tion on a single radar utilizing AoA is evaluated. The +evaluation is conducted at an open field with localiza- +tion distance ranging from 20 to 25 m and AoA ranging +from −60◦ to 60◦, as depicted in Figure 22(a). For each +location, a total of 30 experimental trials are performed +at 6 different locations. Figure 22(b) and (c) show the +CDF of distance and angle error, where the median er- +rors are 2.7 mm and 4.7◦, and the 90th percentile errors +are 9.04 mm and 8.4◦. Collectively, the median 2D er- +ror results in 2.08 m, whose accuracy can be improved +with a larger antenna array. +5.5 +3D Localization +In order to verify the 3D localization performance +of Hawkeye, a localization experiment is conducted in +an indoor concert hall with three radars fixed to a wall. +The radar positions are described in Figure 23(a), where +the three radars are located at (1.8, 1.25, -1.2), (1.8, - +1.25, -1.2) and (0, -1.25, -1.2) coordinates, assuming +10 + +Hawkeye Range +(a) +Tag +58.9mm +225 +Range w.o. f.. elimination +Ice pack +150.95 +lOscillatorFrequency +(zH +220 +150.94 +(cm +Arduino +150.93 +176.8mm +215 +Range +150.92 +llator +(b) +210 +Tag +Hand +150.91 +warmer +205 +150.9 +(c) +200 +Arduino +9.5 +23.7 +38.4 +Oscillator Temperature (°C)(a) +(b) +TagLocations +x +0.75 +DF +X +0.5 +Y +m +0.25 +2D +D +0 +0.1 +10 +100 +1 +■ +Radar +Radar +Localization Error (mm)(b) +1 +(a) +Tag Locations +0.75 +CDI +0.5 +25m +0.25 +20m +0 +25m +0.1 +10 +100 +(c) +Distance Error (mm) +1 +0.75 +30° +CDF +0.5 +25m +0.25 +Radar +0 +20m +0 +10 +20 +30 +40 +50 +60 +Angle Error (°)(a) 3D setup. +(b) Localization result. +Figure 23: (a) Experiment setup of 3D localization ex- +periment, conducted within a indoor concert hall. (b) +An exemplary localization result at four tag locations, +where the estimated points within the median error are +plotted. +Figure 24: The CDF of 3D localization error. +(0, 0, 0) is the tag center. Tag control signal of 1.15 V +is fed with a signal generator [21] for 3D localization +experiment. Total 1840 experimental trials at 23 po- +sitions in xz-plane are conducted utilizing the linear +stage. Figure 24 demonstrates the CDF of 3D local- +ization error, presenting subcentimeter median error of +8.9 mm and 90th percentile error of 13.8 mm. For each +x, y, z dimensions, the median error is 5.9 mm, 2.7 mm, +3.4 mm and 90th percentile error is 11.2 mm, 8.5 mm, +8.8 mm each. Figure 23(b) depicts an exemplary lo- +calization result at four tag locations, where estimated +points within the median error are plotted in 3D. The +successful subcentimeter 3D localization in indoor space +proves the retro-reflective performance of Hawkeye tag +in both azimuth and elevation plane, while establish- +ing a solid foundation on the practicality of Hawkeye +localization. +Simultaneous Localization of Mobile Tags. We evalu- +ate Hawkeye’s ability to simultaneously localize mobile +tags by attaching five Hawkeye tags to the body cen- +ter, both legs and both arms of a humanoid robot [14]. +Each tag concurrently modulates with unique fm be- +tween 150 kHz and 151 kHz. The experiment is con- +ducted in an indoor concert hall with the same settings +as Figure 23(a). The robot has dimension of 48×36cm, +operating with 16 servo motors. The radar is set to have +1024 samples per chirp with 2048 chirps for interroga- +tion signal. As depicted in Figure 25, three different +actions of lift arms, sit down and spread legs are cap- +tured with the maximum moving speed of 17cm/s. A +total of 90 experimental trials are conducted per action, +Figure 25: Robot movements and the corresponding lo- +calization results. +Figure 26: The CDF of 3D localization error from mov- +ing robot. +with three different robot positions. As depicted, the +three robot actions are captured with 23.5 localization +FPS, where a ground truth measured by OptiTrack is +provided together. The CDF is presented in Figure 26, +where the median error of 3D localization is 7.66 mm +and 90th percentile error is 15.96 mm. For each x, y, z +dimension, the median error is 4.05 mm, 3.53 mm, +2.59 mm and 90th percentile error is 11.13 mm, 9.07 mm, +8.31 mm each. This verifies Hawkeye’s one-shot local- +ization of mobile tags in practice. +5.6 +Large-scale Localization +To verify the large-scale support of Hawkeye, a si- +multaneous, 3D localization experiment consisting of +100 tags is conducted in an indoor environment. Fig- +ure 28(a) depicts the arrangement of 100 tags, where +they are deployed as 10 by 10 on a acrylic board. The +tags are densely deployed with intervals of 5 mm to +demonstrate operation in harsh environments (e.g., items +stacked up in the warehouse) where substantial cou- +pling between tags exists. The 100 tags concurrently +modulates with unique fm in between 100 kHz and 250 +kHz. For 3D localization, three radars are attached to a +wall as depicted in Figure 29. Each radars are located +at (3, 1.3, −3), (−3, 1.3, −3) and (0, −1.2, −3) coordi- +nates, assuming (0, 0, 0) is the 100 tags center. +Fig- +11 + +y +Radar +Radar +.... +2.5m +2.5m +Tag +Linear +Stage +Radar +1.8m +1.8m +1.2mGround Truth +50 +Hawkeye +40 +(mm) +30 +20 +N +10 +20 +0 +10 +0 +-20 +-40 +-60 +-80 +0 +-100 +y(mm) +x (mm)1 +0.75 +X +DF +0.5 +Y +Z +0.25 +3D +0 +0.1 +1 +10 +100 +Localization Error (mm)cm +6 +3-0.2 +0.2 +-0.2 ++ +(length, +-0.1 + +0.1 +0 +.0 +0 +m) +m) +0.1 +0.2 +.0.2 +0.2 +0.2 +0.2 +0.2 +(height, +(height, +(height, +0 +0 +0 +-0.1V +-0.1y +Ground Truth +Robot Body +-0.2 +-0.2 +Right Arm +0 +0 +Left Arm +0.1 +0.1 +0.2 +0.2 +Right Leg +(a) Lift arms +(b) Sit down +z (width, m) +(C) Spread legs + Left Leg1 +0.75 +CDF +X +0.5 +Y +0.25 +3D +0 +0.1 +1 +10 +100 +Absolute Error (mm)Figure 27: Sinc envelope of 100 tags during the large-scale simultaneous localization experiment. +Figure 28: (a) The localization result of 100 tags de- +ployed 10 by 10 on a acrylic board projected on the +experiment photo, and (b) the accurate localization re- +sult of Hawkeye at scale. +Figure 29: Experiment setup of large-scale simultaneous +localization. The experiment is conducted indoors, with +5 mm spacing between the 100 tags. +Figure 30: The CDF of large-scale localization error at +Hawkeye. +ure 27 demonstrates the 100 tags localization process- +ing result, where the successful localization of each tags +resulting in 100 different envelope sincs are visible. Fig- +ure 28(b) depicts the the localization result of Hawkeye +at scale, where all estimation results are shown as red +cross around the ground truth. The CDF of large-scale +3D localization error is shown at Figure 30, showing +median error of 19.9 mm and 90th percentile error of +48.4 mm. For each x, y, z dimension, the median error +is 9.8 mm, 16.1 mm, 6.4 mm, and 90th percentile error +is 20.5 mm, 41.1 mm, 15.5 mm each. The concurrent +localization of 100 tags verifies Hawkeye scalability at +realistic settings. +6. +RELATED WORK +mmWave Systems and Sensing. mmWave systems have +been proposed to exploit the large bandwidth [26, 31, +64, 12]. +mmWave radars also utilize extensive band- +width to achieve higher sensing accuracy [23, 9, 52, 28]. +Backscatter. +Backscatter offers extremely low-power +signaling for power constrained scenarios [10, 16, 67, +36]. Backscatter system also facilitates low-power sens- +ing, including RFID-based approaches [6, 59, 44, 30]. +Backscatter/RFID Localization. As one of the repre- +sentative implementation of backscatter systems, RFID- +based localization techniques have long been discussed +in the community [18, 20, 51, 57, 61]. Initial studies +measure the amplitude, phase and the angle of arrival +of the received signal [2, 3, 29, 65, 25, 66], which suffered +from multi-path effects. Recent studies still suffer from +low accuracy in practical scenarios due to limited band- +width of RFID, including [13, 58, 48, 60, 62] which re- +quire knowledge on motion or reference tags to mitigate +multi-path effects. While [33] emulates large bandwidth +to achieve sub-centimeter accuracy, range is limited to +room-scale scenarios to comply with FCC regulations. +[55] utilize large bandwidth of mmWave FMCW radar +to achieve 100 m range, but is still essentially limited +to FMCW range resolution. +7. +CONCLUSION +This paper presents Hawkeye, a mmWave backscat- +ter localization that can achieve subcentimeter median +accuracy at hectometer range, while using an afford- +able commodity radar (∼200 USD [17]) for the reader. +Hawkeye simultaneously localizes 100 tags in only 33.2 ms, +and is capable of supporting up to 1024 tags in the- +ory. Our design consists of (i) a new planar Van Atta +Array tag that retro-reflects in both azimuth and ele- +vation, combined with a low-loss FSK modulator, and +(ii) a novel localization algorithm that achieves ×60 the +localization performance of FMCW radar, while being +immune to the tag oscillator frequency offset. Collec- +tively, Hawkeye take a solid step towards bringing per- +vasive tag deployment and localization to practice. +12 + +1 +0 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +Frequency (kHz) +2.95 +3.44 +3.93 +4.42 +4.91 +5.40 +5.89 +Distance (m)Ground Truth +Ground Truth +Hawkeye +Hawkeye +40 +20 +(cm) +0 +-20 +280 +-20 +0 +300 +20 +40 +320 +x (cm) +z (cm) +(a) +(b)Radar +y +100 Tags +Radar +2.5m +2.5m +3ml +Radar +3m0.75 +X +CDF +0.5 +Z +0.25 +3D +0 +0.1 +1 +10 +100 +Localization Error (mm)8. +REFERENCES +[1] Analog Devices. EVAL-TINYRAD. +https://www.analog.com/en/design- +center/evaluation-hardware-and- +software/evaluation-boards-kits/eval- +tinyrad.html. +[2] D. Arnitz, K. 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In +Proceedings of the 20th Annual International +Conference on Mobile Systems, Applications and +Services, pages 330–342, 2022. +15 + diff --git a/7tE0T4oBgHgl3EQffQCV/content/tmp_files/load_file.txt b/7tE0T4oBgHgl3EQffQCV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fed1b7b7902f3ce97342414272b7b3c5f3d8ed54 --- /dev/null +++ b/7tE0T4oBgHgl3EQffQCV/content/tmp_files/load_file.txt @@ -0,0 +1,1350 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf,len=1349 +page_content='Hawkeye: Hectometer-range Subcentimeter Localization for Large-scale mmWave Backscatter Kang Min Bae†, Hankyeol Moon†, Sung-Min Sohn§, Song Min Kim¶ Korea Advanced Institute of Science and Technology (KAIST) , §Arizona State University {bkm2259, moonkyul1, songmin}@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='kr, smsohn@asu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='edu ABSTRACT Accurate localization of a large number of objects over a wide area is one of the keys to the pervasive interaction with the Internet of Things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This paper presents Hawkeye, a new mmWave backscatter that, for the first time, offers over (i) hundred-scale simultaneous 3D localization at (ii) subcentimeter accuracy for over an (iii) hectometer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye generally applies to indoors and outdoors as well as under mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag’s Van Atta Array design with retro-reflectivity in both elevation and azimuth planes offers 3D localization and effectively suppresses the multi- path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye localization algorithm is a lightweight signal processing compatible with the commodity FMCW radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' It uniquely leverages the interplay between the tag signal and clutter, and leverages the spetral leakage for fine-grained po- sitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Prototype evaluations in corridor, lecture room, and soccer field reveal 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm median accuracy at 160 m range, and simultaneously localizes 100 tags in only 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye is reliable under temperature change with signifi- cant oscillator frequency offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' INTRODUCTION Precise interaction with a large number of objects spread over a region has long been a vision for the IoT, where accurate localization is one of the essen- tial features for an immersive experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Backscatter possess great potential towards this goal, with the low- cost and ultra low-power tags that can be massively de- ployed over a large area with the minimum deployment cost and maintenance efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Localization of large-scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', hundreds to thousands) tags with subcentimeter accuracy installed over an area (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', hectometer-range) would offer benefits to a wide range of applications in- cluding asset tracking, inventory management, ware- house automation, smart factories, virtual/augmented reality, and structural health monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To this end, backscatter (including RFID) localiza- tion has been extensively studied in sub-6GHz bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' However, their accuracy, scalability, and range are fun- damentally throttled by the hard bandwidth constraint †Co-primary Student Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' ¶Song Min Kim is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 1: Hawkeye is tested under large-scale (left), over a long range (center), and on mobile objects (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', tens of KHz for UHF RFID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This limits their performance to tens of cm accuracy [4, 8, 37, 45, 65], restricts the deployment scenarios by requiring fixed movement trajectories [42, 50, 60, 62], or requires de- ploying dense reference tags with prior knowledge [5, 13, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' A recent line of research, RFind [33] and Tur- boTrack [32], resolve the bandwidth issue by enabling RFID to emulate the wide bandwidth of 220 MHz that extends beyond the ISM band, to achieve subcentime- ter accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' However, the range is bounded to sev- eral meters to remain compliant to the FCC regulations, limiting the usage to a room-scale and requiring a cus- tomized reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The latest work of millimetro [55] takes a fundamental approach of exploring the rich, 250 MHz bandwidth in the 24 GHz mmWave band, by utilizing FMCW radar and backscatter tag to reach over 100 m range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' However, the median accuracy of millimetro is limited to 15 cm, which is essentially the accuracy of 24 GHz FMCW radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This paper presents Hawkeye (Figure 1), a mmWave backscatter localization with the empirical performance of (i) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm median accuracy (ii) at 160 m range (@1D), (iii) simultaneously localizes 100 tags in only 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 ms (scales up to 1024 tags in theory), and (iv) uses an affordable commodity radar (∼200 USD [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye blends a new backscatter tag for efficient sig- nal delivery and lightweight radar-side signal processing for accurate and rapid localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye backscat- ter tag is a planar Van Atta Array (VAA) combined 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='02402v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='SP] 6 Jan 2023 Large Scale Long Range Sub-cm (100 Tag) (160m) &Mobile 1153Systems Accuracy @ 5 m Range Bandwidth Simultaneous Localization Fixed Trajectory Hawkeye 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm 180 m 250 MHz 100 Tags (1024 in theory) No Millimetro [55] 78 mm 180 m 250 MHz 6 Tags (106 in theory) No RFind [33] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 mm 6 m 220 MHz No No TurboTrack [32] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 mm 10 m 100 MHz 2 Tags No Tagoram [60] 10 mm 12 m 6 MHz No Yes Table 1: Comparison with the state-of-the-arts with a power-efficient low-loss FSK modulator using hy- brid coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag retro-reflects in both azimuth (90◦ FoV) and elevation (140◦ FoV) to enable 3D local- ization and effectively suppresses multipath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The de- sign is robust against oscillator frequency offset, with only 4 mm localization error across low (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='45◦C), high (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='43◦C), and room (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7◦C) temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is an essential property for practical subcentimeter localiza- tion under disparate deployment scenarios, using low- end tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Furthermore, one-shot interrogation localizes up to 1024 tags (evaluated with 100) simultaneously, supporting large-scale rapid localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Table 1 sum- marizes the comparison to the state-of-the-art backscat- ter localization systems, showcasing that Hawkeye is uniquely positioned to achieve high scalability, long range, and precision at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye exploits the interplay between Hawkeye back- scatter FSK signal and the chirp-based Frequency Mod- ulated Continuous Wave (FMCW) radar to improve the localization performance by over ×60 over using the FMCW alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag is tuned to demonstrate S11 of -10 dB throughout the entire 250 MHz bandwidth, where FSK modulation is performed by the combina- tion of reflective network and low-loss 90◦ hybrid cou- pler co-optimized for efficient VAA reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The use of the VAA, along with the severe signal attenuation of the mmWave, naturally suppresses the multipath intef- erence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To the best of our knowledge, Hawkeye tag is the first planar VAA mmWave backscatter design with modulation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The radar-side subcentimeter localization is designed as a lightweight post-processing on top of FMCW demodulation, without requiring any change to the commodity FMCW radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Specifically, Hawkeye localization algorithm is built on the recent technique of HD-FMCW [11] that isolates the tag FSK signal from the clutter noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The key insight of Hawk- eye is to leverage the relationship between the tag sig- nal and the clutter, and the spectral leakage signature embedded in the tag signal, from which the precise loca- tion can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye supports single radar or multilateration positioning, where 1D-3D localization was evaluated throughout indoors (corridors and lec- ture rooms), outdoors (soccer field), NLOS, and varying temperatures to demonstrate practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye is an accurate, long-range, and large-scale lo- calization for mmWave backscatter, essentially enabling tracking many objects spread over an area, ranging from everyday spaces like homes and offices, to industrial sec- tors such as inventories and warehouses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye is kept economic with low-cost tags and compatibility to affordable commodity radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To sum up, we believe Hawkeye takes a solid step towards bringing pervasive tag deployment and localization to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Our con- tribution is three-fold: We design Hawkeye, a mmWave backscatter-based subcentimeter localization that works over hecto- meter-range and simultaneously localizes over a thousand tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To the best of our knowledge, we design the first planar VAA mmWave backscatter with modula- tion capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We prototype Hawkeye tags on Rogers RO4003C substrate for antenna with planar VAA structure, VXCO-based control board on the PCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hundred tags were produced for large-scale simultaneous 3D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Readers were implemented on com- modity 24GHz radars [1, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We will release Hawkeye’s source code and HFSS tag design file upon acceptance, for facilitating community’s future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' BACKGROUND This section provides the technical background for Hawkeye, followed by the design overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' FMCW Radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' An FMCW radar leverages chirp, whose frequency linearly increases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Objects in the radar’s vicinity reflect the transmitted chirp, which re- turns to the radar with a round-trip propagation delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' FMCW radar mixes the transmitted chirp with the re- ceived chirp (with propagation delay) to produce an IF signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The range is measured by performing FFT on the IF signal, where each reflected object is represented as a signal with range frequency fr proportional to the propagation delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Planar Van Atta Array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Planar Van Atta array (VAA) passively reflects back the signal to the direction of ar- rival, achieving retro-reflectivity in both azimuth and elevation planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' It is a simple antenna array structure where centrosymmetric pairs of the antenna in 2D plane are interconnected by transmission lines (TLs) with a length difference equal to λg (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', guided wavelength, which is the wavelength of EM wave in the dielectric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 2 Figure 2: Planar Van Atta array The centrosymmetrically intercon- nected antenna pair flips the inci- dent signal’s phase sequence, which directs the signal to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The phase induced by the TLs does not affect the radiation direction, be- cause it is applied equally to all lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As an illustrative example in Figure 2, consider a 2×2 planar VAA, where the signal comes in the azimuth plane with the phase sequence of [−2ϕ, −ϕ] at both [A,B] and [B’,A’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' After the propaga- tion in TL, the phase sequence is inverted and produces a reflected signal with a phase sequence of [−ϕ, −2ϕ] at both [A,B] and [B’,A’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This makes the reflected wave back to the incident angle achieving retro-reflectivity in the azimuth plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Retro-reflectivity in the elevation plane is also achieved in the same manner, by flipping the phase sequence at [A,B’] and [B, A’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 3: Hawkeye tag with key geometrical parameters optimized for the 24 GHz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Centrosymmetric an- tenna pairs are interconnected by TLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' All dimensions are in mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) (b) Figure 4: Comparison of measured normalized mono- static RCS of Hawkeye tag and equal-sized flat plate with the same substrate (a) in azimuth plane and (b) in elevation plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In each cases, we achieve over 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 dB and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 dB beamforming gain over the flat plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' HAWKEYE TAG DESIGN Figure 3 presents Hawkeye tag prototype, which uniq- uely blends a 4×4 planar Van Atta array with FSK modulation which serves as a basis to long-range sub- centimeter 3D localization with effective multipath sup- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In the following, we present detailed structure and design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 Retro-reflectivity via Planar VAA As depicted in Figure 4, Hawkeye VAA achieves retro- reflectivity with an average beamforming gain of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 dB with FoV (10dB beamwidth) of 90◦ and 140◦ in az- imuth and elevation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To meet the VAA condition, centrosymmetric antenna pairs are intercon- nected via TLs, whose lengths are based on the guided wavelength of λg = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='56 mm, derived from the 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='125 GHz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', the center frequency of 24GHz band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Specif- ically, the length differences of TLs are multiples of λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To limit the phase misalignment within the 250MHz bandwidth, the difference between the maximum and the minimum TL lengths is capped to 9λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This trans- lates to the maximum phase misalignment of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9◦, or equivalently, beamforming power loss of only 5×10−2dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) S11 (b) Smith chart Figure 5: Measured antenna performance of Hawkeye tag, with (a) S11 and (b) Smith chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) (b) Figure 6: (a) Magnified view of Hawkeye tag with key geometrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' All dimensions are in mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) Equivalent circuit of one pair of antenna, equipped with a 90◦ hybrid coupler and a reflective network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag adopts inset-fed rectangular microstrip patch antenna with the patch, edge notch depth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', the antenna feeding point [38]), and TL dimensions op- timized to 50 Ω impedance matching at 24GHz, through HFSS parametric sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 5 demonstrates the antenna performance measured via VNA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 5(a) depicts the antenna response (S11) of −10 dB through- out the 24 GHz band, and Figure 5(b) shows the cor- responding impedance matching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The parame- ters are shown in Figure 6(a) on Rogers RO4003C sub- strate (dielectric constant ϵr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='55, dissipation factor tanδ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 × 10−3), where the top layer holds Hawkeye tag circuit while bottom layer is ground plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Lastly, 4×4 antenna elements are chosen to balance between the FoV of the retro-reflectivity and the tag size – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=", 3 Reflected wave Incident wave A 29 B B A' 2Φ 11-2966 L118." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='856 5111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='341 L103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='826 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='311 66 5 L52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='280 15 Biaspin L59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='781 L67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='281 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7810 Normalized RCS (dB) 20 Avg 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2dB 40 60 Hawkeye Flat Plate 90 60 30 0 30 60 90 Azimuth (degrees)0 Normalized RCS (dB) 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 Avg 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9dB 60 Hawkeye Flat Plate 90 60 30 0 30 60 90 Elevation (degrees0 10 0 -20 30 40 23 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 24 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 25 Frequency (GHz)j50 j25 j100 j250 j10 U 0 8 10 2550 100250 j250 j10 j100 j25 j504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 4 RF choke Inset-fed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 2 Patch Antenna 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 Reflective 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='64 Network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='355 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='45 Via Hybrid Coupler PIN 13 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='45 DiodeVAA 90° Hybrid Coupler Reflective Network Low Bias : 10000Q 00o° Bias Low Bias High Bias : 62 Z1 000 ① ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' OH High Bias Z2 Z2 High Bias 4 PiN diode Z1 Low Bias Low Bias : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='023pF High Bias : OpFthe maximum beam inclination1 increased beyond 4×4 becomes marginal, relative to the exponentially increas- ing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Low-loss Modulator for Planar VAA An efficient modulator directly affects the SNR of the tag signal, which determines the tag’s detection ro- bustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Given Hawkeye operating throughout the 250 MHz bandwidth in the 24 GHz, this section discusses how it effectively performs FSK modulation without af- fecting the retro-reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 6(b) presents the equivalent circuit representation of an antenna pair, cor- responding to Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag consists of 8 such antenna pairs, each equipped with a 90◦ hybrid coupler and a reflective network for FSK modulation via periodic 180◦ phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As shown in Figure 6(a), both the coupler and the reflective networks have mi- crostrip architecture with a couple of PIN diodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This structure enables Hawkeye to maximize SNR by (i) low- loss characteristic of the 90◦ hybrid coupler in combina- tion with symmetric reflective networks, and (ii) keep- ing the retro-reflectivity intact whilst FSK modulation via carefully designed bias and compensation of the cou- pling effects, avoiding leakage or distortion of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Our design collectively brings low-loss FSK modulation, laying a solid foundation for hectometer-range support, in combination with the unique localization algorithm in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Low-loss Phase Shifting using Hybrid Coupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawk- eye performs energy-efficient FSK via (i) 180◦ phase shifts with (ii) Hybrid Coupler, minimizing the backscat- ter reflection loss at the Hawkeye tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As in Figure 6(b), the phase is 180◦ flipped depending on the length of the signal path at the reflection network, controlled by the ON (high bias) and OFF (low bias) states of the PIN diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This retains the incident signal power for low- loss modulation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' many state-of-the-art backscatters modulate via SPDT switches, to toggle between absorp- tion and reflection states [11, 39, 55] – essentially sac- rificing the half of the interrogation signal power (dur- ing absorption)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For retro-reflectivity, the modulated interrogation signal flows into the other side of the an- tenna pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is achieved with low-loss via a hybrid coupler and a reflective network in Figure 6(b), perform- ing as the impedance matched TL (details in the later part of the section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Alternative low-loss modulator is the switched line phase shifter that switches between two TLs with different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' However, this design requires 2× diodes than the hybrid coupler [24] to in- duce higher power consumption and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' While the switched line phase shifter has advantage in the form- factor, this advantage is insignificant in mmWave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 1Max beam inclination for antenna array is 90◦−47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='83◦× (λ/Md)1/2 [19], where M, λ, and d are # of antenna ele- ments, wavelength, and antenna spacing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 7: The measured response from the fabricated modulator unit throughout the 24GHz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) Loss for low and high biases, (b) phase shift between low and high biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To understand the operation of Hawkeye phase shifter, let us refer to Figure 6(b) where the incident interro- gation signal is received at the upper antenna (port 1⃝) and flows into the lower antenna (port 4⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To maintain the retro-reflectivity alongside FSK modula- tion, the hybrid coupler performs low-loss signal trans- fer from the port 1⃝ to port 4⃝, with the reflective net- work in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For this, the hybrid coupler acts as an impedance-matched TL with low insertion loss [24], by the following operation: The incident signal is divided into four paths by the coupler, each flowing into reflec- tive networks and bouncing off back to the coupler – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', path1: 1⃝ → 2⃝ → 1⃝, path2: 1⃝ → 3⃝ → 1⃝, path3: 1⃝ → 2⃝ → 4⃝, and path4: 1⃝ → 3⃝ → 4⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hybrid coupler with four λg/4 TLs yields 180◦ phase difference between path1 and path2, and the same phase between path3 and path4 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' By keeping the signal amplitudes in the four paths identical, path1 and path2 cancel each other at port 1⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Therefore, no signal is radiated on the upper antenna, thereby minimizing the reflection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', S11=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' On the other hand, path3 and path4 are constructively added at port 4⃝ for maximized radia- tion on the lower antenna (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', S41 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In summary, the incident signal is maximally delivered from the up- per to the lower antenna, or equivalently, the insertion loss is minimized when the signal amplitudes in the four paths are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The path amplitude ratios are com- puted as path1/path2 = Z2 2 and path3/path4 = Z2/Z2 when Z1 = Z2/ � Z2 2 + 1 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For the ratios of 1, we get Z1 = 1/ √ 2 and Z2 = 1, that is, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3Ω and 50Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' VNA measurement in Figure 7(a) shows the low-loss of under 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='17 dB, where the modulator was isolated from the tag (DUT in the figure) for precise measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Incorporating the Modulator and Planar VAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For pre- cise 180◦ phase shifts throughout the entire 250 MHz in the 24 GHz we leverage an equivalent circuit in Fig- ure 6(b) with the PIN diode MADP-000907-14020 rep- resented in corresponding R, L, and C as per the datash- eet [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Also, to maintain VAA retro-reflectivity while combining with the FSK modulator, RF leakage through the bias line (driving the PIN diodes) should be avoided 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 210 (a) (q) 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8° Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Loss=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='17dB SSOT 180 177° High Blas Measured DUT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 Low Bias 150 --Ideal shift 24 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 24 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 Frequency (GHz) Fregquency (GHz)– the leakage would distort the phase and corrupt the retro-reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye adopts λg/4 open stub (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', an open-ended microstrip line [15]) as an RF choke to prevent RF leakage, which is represented as an induc- tor in the equivalent circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The parameter values are found through an extensive HFSS parameter sweep sim- ulation, including the TL lengths and gaps between TLs to compensate for the coupling effect between the mod- ulator and the TLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The parameter values are shown in Figures 3 and 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As in Figure 7(b), Hawkeye modu- lator yields an accurate FSK with the maximum error of 3◦ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', [177◦, 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8◦]), indicating only 3 × 10−3dB power loss throughout the entire 250 MHz in the 24 GHz band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The RF signal was effectively isolated by greater than 30dB by the RF choke to achieve FSK modulated retro-reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 8: Delay profiles in an (a) office, and a (b) hallway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Environment delay profile demonstrates the multipath-rich indoor scenario with the delay spanning over 230us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye effectively suppresses the multi- path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', delay) via retro-reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 Indoor Multipath Suppression Retro-reflectivity of Hawkeye effectively suppresses the multipath, enabling robust indoor localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 8 presents the delay profile measured from two multipath- rich indoor settings of an office and a hallway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In both scenarios Hawkeye significantly reduces the delay spread and limits the multipath signal power to be 20dB or more below the LOS signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is because, under retro-reflectivity, the received NLOS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', multipaths) signals are strictly limited to the direction to which the signals are sent (instead of all directions without retro- reflectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Thus, higher the retro-reflectivity, less the delay spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The multipath suppressed by Hawkeye tag in Figure 8 induces only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm (office) and 2 mm (hall- way) error in Hawkeye localization (Section 4), enabling subcentimeter indoor positioning as demonstrated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) FMCW (b) HD-FMCW [11] Figure 9: IF of FMCW and HD-FMCW with tag FSK at 40 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' HD-FMCW isolates tag signal from clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' HAWKEYE LOCALIZATION In this section, we describe how Hawkeye achieves subcentimeter localization at hectometer-range, using the commodity radar and Hawkeye tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Essentially, it is an extremely accurate ranging design that can be extended to 2D/3D localization with mutliple (multi- lateration) or single radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' At a high level, Hawkeye leverages the spectral leakage signature of a tag signal to extract super-resolution range frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Our design leverages a recent technique of HD-FMCW presented in OmniScatter [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We first provide a brief primer on HD-FMCW, followed by the subcentimeter localization supporting mobile tags and large-scale simultaneous lo- calization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' HD-FMCW Primer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' A recent technique of HD-FMCW, with a light add on signal processing on commodity FMCW radars, effectively isolates the FSK signal from the clutter noise in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Compared to the original FMCW which uses a single-chirp symbol, HD-FMCW leverages multiple (periodic) chirp symbol, s(t) = c(t) ∗ �N n=1 δ(t − nT), where c(t) is a chirp with duration T, N is the number of chirp repetitions, and ∗ is the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This interrogation signal, when reflected from the clutter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', clutter noise), simply becomes s(t − ∆t) where ∆t is the round trip propa- gation delay between the radar and the clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Since it is simply a time-shifted version of the interrogation signal, it maintains the period of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is therefore represented as peaks on the multiples of 1 T Hz frequency bins in the IF signal, where all other bins in between are left zero2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Note that, this applies to clutter noise from all sources – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', all noises are concentrated on the same set of frequency bins, leaving other bins zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' On the contrary, the interrogation signal reflected off the tag (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', the tag signal) is not only time-shifted, but also modulated by the tag FSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Specifically, the tag signal is s(t − ∆t) � �� � Interrogation signal (period T ) ej2πfmt � �� � FSK (period 1 fm ) (1) where fm is the modulation frequency of the FSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This yields a new period, other than T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', the least com- mon multiple of T and 1 fm ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Therefore, in IF, tag signal and clutter noise are placed in different frequency bins – isolating the tag signal from all environmental clutter noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The separation of tag and clutter is demonstrated at Figure 9, where HD-FMCW (Figure 9(b)) isolates the tag signal unlike the original FMCW (Figure 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 2From the Fourier’s Theorem [7], an arbitrary signal with period T seconds in the time domain is represented as the multiples of 1 T Hz in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5 Hawkeye Delay Profile Environment Delay Profile (w/o Hawkeye Amplitude (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Normalized 0 0 0 50 150 200 50 150 200 100 0 100 Delay (ns) Delay (ns)I Clutter Tag Signal 30 20 10 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 125 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 Frequency (kHz)1000 Amplitude (mV) Clutter Tag Signal 30 20 10 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 125 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 Frequency (kHz)Figure 10: Hawkeye one-shot localization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The fm is determined in real time (a) to be detached from IF, where IFFT is applied to (b) the isolated clean signal with only fr (including the spectral leakage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' At (c) time domain of the signal, (d) a subset with duration T is zero-padded, where (e) FFT is performed to reveal the envelope sinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The frequency of the maximum peak amplitude is chosen as fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 11: The time-frequency representation of tag sig- nal at HD-FMCW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The ∆t is the round trip propaga- tion delay between radar and tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 Subcentimeter-accuracy Localization Hawkeye localization uniquely leverages the relation- ship between the isolated tag signal and the clutter noise in the HD-FMCW, and the spectral leakage signature embedded in the tag signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 11 illustrates the re- flected tag signal in relation to the interrogation signal, where the offset between the two signals is the combi- nation of fr and fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The range frequency fr is from the tag-radar propagation delay ∆t whereas fm is the FSK modulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In other words, fr indicates the true position of the tag and therefore, accurate lo- calization translates to the problem of finding fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is achieved in two steps: (i) Removing the effect of fm from the relationship between the tag signal and clutter noise, and (ii) precise estimation of fr from the spectral leakage signature, which we discuss in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Eliminating Tag Modulation fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The first step to achiev- ing accurate estimate of the range frequency fr is to eliminate the effect of fm in the IF signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that, in practice, fm is not known in advance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' It needs to be detected in real time due to the instability of the oscillator speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For instance, a crystal oscilla- tor can have 500 ppm variance under different environ- ments [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This indicates 25 Hz error at 50 kHz, which can translate to a vast amount of over 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 cm localiza- tion error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye is inherently robust to the oscillator variance, as it detects the accurate fm on the fly, with- out any prior knowledge on the modulation speeds or the environment under which the tags are installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In order to precisely identify fm, we begin from the fact that fr originates from the propagation delay be- tween the radar and tag, or equivalently, s(t − ∆t) ( Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 1) that has the period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' On the contrary, fm stems from the FSK signal of ej2πfmt with period 1 fm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As a result, the IF signal is represented as peaks at frequencies with 1 T Hz interval (from s(t − ∆t)), with offset of fm (from ej2πfmt) as depicted in Figure 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Thus, the fm can be precisely determined in real time, simply from the offset from n T Hz, (n ∈ N) frequency bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that those frequency bins hold the clut- ter noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Therefore, to remove fm, the clutter noise are first nullified to zero, and then the tag signal is shifted to the nullified frequency bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 10(b) depicts the resulting clean signal with only fr, free of contamina- tion from fm and clutter noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This is a key technique for subcentimeter localization unaffected from the tag clock offset prevalent in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Extracting Super-resolution Range Frequency fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawk- eye subcentimeter localization is achieved by accurately identifying the fr (indicating tag-radar distance) with boosted frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This deviates from the conventional HD-FMCW, where the fr is represented with frequency resolution of 1 T Hz – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', identical to the original FMCW frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To do so, Hawk- eye exploits the spectral leakage in the discrete Fourier Transform (DFT), where the DFT of a signal with the period of T and frequency of fr is represented as peaks at the multiples of 1 T Hz whose envelope follows the sinc function centered at fr – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', Tsinc(πT(f − fr)) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Therefore, accurately deriving fr becomes the problem of fine-grained identification of the envelope sinc func- tion from which the center frequency can be pinpointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To achieve this, Hawkeye zero-pads a subset with dura- tion T to increase the period to Tpad (≫ T) in the time domain, as depicted in Figures 10(c),(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 10(e) demonstrates the zero-padding result, where the peaks in the frequency domain are densified to precisely re- veal the envelope sinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In our experiment we set Tpad = 128T to keep the computation overhead low while achieving the subcentimeter localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Given the dense peaks, the center frequency fr is simply found as the frequency with the maximum peak ampli- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 6 Zero-Padding fm Elimination IFFT FFT 26 m Estimated fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 20 20 Clutter Zero-padded Tag Tag w/o fm Signal frm 5 Tag w/o fm Period Tpad 10 Hz Hz T 50 50 Period T ５ 50 Zero-Padding 0 0 0 1 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 2 6 2 4 5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 Time (ms) Frequency (kHz) Frequency (kHz) Time (ms) ×10 ×10 Frequency (kHz) (a) (b) (c) (d) (e)Interrogation Signal Frequency Delayed Signal (t) Tag Signal At TimeAn extensive experiment reveals Hawkeye median range error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm, over ×60 improvement compared to the original FMCW with 15 cm median error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' With the super-resolution fr acquisition mechanism, Hawkeye achieves subcentimeter localization up to 160 m out- doors, and 80 m indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that entire Hawkeye localization algorithm has the computation complexity of O(N log N) (for FFT/IFFT) where N is the number of samples – retaining the complexity of the original FMCW that mandatorily runs FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 12: IF signal of a mobile tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) The tag sig- nal has offset of fm + fd from clutter noise, where (b) varying fr incurs frequency dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Mobile Tags On the contrary to the static tags, a mobile tag in- duces Doppler frequency fd and time-varying range fre- quency fr(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 12(a) illustrates the IF of the mobile tag (↔ Figure 10(a) for static tag), where fd is added on top of fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The fd is effortlessly removed together with fm by simply following the fm elimina- tion mechanism discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' On the other hand, time-varying range frequency, fr(t), causes frequency dispersion of the peak as shown in Figure 12(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tracks fr(t) with subcentimeter accuracy, through fine-grained temporal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For mobile localization, we begin by distinguishing the mov- ing tags from the static ones via frequency dispersion, proportional to the tag velocity3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For subcentimeter lo- calization, we define mobile tags as those with > 1 cm movement within a symbol, revealed to be the frequency dispersion of ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 Hz according to our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Then, the movement is tracked by the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Extracting Time-varying Range Frequency fr(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Lo- calization of mobile tags essentially follows the same design principles as Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1, where mobile tag signal fm is eliminated to run IFFT (Figures 10(a)-(c)), recon- structing the range frequency in the time domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', fr(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Subsequently, each subset with duration T of fr(t) can be zero-padded to reveal the precise location at the corresponding time (Figures 10(d),(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawk- eye mobile localization can be configured for balance between time granularity and computation overhead, by choosing the location update interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For instance, zero-padding can be applied on fr(t) with a 520 T inter- val to provide 60 localization updates per second (under 3This is known as the dispersion factor [22] in radar con- text T = 32 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Our evaluations show 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 mm median error for a humanoid robot with 17 cm/s speed, providing evi- dence for mobile tag localization Hawkeye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that the minor modification of Hawkeye sustains O(N log N) computation complexity for mobile localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) IF for 5 tags (b) Localization Figure 13: Five tags (a) IF from a single interrogation and (b) its localization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 Large-scale One-shot Localization The lightweight localization of Hawkeye can be di- rectly extended to large-scale, for simultaneous local- ization of mobile and static Hawkeye tags with a single interrogation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For instance, localizing 100 tags takes less than 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 ms end-to-end (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 ms interrogation + 30 ms processing time) on a mediocre desktop PC (i7- 8700, 32 GB RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We verify simultaneous localization of 100 tags in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6, where each tag is identified according to its unique modulating frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The lo- calization runs iteratively for each tag, to eliminate tag modulation before extracting accurate range frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 13(a) illustrates simultaneous localization of 5 mobile and static tags, where the tags with 200, 500, 890 Hz modulation are mobile and 350, 770 Hz modulation are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 13(b) depicts successful localization of each tag, where individual tag signals are distinguished according to the modulation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that Hawkeye supports up to 1024 tags under 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 ms in- terrogation signal (T = 32 µs and N = 1025), which translates to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Hz interval between each tag IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The ample ID space tolerates over 500 ppm frequency offset in low-end crystal oscillators, demonstrating the scalability of Hawkeye in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The large scalability, in combination with long-range localization and retro- reflective tag, offers a wide-area coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 Radar Setup Multilateration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye is capable of supporting seam- less 2D/3D tag localization, where multiple Hawkeye radars concurrently interrogate Hawkeye tags for multi- lateration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Essentially, concurrent interrogation is made possible by Hawkeye plannar VAA tag which retro-reflects interrogation signal back to the source radar, efficiently avoiding tag signal interference amongst radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hence, Hawkeye radars can be set up for multilateration with- out the need for access control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Furthermore, Hawkeye radars can be time-synchronized to support 2D/3D lo- calization for mobile tags, utilizing the Network Time 7 Clutter 150 Tag (m ude 100 mplitu Frequency 50 Dispersion A 0 4 6 8 Frequency (kHz) (q) (a)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 Clutter Tag 1 (mobile) (m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Tag 2 (static) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 e Tag 3 (mobile) Amplitud Tag 4 (static) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 Tag 5 (mobile) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 A 0 4 5 6 7 8 Frequency (kHz)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='558 mo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 (mV) Tag 1 Tag 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Tag 3 Amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 Tag 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 Tag 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='203 m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='081 m0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 0 4 5 6 7 8 Frequency (kHz)Protocol [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The protocol provides sub-millisecond accuracy in local area networks, where a millisecond er- ror translates to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 mm localization error for a typical human running speed of 13 kmph, sustaining subcen- timeter accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Single Radar Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Single radar 2D/3D localiza- tion can be achieved by utilizing the AoA of the MIMO radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Compared to multilateration, single radar local- ization trades off accuracy for lower deployment cost (less number of radars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Localization error from the AoA inaccuracy is amplified over distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For instance, an AoA error of 5◦ causes 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 m localization error at 100 m (100 m × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='087 rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Single radar localization is demonstrated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 14: Hawkeye tag evaluation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We utilize a separate control board to provide the control signal to tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag is mounted on a linear stage with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='01 mm resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' EVALUATION This section presents the implementation details and evaluation results of Hawkeye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 Implementation Hawkeye radar is implemented on Eval-TinyRad (Ana- log Devices) [1] commodity 24GHz radar, where the op- eration of Hawkeye localization is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The radars provide the IF data to PC, where it is collected to per- form Hawkeye localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To deliver the control sig- nal to Hawkeye tag, a fabricated control board with VCXO (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', Voltage Controlled Crystal Oscillator) is used with a small form factor of 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='67 mm×38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='354 mm, as depicted in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The board uses Skyworks 515NDAM 134200BAG [53] oscillator for an accurate fm generation with 20 ppm variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To control the frequency of the VCXO, a variable resistor is combined with a coin cell battery, where a LDO voltage regula- tor (Toshiba TAR5SB33 [56]) is utilized to stabilize the voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Arduino Uno is also implemented as a control board for large-scale localization, where it provides a wider range of fm (4 MHz bandwidth, 0−4 MHz) com- pared to the VCXO (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Hz bandwidth, 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1748 − 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2252 kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Tag Power Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag is composed of 16 PIN diodes (Macom MADP-000907-14020), with a separated control board for operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag power consumption is highly variant according to the control signal voltage, as the diodes consume more power at higher control voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 15 provides our evalua- Figure 15: The power consumption (excluding the con- trol board power) is plotted versus the SNR at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 m tag-radar distance with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='45 dBm transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The power consumption ranges from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 µW to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='68 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (a) (b) Figure 16: For the ground truth of localization, we uti- lize a (a) laser distance meter with 1 mm resolution mounted on a tripod, then (b) mount the radar on the tripod for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' tion on the power consumption vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' SNR at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 m tag- radar distance with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='45 dBm interrogation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The power consumption is calculated utilizing the diode IV data [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The results demonstrate the control signal voltage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 V (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', tag operating at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 µW power) is sufficient for the operation of Hawkeye tag at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 m distance with over 20 dB SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Meanwhile, the control board power consumption is analyzed by simulating an IC using the Libero SoC SmartPower [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The simula- tion consists of a ring oscillator and a modulator circuit, where the power consumption results in 2 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Thus, Hawkeye tag can run with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 µW power, which can be easily operated by energy harvesting [63], or with a coin cell battery of 1000 mAh for 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Evaluation settings By default, we conduct evaluations using Analog De- vices Eval-Tinyrad as Hawkeye radar, with the specific radar parameters set as follows: bandwidth 250 MHz (24 GHz to 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 GHz), transmit power 8 dBm, IF sampling frequency 1 MHz, and 8192 samples per chirp with 2048 chirps for interrogation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We use a sin- gle transmit antenna and a single receive antenna for omni-directional Hawkeye operation, unless otherwise mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We note that the PCB fabrication and sol- dering error may result in separate distance offset per tag, which we calibrate by measuring the distance offset of a tag with known distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In all evaluations, Hawk- eye tag is mounted on a acrylic plate to avoid unneces- sary electric coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The Arduino Uno is utilized for our control board, which supplies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 V control voltage to Hawkeye tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 8 Tag Hawkeye Tag 吧 66 mm Control Board Ctrl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='67 mm 品 Sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Linear Stage 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='354 mm 66 mm40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 Voltage ( SNR (dB) 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 SNR Ctrl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Sig 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='95 Voltage 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 20 101 102 103 104 Power Consumption (uW)Tag LaserDistance MeterTag Radar(a) The tag deployment at Hawkeye 1D localization ex- periment, performed over 180 m at every 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) Box plot of Hawkeye 1D localization result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (c) Detection rate of Hawkeye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tags achieve stable localization up to 160 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 17: The 1D localization evaluation setup and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In order to obtain precise ground truth of localization, laser distance meter with 1 mm resolu- tion up to 200 meters range (Leica DISTO D510 [27]) is utilized at radar side, as depicted in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag-radar distance is first measured on the laser distance meter mounted on a tripod, where subsequently Hawk- eye radar is mounted on the same tripod for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' At tag side, a linear stage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='01 mm resolution up to 150 mm range (Soar STMX1020-D [54]) is utilized as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag is mounted on the linear stage, where the linear stage relocates the tag for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The ground truth error caused by the laser distance meter does not exceed 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For mobile tag evaluation, we utilize OptiTrack PrimeX [47] with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 mm 3D accuracy for ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 1D Localization To verify Hawkeye’s subcentimeter accuracy at hec- tometer range, a 1D localization experiment is conducted at a soccer field, where the measurement is conducted up to 180 meters in a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye tag is located at every 20 meters as shown in Figure 17(a), where a total of 100 experimental trials are performed at 20 different locations within each 20 m position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Fig- ure 17(b) demonstrates Hawkeye 1D localization perfor- mance, where the edge of the box indicates the 75th and 25th percentile error, while the whiskers indicate the 90th and 10th percentile error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Data outside the 90th and 10th percentile error is considered as outliers, which are marked as red dots outside the whiskers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye achieves subcentimeter accuracy with 90th percentile er- ror up to 100 m, where the error is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Further- more, the subcentimeter accuracy is sustained up to 160 m with 50th percentile (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', median) error, where the median error is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm at 160 m, demonstrating successful hectometer range subcentimeter localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For control signal, Arduino Uno produces 150 kHz fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 17(c) shows the detection rate of Hawkeye, where we achieve 100 % detection rate up to 140 m, which is decreased to 96 % and 54 % at 160 m and 180 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The subcentimeter accuracy at hectometer range proves the robustness of Hawkeye tag, in combination with the ef- ficiency of Hawkeye localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 18: (a) Indoor hallway experiment setup of Hawkeye, where the tag is placed at every 20 m from the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) Box plot of hallway experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Indoor 1D Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We further evaluate 1D local- ization at multipath rich hallway to verify Hawkeye op- eration in indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As depicted in Figure 18(a), the hall- way experiment is conducted up to 80 m in a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag is located at every 20 m, where 50 ex- perimental trials are conducted at 10 different locations within each 20 m positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 18(b) plots the box plot of the indoor 1D localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The median error stays below 4 mm throughout 80 meters, validating Hawkeye’s subcentimeter localization even in indoors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The detection rate stayed at 100 % up to the 80 m range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This results totally prove the multipath sup- pression of Hawkeye tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 19: Experiment setup for testing the effect of blockage in the indoor environment (a) without block- age (LOS) (b) with cardboard (c) with plywood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' LOS Cardboard Plywood Median (mm) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 90th percentile (mm) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 Table 2: 1D localization accuracy under blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Robustness Against Blockage and Temperature Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Localization for IoT applications may face various envi- ronment changes during practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' In order to sub- stantiate Hawkeye operation for pervasive deployment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='we analyze the 1D localization performance impact of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Tag Locations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Radar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='口 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='180m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='160m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='140 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='120m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='100m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='80 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='60 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20m(mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='goth percentile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Median ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='sub-cm error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='sub-cm error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='： ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Range ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Distance(m)100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='96 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Detection rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Distance (m)Error (mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Tag Locations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Radar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Range ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='FTL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='60 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='80 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Distance (m)(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Radar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Plywood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='Cardboard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='TagFigure 20: Experiment setup for testing the effect of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='temperature at indoor environment with (a) ice pack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='(b) hand warmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (c) Hawkeye effectively eliminates the effect of the instability of oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' two common error sources – blockages and tempera- ture variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The experiments are conducted in an indoor concert hall, where the tag-radar distance is set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For each experiment, total 40 experimen- tal trials are conducted at 4 different locations, with Arduino Uno as the control board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 19 shows the setup for the blockage experiments, where card- board and plywood of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 mm and 5 mm thickness is utilized as blockage materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The experiment re- sults are summarized at Table 2, where the cardboard and plywood blockage increased the median localization error by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 mm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The results show that Hawkeye sustains subcentimeter accuracy even un- der NLOS, demonstrating our robustness to the block- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Localization performance under varying temper- ature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', varying fm due to the instability of oscil- lator) is also evaluated, to show Hawkeye’s ability to eliminate the effect of fm for precise localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As shown in Figure 20, the Arduino oscillator temperature is set to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='54◦C, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7◦C and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='43◦C, for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' At each temperature, the fm varied from 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='901 kHz to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='949 kHz, showing high frequency instability of the control board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' To control the temperature of the oscillator, the control board was either surrounded by ice pack or hand warmers, as shown in Figures 20(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 20(c) compares the localization result with and without Hawkeye fm elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Without the fm elimination, the error induced by the tempera- ture at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='54◦C and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='43◦C is 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm and 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Contrarily, with Hawkeye’s fm elimination ap- plied, the error induced by the temperature change at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='54◦C and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='43◦C stays under 4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Altogether, the results prove the performance of Hawkeye at diverse en- vironments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 2D Localization We evaluate hectometer range 2D localization at soc- cer field with track, utilizing two radars and a single tag for multilateration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As demonstrated at Figure 21(a), the two radars are located at 100 m distance from the tag, where the radar interrogates the tag with 30◦ inci- dence angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The distance between the radars is 100 m, Figure 21: (a) The experimental setup of 2D localiza- tion experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tag and two radars form an equi- lateral triangle with a side of 100 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) The CDF of Hawkeye 2D localization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' and the tag modulation fm is 150 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The exper- iment consists of total 400 experimental trials, where the tag localization is conducted at 20 positions on the linear stage moving towards positive y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Fig- ure 21(b) plots the CDF of 2D localization error, where Hawkeye achieves subcentimeter median error of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm and 90th percentile accuracy of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Median error along x and y dimensions are 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 mm and 4 mm, while the 90th percentile error is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 mm and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The 2D localization, which inherently requires retro-reflectivity, is made possible by Hawkeye tag’s high retro-reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The subcentimeter 2D localization at hectometer range demonstrates the capability of Hawk- eye at practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 22: The CDF of Hawkeye 2D localization error with single radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Single Radar 2D Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye 2D localiza- tion on a single radar utilizing AoA is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The evaluation is conducted at an open field with localiza- tion distance ranging from 20 to 25 m and AoA ranging from −60◦ to 60◦, as depicted in Figure 22(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For each location, a total of 30 experimental trials are performed at 6 different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 22(b) and (c) show the CDF of distance and angle error, where the median er- rors are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7◦, and the 90th percentile errors are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='04 mm and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Collectively, the median 2D er- ror results in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='08 m, whose accuracy can be improved with a larger antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 3D Localization In order to verify the 3D localization performance of Hawkeye, a localization experiment is conducted in an indoor concert hall with three radars fixed to a wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The radar positions are described in Figure 23(a), where the three radars are located at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8, - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2) and (0, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2) coordinates, assuming 10 Hawkeye Range (a) Tag 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9mm 225 Range w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='. elimination Ice pack 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='95 lOscillatorFrequency (zH 220 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='94 (cm Arduino 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='93 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8mm 215 Range 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='92 llator (b) 210 Tag Hand 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='91 warmer 205 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 (c) 200 Arduino 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 Oscillator Temperature (°C)(a) (b) TagLocations x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 DF X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Y m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 2D D 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 10 100 1 ■ Radar Radar Localization Error (mm)(b) 1 (a) Tag Locations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 CDI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 25m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 20m 0 25m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 10 100 (c) Distance Error (mm) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 30° CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 25m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 Radar 0 20m 0 10 20 30 40 50 60 Angle Error (°)(a) 3D setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) Localization result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 23: (a) Experiment setup of 3D localization ex- periment, conducted within a indoor concert hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (b) An exemplary localization result at four tag locations, where the estimated points within the median error are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 24: The CDF of 3D localization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' (0, 0, 0) is the tag center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Tag control signal of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='15 V is fed with a signal generator [21] for 3D localization experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Total 1840 experimental trials at 23 po- sitions in xz-plane are conducted utilizing the linear stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 24 demonstrates the CDF of 3D local- ization error, presenting subcentimeter median error of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm and 90th percentile error of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For each x, y, z dimensions, the median error is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 mm, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 mm and 90th percentile error is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 mm, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 23(b) depicts an exemplary lo- calization result at four tag locations, where estimated points within the median error are plotted in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The successful subcentimeter 3D localization in indoor space proves the retro-reflective performance of Hawkeye tag in both azimuth and elevation plane, while establish- ing a solid foundation on the practicality of Hawkeye localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Simultaneous Localization of Mobile Tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' We evalu- ate Hawkeye’s ability to simultaneously localize mobile tags by attaching five Hawkeye tags to the body cen- ter, both legs and both arms of a humanoid robot [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Each tag concurrently modulates with unique fm be- tween 150 kHz and 151 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The experiment is con- ducted in an indoor concert hall with the same settings as Figure 23(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The robot has dimension of 48×36cm, operating with 16 servo motors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The radar is set to have 1024 samples per chirp with 2048 chirps for interroga- tion signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As depicted in Figure 25, three different actions of lift arms, sit down and spread legs are cap- tured with the maximum moving speed of 17cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' A total of 90 experimental trials are conducted per action, Figure 25: Robot movements and the corresponding lo- calization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 26: The CDF of 3D localization error from mov- ing robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' with three different robot positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As depicted, the three robot actions are captured with 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 localization FPS, where a ground truth measured by OptiTrack is provided together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The CDF is presented in Figure 26, where the median error of 3D localization is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='66 mm and 90th percentile error is 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='96 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For each x, y, z dimension, the median error is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='05 mm, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='53 mm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='59 mm and 90th percentile error is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='13 mm, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='07 mm, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='31 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' This verifies Hawkeye’s one-shot local- ization of mobile tags in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 Large-scale Localization To verify the large-scale support of Hawkeye, a si- multaneous, 3D localization experiment consisting of 100 tags is conducted in an indoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Fig- ure 28(a) depicts the arrangement of 100 tags, where they are deployed as 10 by 10 on a acrylic board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The tags are densely deployed with intervals of 5 mm to demonstrate operation in harsh environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=', items stacked up in the warehouse) where substantial cou- pling between tags exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The 100 tags concurrently modulates with unique fm in between 100 kHz and 250 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For 3D localization, three radars are attached to a wall as depicted in Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Each radars are located at (3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3, −3), (−3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='3, −3) and (0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2, −3) coordi- nates, assuming (0, 0, 0) is the 100 tags center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Fig- 11 y Radar Radar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='. 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5m Tag Linear Stage Radar 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2mGround Truth 50 Hawkeye 40 (mm) 30 20 N 10 20 0 10 0 20 40 60 80 0 100 y(mm) x (mm)1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 X DF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Y Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 3D 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 1 10 100 Localization Error (mm)cm 6 3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 + (length, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='0 0 m) m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 (height, (height, (height, 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1y Ground Truth Robot Body 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Right Arm 0 0 Left Arm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Right Leg (a) Lift arms (b) Sit down z (width, m) (C) Spread legs Left Leg1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 CDF X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 3D 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 1 10 100 Absolute Error (mm)Figure 27: Sinc envelope of 100 tags during the large-scale simultaneous localization experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 28: (a) The localization result of 100 tags de- ployed 10 by 10 on a acrylic board projected on the experiment photo, and (b) the accurate localization re- sult of Hawkeye at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 29: Experiment setup of large-scale simultaneous localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The experiment is conducted indoors, with 5 mm spacing between the 100 tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Figure 30: The CDF of large-scale localization error at Hawkeye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' ure 27 demonstrates the 100 tags localization process- ing result, where the successful localization of each tags resulting in 100 different envelope sincs are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Fig- ure 28(b) depicts the the localization result of Hawkeye at scale, where all estimation results are shown as red cross around the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The CDF of large-scale 3D localization error is shown at Figure 30, showing median error of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 mm and 90th percentile error of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' For each x, y, z dimension, the median error is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 mm, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 mm, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='4 mm, and 90th percentile error is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 mm, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 mm each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' The concurrent localization of 100 tags verifies Hawkeye scalability at realistic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' RELATED WORK mmWave Systems and Sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' mmWave systems have been proposed to exploit the large bandwidth [26, 31, 64, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' mmWave radars also utilize extensive band- width to achieve higher sensing accuracy [23, 9, 52, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Backscatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Backscatter offers extremely low-power signaling for power constrained scenarios [10, 16, 67, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Backscatter system also facilitates low-power sens- ing, including RFID-based approaches [6, 59, 44, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Backscatter/RFID Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' As one of the repre- sentative implementation of backscatter systems, RFID- based localization techniques have long been discussed in the community [18, 20, 51, 57, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Initial studies measure the amplitude, phase and the angle of arrival of the received signal [2, 3, 29, 65, 25, 66], which suffered from multi-path effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Recent studies still suffer from low accuracy in practical scenarios due to limited band- width of RFID, including [13, 58, 48, 60, 62] which re- quire knowledge on motion or reference tags to mitigate multi-path effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' While [33] emulates large bandwidth to achieve sub-centimeter accuracy, range is limited to room-scale scenarios to comply with FCC regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' [55] utilize large bandwidth of mmWave FMCW radar to achieve 100 m range, but is still essentially limited to FMCW range resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' CONCLUSION This paper presents Hawkeye, a mmWave backscat- ter localization that can achieve subcentimeter median accuracy at hectometer range, while using an afford- able commodity radar (∼200 USD [17]) for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Hawkeye simultaneously localizes 100 tags in only 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 ms, and is capable of supporting up to 1024 tags in the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Our design consists of (i) a new planar Van Atta Array tag that retro-reflects in both azimuth and ele- vation, combined with a low-loss FSK modulator, and (ii) a novel localization algorithm that achieves ×60 the localization performance of FMCW radar, while being immune to the tag oscillator frequency offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' Collec- tively, Hawkeye take a solid step towards bringing per- vasive tag deployment and localization to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content=' 12 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='2 Frequency (kHz) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='93 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='42 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='89 Distance (m)Ground Truth Ground Truth Hawkeye Hawkeye 40 20 (cm) 0 20 280 20 0 300 20 40 320 x (cm) z (cm) (a) (b)Radar y 100 Tags Radar 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5m 3ml Radar 3m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='75 X CDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='5 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='25 3D 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE0T4oBgHgl3EQffQCV/content/2301.02402v1.pdf'} +page_content='1 1 10 100 Localization Error (mm)8.' metadata={'source': 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--git a/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/2301.01230v1.pdf.txt b/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/2301.01230v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..414bd00304e09dcaf8e84691b16c1efb750c7d56 --- /dev/null +++ b/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/2301.01230v1.pdf.txt @@ -0,0 +1,1302 @@ + +1 + + + + + + + +A Search for Transient, Monochromatic Light +from the Galactic Plane + +Geoffrey W. Marcy1* & Nathaniel K. Tellis2 + +1 Center for Space Laser Awareness, 3388 Petaluma Hill Rd, Santa Rosa, CA, 95404, USA +2RocketCDL + +Accepted xxx Received xxx + + + +ABSTRACT +The Galactic Plane was searched for transient, monochromatic light at optical and near-IR wavelengths to detect pulses shorter +than 1 sec. An objective-prism Schmidt telescope of 0.28-meter aperture and a CMOS camera were used to observe 973 square +degrees, with 8864 exposures of 1-sec each, within a strip 2.1 deg wide along the Galactic Plane, from Galactic longitude -4 +deg to +248 deg. All exposures were analyzed for transient, monochromatic sources using a “difference image” algorithm that +yielded 11 candidate sources. All 11 sources were found to be associated with either astrophysical emission-line objects or +aircraft with sub-second blinking lights. Our survey “rediscovered” many Wolf-Rayet stars, M dwarf flare stars, and planetary +nebulae. It also identified an aircraft, of unknown type, that apparently had a nearly monochromatic lamp and a xenon lamp. +This survey would have revealed optical and near-IR pulses having a power of ~180 GW (wavelength dependent) if emitted by +a 10-meter aperture laser located 1 kiloparsec away. These non-detections of laser pulses from the Galactic Plane, including a +10-degree region toward the Galactic Centre, add to the non-detections from more than 5000 nearby stars. Indeed, all-sky +surveys for emission-line objects (e.g., ionized gas, supernovae remnants, and active galactic nuclei) would have revealed +lasers of a wide range of average brightness, wavelength, and cadence. The absence of beacons reveals more of a SETI desert, +notably at the intensely surveyed optical and radio wavelengths. + +Key Words: Transients, Extraterrestrial intelligence, Galaxy, Techniques: Spectroscopic + +1 +INTRODUCTION + +Surveys of the sky with time resolution have serendipitously revealed unexpected time-variable, dynamical astrophysics. +Historic examples include eclipsing binary stars, Cepheid and RR Lyrae pulsating stars, flare stars, cataclysmic variables, and +supernovae. More recent examples include previously unknown classes of objects such as pulsars, x-ray binaries, kilonovae, +gamma ray bursts, and fast radio bursts (FRB). Surveys at radio wavelengths have enjoyed a natural propensity at finding +unexpected sub-second transients because of the necessity to record voltage every ~10-6 s or ~10-9 s (at MHz or GHz +frequencies), leading to the unanticipated discoveries of pulsars and FRBs. These fortuitous successes motivate searching +unexplored domains of time and wavelength. + +Very few all-sky surveys at ultraviolet, optical, and IR wavelengths have been done with exposure times shorter than 20 s, and +fewer still with exposures shorter than 1 s. Long exposure times cause sub-second flashes to be diluted relative to the +background night sky brightness, making them less visibl. The LaserSETI program at the SETI Institute employs millisecond +exposures to overcome such dilution (Gillum 2022). The recent development of astronomy-quality CMOS sensors offers sub- +second readout times, small pixels of a few microns size, and large pixel arrays >50 megapixels, providing access to transient +phenomena having sub-second time scales. Here we report a search for monochromatic, sub-second pulses of optical and near +IR light, motivated both by the unexplored domain of time and wavelength and by a speculative model of interstellar +communication. + +The Milky Way Galaxy may contain spacecraft, communication relay stations, or home star systems that communicate by +lasers (e.g., Bracewell 1960,1973; Freitas 1980; Maccone 2021; Gillon 2014, Hippke 2020, 2021abc; Gertz 2018, 2021, Gertz +& Marcy 2022). Indeed, lasers are already widely used for communication by Earth-orbiting satellites because they offer +narrow-beams for privacy, high bit rate, and minimal payload mass. Similarly, laser communication, or next-generation +coherent quantum communication methods, may be used for interstellar communication (Schwartz and Townes 1961, +Zuckerman 1985, Hippke 2018, 2021abc), perhaps using repeater nodes (Gertz & Marcy 2022). + +Laser light may be identified in telescopes by its narrow range of wavelengths, hereafter “monochromatic” (e.g., Naderi et al. +2016, Su et al. 2014, Wang et al. 2020), regardless of the unknown duration or cadence of the pulses. Previously, we have + + +2 +searched for monochromatic light from more than 5000 individual stars of all masses, ages, and chemical compositions +(O,B,A,F, G, K, and M) using high-resolution optical spectra. These extensive searches yielded no laser detections and no +viable candidates (Reines & Marcy 2002; Tellis & Marcy 2017, Marcy 2021, Marcy et al. 2022, and Tellis 2022, private +communication). These laser searches employed spectra of high resolution, l/Dl > 60000, in the wavelength range l = 3600 to +9500 Å. The detection threshold of laser power was 50 kW to 10 MW, assuming a diffraction-limited laser emitter consisting +of a benchmark 10-meter aperture. We also searched a 10x10 degree field at the Milky Way Centre, albeit at low spectral +resolution, yielding no laser detections (Marcy et al. 2022). We also searched the solar gravitational lens focal points for +nearby stars (Marcy et al. 2021). No monochromatic light was found, neither sub-second pulses nor continuous emission. + + +Searches for narrowband radio waves from technological entities also make use of the “monochromatic” nature of a signal to +distinguish it from ordinary astrophysical sources and to promote the candidacy of technological signals (e.g. Isaacson et al. +2017, Enriquez et al. 2017, Price et al. 2020, Wlodarczyk-Sroka, Garrett, & Siemion 2020, Sheikh et al. 2021, Garrett & +Simion 2022). There is no guarantee that interstellar communication will be nearly monochromatic or multi-bandpass. But that +characteristic offers a search property that excludes the vast majority of false positives that are either astrophysical or terrestrial +noise. + +Without spectroscopic information, other searches for technological signals have been done by hunting for sub-second optical +pulses, e.g., Wright et al. (2001); Howard et al. (2004); Stone et al. (2005); Howard et al. (2007), Hanna et al. (2009), +Abeysekara et al. (2016), Villarroel et al. (2020, 2021). No definitive optical pulses were found, but some candidates emerged. +Next generation searches for sub-second optical pulses are planned (Maire et al. 2020). Here we describe a search for sub- +second pulses of monochromatic light along the Milky Way Plane using a special optical system designed to optimize this +search. + + +2 +OBJECT-PRISM OBSERVATIONS OF THE MILKY WAY PLANE + +We used the objective prism Schmidt telescope operated by the Center for Space Laser Awareness and described in +Marcy et al. (2021, 2022) and at www.spacelaserawareness.org . In brief, the telescope is a modified Schmidt design +with aperture diameter 0.28 m and a 7-degree wedge prism to produce spectra of low resolution, R~100, of every point within +the 2x3 deg field of view. The CMOS camera at the focal plane contains 9400 x 6600 pixels, each 3.7 microns with a quantum +efficiency over 80% between 500 – 800 nm and lower QE extending to 370 nm and 950 nm. We operated with an exposure +time of 1.0 sec and dead time of < 0.01 sec. + +The system is optimized to detect monochromatic pulses of optical light (see Marcy et al. 2021, 2022) having pulse duration +less than 1 sec. Monochromatic emission has the shape of a two-dimensional PSF, with a FWHM ~ 5.5 pixels, allowing +efficient search algorithms. In contrast, direct hits by particles or gamma rays (cosmic rays) make “dots” sharper than the PSF, +allowing discrimination. Another nemesis comes from reflections off satellites (Corbett et al. 2021, Nir et al. 2021) which +exhibit the solar spectrum, obviously not monochromatic. The optical design is remarkably similar to the objective prism +telescopes of the Harvard Observatory (Pickering 1912; Fleming 1917), albeit with 50x the QE. + +We performed multiple performance tests of the QHY600 CMOS camera, finding read noise is ~2 photons (RMS), the dark +noise <0.1 e-/s per pixel, and the response is linear within 0.3% over a dynamic range of a few to 56,000 photo-electrons. +We operate with modest thermoelectric cooling to -20C. The properties of the QHY600M are comparable to CCDs (Gill et +al. 2022, Betoule et al. 2022), but offer frame rates up to 30 fps. We can detect light pulses having sub-second duration +with minimal contamination from the background “noise” of stars, galaxies, and sky. From our observing station at Taylor +Mountain in California, the sky produces ~40 photons/pixel during 1 sec coming mostly from Santa Rosa city lights. + +Figure 1 shows a sum of 10 images obtained with the objective prism telescope system and QHY600 CMOS camera. This +image is centred at Galactic Longitude 38 deg and Galactic Latitude 0 deg. The image shows hundreds of stellar spectra +oriented vertically, each spanning wavelengths 380 – 950 nm spread over 1200 pixels, with long wavelengths downward and +North up. In the 1 sec exposures, the faintest stars have Vmag=13 with signal-to-noise ratios of ~10 per pixel. Stars brighter +than Vmag = 2.5 saturate the sensor with >56000 photons/pixel. A monochromatic, spatially unresolved point source would +appear as a two-dimensional “dot” with a PSF shape. Sub-second monochromatic pulses would appear in only one image as a +PSF-shape “dot” within a sequence of images. This survey searches for cadences of at least one pulse every 10 minutes, the +duration of the observing sequence of a given field. Continuous sources and high cadence sources, >1 Hz, would appear in all +images in a sequence of 600 images. We judge the PSF by the width of the spectra in the spatial direction that is dominated by +both seeing and optical imperfections in the prism, yielding a PSF width of typically 6 to 8 arcseconds corresponding to a +FWHM 5 to 6 pixels. + + + + +3 + + + + + + + +Figure 1. The sum of 10 images (1 sec each) from the objective prism system with a field of view of 3.1 x 2.1 deg, 9500 x 6300 +pixels, each subtending 1.3 arcsec on the sky. The stellar spectra span wavelengths 370 – 950 nm with longer wavelengths +downward and north up. This image is centred at Galactic longitude 38 deg, at RA=19h 00m, DEC=+4o 30’. The stellar +spectra come from stars of magnitude 8 – 14. Monochromatic emission would appear as a PSF-shape “dot”. + + +5.5 +5.0 +[deg] +DEC +4.5 +4.0 +3. +6 +4 +2 +0 +2 +4 +6 +RA +一 +19hr 00min +min +4 + +Figure 2. Spectrophotometry of Vega vs wavelength obtained with the objective prism telescope and CMOS camera. The spectrum +establishes the spectrophotometric sensitivity, wavelength scale, and spectral resolution, R~100, of all spectra that have a length 1200 pixels. +Prominent absorption lines in Vega and strong atmospheric lines are labelled. Exposure time was 0.5 sec. + + +Figure 3. A spectrum of the planetary nebula NGC7027 with the objective prism system and 5 sec exposure total. It shows emission lines +common from ionized gas. The wavelength scale is based on a spectrum of Vega, with a zero point set by Ha here. Laser emission is +easily distinguished from known astrophysical sources by their pattern of known emission lines. + +Figures 2 and 3 show spectrophotometry of Vega and NGC7027 obtained with the objective prism system. The reduction to +one dimensional spectra was accomplished by a simple summing of the photons along the spatial width at each wavelength. +The spectrum of Vega (0.5 s exposure) shows the Balmer lines up to H11, along with telluric lines (Fig.2). The wavelength +calibration was done with a 7th order polynomial fit to 14 pixel positions and the corresponding wavelengths in that Vega +spectrum. The prism creates a highly nonlinear wavelength dispersion. The resulting Vega spectrum in Fig. 2 shows that +stellar spectra can be classified and that emission-line spectra can be identified, to distinguish them from non-astrophysical + +NGC7027 10x0.5 sec +nm +2021 July 20 +500.7 +4×10* +Pixel ++ +3×104 +α +per +n +H +9 +Photons +5 +9 +4 +2x104 +u +732.5 nm +m +[ArIV] +uu +n +713.5 +0 +6. +330. +1×10* ++ +587. +6 ++ +5 +Arl +H +H +I +H +S +400 +500 +600 +00 +800 +006 +Wavelength +1 [nm] +5 +sources. The spectrophotometry of Vega in Figure 2 is given in photons per nm per sec detected with our objective prism +system, allowing this spectrophotometry to map magnitude to photons per nm per sec of other sources. The monotonic +decrease in photons detected for wavelengths shortward of 440 nm is due to decreasing quantum efficiency of the CMOS +sensor. + +The spectral and spatial resolutions are set by the PSF that has FWHM ~5.5 pixels, dominated by seeing and optical aberrations +in the prism. The spectrum of NGC7027 (Figure 3) shows the usual emission lines from ionized gas at 10000K (e.g. Zhang & +Li 2003). The two [OIII] lines at 495.9 and 500.7 nm are barely resolved. This modest spectral resolution, 2 to 10 nm from +380 to 950 nm, allows the identification of stars, galaxies, ionized gas, asteroids, aircraft, and satellites, to distinguish them +from less common phenomena. + +Reflected sunlight and glints from orbiting satellites are easily identified by their solar spectrum. This allows light pulses from +Earth-orbiting satellites and discarded rocket boosters to be immediately distinguished from extraterrestrial laser pulses. The +spectroscopic information allows instant discrimination of both astrophysical and terrestrial sources from monochromatic +extraterrestrial sources. However, laser pulses from human-made satellites could be indistinguishable from extraterrestrial +laser pulses. Satellite-born laser pulses that are sufficiently brief to avoid detecting an orbital “streak” on the image, i.e. less +than ~1 arcsecond, can masquerade as extraterrestrial laser pulses. + +Figure 4 shows the RA and DEC of the fields we observed, each field having of angular size 3.2 x 2.1 deg located along the +Galactic Plane. Also shown on Figure 4 is the 14 x 10 deg region we previously observed near the Galactic Centre, as labelled. +The total coverage is 973 square degrees along the Galactic Plane. Missing is the Galactic Plane south of 34 deg, inaccessible +to our observatory in Northern California. The field toward the Galactic Anti-Centre, at RA = 5h 45m 37s DEC=+28 56’, was +observed four times, as that direction is special. At the Anti-Centre, laser guide stars or communication lasers may be pointed +toward us, but actually intended for the Galactic Centre. + +Each field was observed with 600 consecutive 1-sec exposures. This survey detects arrival of at least one photon pulse every +10 minutes. The fields overlap to provide both complete coverage of the region and security against algorithmic or optical +poverty at the edges of the field such as from poor background assessment or vignetting. + +Figure 4. The 124 fields, each 3.2x2.1 deg, observed along the Galactic Plane in this objective prism survey. Also shown are +the previously observed fields near the Galactic Centre, for a total of 973 square degrees surveyed. Each field was observed + +60 +40 +Galactic +Anti-Centre +(deg) +DEC +20 +Galactic +Centre +0 +5 +10 +15 +20 +RA (hr) +6 +with 600 consecutive 1-sec exposures, giving time-resolved spectra of all points in the sky to reveal sub-second or +continuous monochromatic optical pulses. +3 THE DIFFERENCE-IMAGE ALGORITHM +We search for monochromatic emission that appears as a transient PSF-shape “dot” in the images. We employ a difference- +image technique described in Marcy et al. (2022), similar to the difference algorithm in Vasilyev et al. 2022. In brief, the +algorithm operates on a set of 600 exposures, each 1 s, for a specific 2x3 deg field. The algorithm takes the average of six +“bookend” images (three prior and three subsequent) surrounding a given “target” image and subtracts the average bookend +image from the target image to yield the “difference image”, having pixel values near zero. Residuals are due to Poisson noise +of the arrival of photons and to the variations in atmospheric “seeing” from image to image that compromises the subtraction of +stellar spectra. We suppress these residuals by performing a 50-pixel boxcar smoothing of the difference image along the +direction of dispersion of the spectra, and we subtract that smoothed version from the original difference image (see Marcy et +al. 2022). This process subtracts the residual continuum of each star spectrum. Narrow emission lines in the target image that +are not in the bookend images will persist in the difference. +This difference-image algorithm yields any monochromatic point sources that were present in each image but not present (or +only weakly present) in the average of the six “bookend” images. Examples of this process are shown in Figure 5 of Marcy et +al. (2022), and we used the same algorithm here. Each target difference-image (9500x6300 pixels) was examined blindly by +this algorithm, yielding PSF-like candidates. +Emission-line sources may be coincident with stellar spectra or they may be located in between them, and the difference-image +algorithm suppresses light from both stars and the sky. The algorithm further demands that the candidate point sources must +have a 2D shape consistent with the instantaneous point spread function (PSF), as measured by the spatial profile of the stellar +spectra determined by cross-correlation. For each exposure, the algorithm measures the FWHM of the spatial profile of stellar +spectra, commonly 5 to 6 pixels (6 to 8 arcseconds), caused by seeing and optical aberrations in the prism. We note that +“cosmic-ray” particles that hit the CMOS sensor are immediately rejected, as they affect only a few neighboring pixels, +inconsistent with the smooth Gaussian-like shape of the PSF with 5.5 pixel FWHM. + +The algorithm is designed to detect sub-second monochromatic pulses. However, if the cadence of light pulses is more +frequent than 1 pulse per second, the image-difference algorithm will be compromised in detecting them because the pulses +appear in both the target image and the bookend reference images. If the pulse amplitudes are not constant or if the seeing +changes over a period of 7 s, the pulses may still be detected. Cadences slower than ~1 pulse per second will yield individual +frames containing the point source and neighboring ones that do not, making the pulse detectable with the difference algorithm. + +Somewhat surprisingly, sources of monochromatic light that are constant in time are usually detected by the difference-image +algorithm despite appearing in both the target and bookend images. The seeing changes on sub-second time scales causes the +captured number of photons to vary by more than 10% from image to image. The intensity of emission lines from astrophysical +objects such as flare stars, planetary nebulae, and Wolf-Rayet stars varied in apparent brightness by 10% - 20% (RMS) among +the 1 s exposures. The emission lines are detected by the difference-image algorithm as if they were transient sources, even +though they are actually steady. Thus, the algorithm actually detects monochromatic point sources no matter if they last less +than 1 s or are continuous in time. + +4 DETECTED MONOCHROMATIC CANDIDATES + +We executed the difference-image algorithm to all 124 fields and their 600 exposures per field along the Galactic Plane region +(Figure 4), yielding 11 monochromatic objects of interest, each requiring visual inspection and assessment. We describe the +monochromatic candidates here. + +4.1 Moving Multiple Flashes: Probably Aircraft + +On 2022 Jan 24, we obtained 600 exposures, 1 s each, at Galactic longitude 92 deg for which the automated difference-image +algorithm gave an alert of several monochromatic sources. Upon inspection, the object clearly had several components, as +seen in Figure 5. That figure shows the seven consecutive raw images, each panel showing the full frame image, 2.1 x 3.2 +deg, with north to the left and longer wavelengths to the right. A few hundred stellar spectra are apparent, of V magnitudes 6 +to 13, fixed in each image. The field has coordinates RA=21h 20m 30s +49o 46’, in the northwest region of the sky at an +altitude 40 deg., and the angular velocity of the object is 0.55 deg/s. + + +7 +An odd-shaped object first appears (barely) in the upper left of the first frame, and it moves down and to the right each +successive image during the full 7 s. Successive locations along a diagonal path, from upper left to lower right, represents time. + +Figure 5. A sequence of seven consecutive full frames, each a 1.0 s exposure, at Galactic longitude 92 deg observed on 2022 Jan 24 at 2:33 +UT. An object enters the field of view at upper left and moves down and to the right each second. During 1.0 s, several flashes occur +having a broad spectrum, 1000 pixels across left-right, and one flash (the brightest) occurs having a smaller range of wavelengths. The +diagonal line in each frame represents a light that is shining continuously during the full 7 s. It is apparently a nearly monochromatic +light because it exhibits very little extent in the wavelength direction (left-right). +A magnified view of the third frame is shown in Figure 6, in which the stellar spectra were subtracted using the previous and +next images, leaving only the moving object. Figures 5 and 6 show that the object travelled from upper left to lower right +during each second, including a continuous diagonal line caused by a light source that stayed on during the entire 7 seconds of +the sequence. Remarkably, this “diagonal light source” has a wavelength extent, left to right length, that is only ~60 pixels +FWHM, corresponding to a wavelength spread of ~22 nm. This light is nearly monochromatic. We do not know its central +wavelength, nor how this continuous light was produced. + +Figures 5 and 6 also show at least six flashes of light that happened during the 1 second exposure. Each flash caused a long +horizontal line showing that the light in the flashes consisted of the full optical wavelengths, 380 – 950 nm, similar to the +background stars. The horizontal lines are narrow, only ~20 pixels wide along the diagonal (time) direction, corresponding to +~1% of the full diagonal length of travel during 1 s. This shows that each flash lasted ~0.010 s. The spectra of all 6 flashes +show spectral structure at the far right end (longest, near-IR, wavelengths), consisting of at least to emission bumps. + +Figure 6. The magnified view of the laser candidate in Figure 5, the 3rd image. Background stars were subtracted using adjacent images. +The diagonal streak is caused by motion of a light source from upper left to lower right and shining during the full one second. Its narrow +width, left-right, shows it was nearly monochromatic, still not understood. Each of the six flashes exhibits a broad range of wavelengths +(left-right extent), with emission lines at the longest wavelengths (at the right end of each flash, consistent with xenon lamps. The +brightest flash (bottom right) is only 400 pixels long, indicating a short wavelength range, but the same emission lines. + +8000F +8000 +2000 +10002000 +Column7000 +6500 +Row +Pixel +6000 +5500 +500 +1000 +1500 +2000 +2500 +3000 +Pixel Column +8 + +The brightest horizontal line located at bottom right in Fig. 6 happened at a time ~0.9 sec during the 1.0 s exposure, judging +from its location along the diagonal path. It contains only 1/3 of the full range of wavelengths. The emission structure at its far +right of its spectrum resembles the emission structure at the far right of the other flashes. Apparently, those wavelengths are at +the far red and infrared end of the spectrum. That brightest flash at bottom right also has a duration of ~0.01 s. One other +attribute of all of the lights is that they continued to illuminate the telescope during the full 7 s. This suggests that all the light +sources emitted a beam that was broad enough in solid angle to keep the telescope bathed in light during the full 7 s of the +object’s motion across 4 deg of the sky. To accomplish this steady bathing, the beams of light must have been many degrees +across at least, and perhaps nearly isotropic. + +We wonder what combination of light sources can create the multi-component images in Figures 5 and 6, and what type of +object they are attached to. A clue comes from spectral features in the full-length spectra. At the long wavelength end of each +spectrum (far right) is clearly some structure, and perhaps broad emission lines. We extracted those spectra from the raw +image by simply adding 10 rows along the full length of the spectrum, one of which is shown in the left panel of Figure 7. The +spectrum shows two broad, strong emission lines in the near-IR, at wavelengths ~830 nm and ~900 nm, along with some +weaker emission lines at 480 nm and ~540 nm. Laboratory spectra of xenon displays its two strongest lines at 850 and 910 nm, +in good agreement with the two strongest seen here in Figure 7 (Povrozin & Barbieri 2016, and +https://mmrc.caltech.edu/Stark/Xe%20lamp%20spectra.pdf). The third strongest emission feature fin the observed object is a +pair of closely spaced lines at ~470 nm (see Figure 7) that also agrees with the lines seen in lab xenon. The spectrum in Figure +7 also has a close pair of lines at 540 nm, which is not in the lab xenon spectrum. This may indicate another gas, besides +xenon, in the lamp, such as mercury. Large airplanes typically have at least eight different external lights, usually composed of +xenon gas, each having different color filters and beam directions, some of which flash at intervals near 1 second. The angular +speed of the object, 0.55 deg s-1 is consistent with that of aircraft by common experience, i.e., a Moon diameter per second. + +Aircraft xenon lamps often have filters to produce “green” and “red” lights. LED lights are now also being used on aircraft. We +wonder if an LED coupled with a narrow band filter could produce the “diagonal” line that was on continuously with its +narrow wavelength range. LIDAR seems unlikely, as the lasers usually operate at 1064 nm, or frequency doubled to 530nm, +both having narrow wavelength ranges of < 1 nm. The observed wavelength range of 22 nm is inconsistent with such lasers. +Alternatively, a xenon lamp that was continuously on, but covered by a narrowband filter, could produce the persistent +diagonal line. There is also the possibility of reflected light, laser or otherwise, off the aircraft. + +In any case, the most likely explanation for this “monochromatic object of interest” is an aircraft with multiple, flashing lights +with different filters, with one lamp shining continuously. Indeed, the northwest region of the sky is the direction of a small +airport, Charles M. Schulz Airport, 18 km away. We don’t know what causes the continuous “diagonal” light. + + + +Figure 7. Photons per pixel vs. wavelength of the two bright flashes in rows 6120 and 5580 in Figure 6. Both spectra show emission lines +at 830 and 910 nm, which match the known strongest emission lines in Xenon gas, commonly used in airplane lamps. The weak emission +lines at 475 nm here also appear in laboratory spectra of Xenon. The spectrum at right is missing light shortward of 600 nm, +undoubtedly caused by a filter that transmits only light longward of 600 nm, making a “red” light. Thus, the lamps are made of xenon +gas, as is commonly used on aircraft. + +On 30 Nov 2021, observations at Galactic longitude 46 deg revealed a new object of interest, detected by the automated +difference-image algorithm. Figure 8 shows a sequence of seven full frames, each 1 second duration. As with the previous +candidate (Figures 5, 6, 7), there are several flashes lasting less than 0.05 sec, several being full spectrum and one being red +and bright. All of them contain the strong emission lines in the near-IR indicating xenon lamps again. A light with a + +2.0×105 +Lamp 5 +Pixel +1.5×105 +Photons per +1.0×10§ +5.0×104 +400 +500 +600 +002 +800 +900 +1000 +Wavelength +[nm]6×10 +Red Lamp +per Pixel +5×105 +4×105 +Photons j +2×105 +1×105 +0 +400 +500 +600 +700 +800 +900 +1000 +Wavelength +[nm] +9 +continuous spectrum is also apparent, again lit continuously, seen as the faint diagonal streak. The angular speed is again ~1 +deg/sec. We presume this object is also an aircraft. + + +Figure 8. A transient found at Galactic longitude 46, seen entering the field in the third frame and exiting in the 5th frame. The full +spectrum flashes of light, and diagonal trajectory, are consistent with the xenon lights on an aircraft +. +4.2 Wolf-Rayet Stars + +At Galactic longitude 8 deg, the automated difference-image code identified a transient emission line among the 600 1-s +exposures. Extraction of the raw image along the full spectrum revealed other emission lines, shown in Figure 9. We +accomplished the wavelength calibration by employing, blindly, the calibration of wavelength vs pixel obtained from the +spectrum of Vega used in Figure 2. The zero-point of a wavelength scale was set by using the pixel at the far infrared end of +the spectrum in the raw image, deemed to be a wavelength of ~950 nm. The uncertainty of that zero-point is ~50 nm, due to +the gradual dimming at the end of the spectrum. This approximate wavelength scale allowed the emission lines to be identified, +within 50 nm, as shown in Figure 9. + +Figure 9. Spectrum of a changing emission line at wavelength 585 nm, identified by the automated difference-image algorithm, in the +field at Galactic longitude 8 deg. The extracted spectrum shows other emission lines, consistent with a Wolf-Rayet star of type WC5. The +coordinates from our image (upper right) suggest this is the known Wolf-Rayet star, “WR111”. + +246 +3000F +2600 +2000 +1500 +1000F +600 +1000 +2000 +3000 +400 +1000 +3000 +40 +1000 +2000 +3000 +400 +1000 +2000 +3000 +4自 +1000 +2000 +8000 +40 +1000 +2000 +9000 +4000 +1000 +2000 +3000 +40005000 +wu +RA=18h08m +nm +587.6 +DEC=-21 15 +581 +4000 +CIV +Hel +Photons +CIII,CIV 465 nm +Hell468.6nm +3000 +P +2000 +Z +CIII +Hell +1000 +400 +500 +600 +700 +800 +006 +Wavelength +[nm] +10 +That approximate wavelength scale showed the similarity of the spectrum with a standard Wolf-Rayet star of type WC5, based +on the catalog Wolf-Rayet spectra given at: https://lweb.cfa.harvard.edu/~pberlind/atlas/htmls/wrstars.html and on the +classification by Smith et al. (1996) and Crowther et al. (1995). Using the lab wavelengths of identified lines, we refined the +zero-point of the wavelength calibration to yield the spectrum shown in Figure 9. We performed astrometry of our image, +yielding coordinates, RA = 18h 08m and DEC=-21d 15’, at which exists the known Wolf-Rayet star, WR 111=HD 165763 +(type WC5). This obviously removes this candidate from further consideration as non-astrophysical. +A Galactic longitude 74 deg, the automated code revealed another emission line that varied in brightness during the 600 1-sec. +It met the criteria of a pulsing laser candidate in 28 of the exposures. Figure 10 shows the full spectrum, revealing the emission +line at 465nm, and also several other emission lines. The line at 465 nm appears to vary in intensity because of seeing changes, +easily verified by the widths in the spatial direction of the spectra. The pattern of emission lines matches that of Wolf-Rayet +stars of type WC8, with coordinates approximately RA=20h 15m and DEC = +36d 38’, with a brightness approximately V = 9, +with a possible identification as HD 192641, a WC7 Wolf-Rayet star. Thus, there is no support for a non-astrophysical +interpretation, laser or otherwise. + +Figure 10. An emission candidate detected automatically by the intensity variations of the emission line at 465nm due to changes in +seeing. This is apparently a WC8 Wolf-Rayet Star of magnitude V~9, at approximate coordinates, RA = 20h 15m , DEC=+36d 38'. We rule +out extraterrestrial technology. + +In the field at Galactic longitude 76 deg, the automated search triggered on another candidate transient +emission line, shown in Figure 11. The spectrum shows it to be a WC7 Wolf-Rayet star, and the +coordinates show that it is HD 192641, an 8th mag WC7 Wolf-Rayet star. The automated search for +transients was triggered by seeing variations. + +3000 +Photons +2000 +Z +1000 +400 +500 +600 +700 +800 +900 +Wavelength +[nm] +11 + +Figure 11. The automated search for transient emission lines discovered this candidate, which is clearly a WC7 Wolf-Rayet star. This +spectrum results from adding 20 1-sec exposures to improve the signal-to-noise ratio. Seeing changes caused the most intense emission +line at HeI 465 nm line to momentarily brighten, triggering the alert. It is likely HD 192641, a WC7 Wolf-Rayet star. + +4.3 M-Type Stars +At Galactic longitude 38, our automated code revealed an apparent, changing emission line at a wavelength of 720 nm. A plot +of the full spectrum, shown in Figure 12 (top), shows a spectral energy distribution that is clearly an M dwarf, with the +characteristic TiO absorption bands at the red and near-IR wavelengths (Leggett et al. 2000). There is a naturally occurring +peak at 720 nm. Examination of a sequence of seven images, shown in Figure 12 (bottom), shows that the momentary +improved seeing conspired to yield an apparent increase in the peak intensity, fooling the code into sensing an emerging +emission line. We see no evidence of a PSF-like line, and instead relegate this candidate to the common occurrence of +momentarily improved seeing at a wavelength with natural high intensity. + + +1.5×104 +Photons +Z +5.0x10° +500 +600 +700 +800 +900 +Wavelength +[nm]1000 +800 +Photons +600 +400 +200 +0 +400 +500 +600 +700 +800 +006 +Wavelength +[nm] +12 + +Figure 12. A candidate emission line at a wavelength ~738 nm (column 2181) at Galactic Longitude 38 deg.. Top panel: the spectrum of +the object, clearly an M dwarf (M4 to M5) with an apparent emission line at 738 nm. Bottom Panel: the seven consecutive raw images +zoomed on the emission. The apparent intensity increase of the emission is just due to momentary improved seeing at a peak of the +spectral energy distribution, between TiO absorption bands. + + +At Galactic Longitude 72 deg on 2021 Dec 02, the automated code identified a monochromatic brightening at wavelength 827 +nm (column 6167), in a star with an M4 spectrum. Figure 13 shows the extracted spectrum vs column #, both for the single +exposure (left panel) that yielded the candidate and for the sum of 20 exposures (right panel) that gives an average spectrum +over 20 sec. The candidate identified by the difference-image algorithm is located at a persistent peak in the spectrum of the +M4 star. On that one exposure there was a momentary enhancement of the intensity that is obviously consistent with the noise +level in that single exposure, thus making the apparent transient emission line to be likely noise. Indeed, examination of the +raw image shown in Figure 14 shows that the enhanced emission was concentrated within 4 pixels, which is inconsistent with +the PSF that has FWHM ~5.5 pixels. Thus, we suggest that this candidate monochromatic emission is simply noise. Follow-up +spectroscopy of this M star may be warranted. + +Unfortunately, the identity of the M star in Figures 13 and 14 remains a mystery. The coordinates from our images are +approximately RA= 20h 13m 10s, DEC=+33d 07’ (eq 2000, epoch 2021). From our image, its magnitude is Rmag ~11, where +two plausible stars reside, HD331958 and TYC 2675-1608-1, neither of which has high quality characterization in the +literature. HD331958 has properties on SIMBAD listed B-V=+1.19 mag, consistent with a K5 star not the M4 spectrum we +see. SIMBAD lists its spectral type as B8, which is inconsistent with both K5 and M4. There are multiple inconsistencies. Its +parallax is 14.16 mas and proper motion is 78.5 mas/yr, implying a transverse space velocity of 26 km s-1, which is consistent +with a Milky Way disk star and with its measured radial velocity of 48 km s-1. The Vmag and parallax imply an absolute +magnitude of MV = 6.62, which is consistent with K dwarf implied by the B-V. However, the spectrum we see is clearly an +M4-M5 dwarf from the obvious TiO bands. Thus, HD 331958 seems unlikely to be the candidate shown in Figure 13. + +The other nearby catalog star of comparable brightness is TYC 2675-1608-1 (20 13 10, DEC=+33 07) with SIMBAD +photometry, V=11.86 and B-V=+1.65 and J-V = +3.28, sufficiently red to be an M4 star. Its listed parallax is 0.623 mas, +implying a distance of 1600 pc. That great distance rules out M dwarf status for a star having Vmag=11.86. One solution that +satisfies the parallax, Vmag, and the observed M4 spectral type is an M supergiant at 1600 pc having MV ~ 0. The measured +proper motion of 12 mas yr-1 and distance ~1600 pc implies a transverse velocity of ~91 km s-1, much larger than the stated +radial velocity of 0.12 km s-1 listed on SIMBAD. Such a mismatch of velocity components raises concerns about a mistake +somewhere. In any case, the apparent emission appears to be noise. Follow-up spectra are certainly warranted to verify that the +apparent emission at 827 nm is indeed merely noise. + +Figure 13. A candidate transient emission line in an M dwarf at Galactic Longitude 72 deg. At left: The extracted spectrum vs. Column #, +showing the location of the candidate emission found by the automated code. At right: The sum of 20 exposures, showing that the +candidate emission (top) resides where there is persistent emission, but is consistent with the noise in a single spectrum, given 600 +exposures to draw from, as shown in Figure 14. + +590 +591 +592 +594 +595 +598 +9850 +3B40 +皖· +9830 +3B20E +1402160 2180 2200 2880 +2140 2160 2180 2800 2880 +2140 2180 2180 2200 2280 +2140 160 2180 2200 2880 +2140 2180 2180 2800 2280 +2140 2160 2180 2800 2880 +8140 8180 8160 2200 2880400 +Candidate +Emission +300 +S +Photons +200 +100 +5950 +6000 +6050 +6100 +6150 +6200 +6250 +Column +[Pixels]6000 +Candidate +Emission +5000 +S +Photons +4000 +3000 +Z +2000 +1000 +5950 +6000 +6050 +6100 +6150 +6200 +6250 +Column +[Pixels] +13 + +Figure 14. The raw image of the emission candidate shown in Figure 11. The bright pixels near the center are not distributed smoothly +over the full vertical length of the PSF in the spatial direction. This indicates the brightening of “emission” is due to noise or a cosmic ray. + +At Galactic longitude 64 deg the automated difference-image code identified candidate transient emission +located on the spectrum of a star. Figure 15 shows a 1.2x1.2 deg zoom with the spectrum of the star in +the NE corner. The spectrum is short in wavelength, and it exhibits four broad, bright wavelength +domains, typical of the molecular bands of M stars with most of the flux longward of 700 nm. +Identification and astrometry of five stars in the vicinity, shown in Fig. 15, shows that the mid-M-type +star is at RA = 19h 45m 38.0s and DEC=+28o 39’ 40” (2000), with an uncertainty of 2 arcmin, and it is +magnitude R~11. The large astrometric and photometric uncertainty is due to the dispersion by the prism +in DEC and to the vignetting in the corner of the image. +The transient emission appears in only one exposure (#588 out of 600) and is located on a broad peak at +745 nm as shown in Figure 16. The stellar spectrum indeed exhibits four broad peaks in the spectral +energy distribution due to the usual absorption by TiO, CaH, CaOH in mid-M dwarfs. Figure 16 shows +the stellar spectrum between wavelengths 710 and 920 nm. There is very little stellar flux detectable +outside that wavelength range. The enhanced emission at 745 nm is apparent in the 1-sec exposure +(shown as squares) relative to the average spectrum (solid line) from 20 exposures, with 10 taken before +and 10 after. +In the right panel of Figure 16, the individual 20 spectra are shown that comprised the average. The +scatter in the number of photons at any given wavelength reveals the noise in the individual 1-sec +exposures. The noise is caused by Poisson fluctuations of photon arrival and seeing variations. The +dispersion of the number of photons, at each wavelength, among the 20 exposures shows the level of +combined noise from Poisson fluctuations and seeing. The enhanced emission in the one exposure at 745 +nm triggered the automated search algorithm (squares), and it is indeed more intense than the average. +However, that enhanced intensity has a magnitude that resides at the end of a distribution of noise rather +than detached from that distribution. Thus, the apparent transient emission is likely a result of rare, but +expected, noise as seen in the other spectra. Further, stellar photospheric flux at 745 nm is naturally the +most intense region of the spectrum, which allows a slight seeing improvement to concentrate the arriving +photons in both the wavelength and spatial directions of the raw image, boosting the peak intensity +momentarily. We are satisfied that this apparent transient emission is merely noise. + + +4196 +4190 +4185 +4180 +4175 +4170 +4165 +6150 +6160 +6170 +6180 +6190 +Pixels +14 + +Figure 15. The sum of 20 consecutive 1-sec images, zoomed on a transient emission-line candidate at Galactic longitude 64 deg, at upper +right. Spectra of stars of brightness Vmag 7 to 14 are visible, with North up, short wavelengths up, and East to the left. The stellar +spectrum with the emission candidate is short because it emits nearly all of its light in the red and infrared. The coordinates of five +identified stars are shown, but the identity of the red star with the candidate emission remains unidentified. The transient emission is +shown in Figure 16, and it is consistent with noise fluctuations from seeing changes and Poisson noise. + + + + + + + + +17 +19h 46 24± 28° 38 1 +Candidate +19h 45㎡ 38" ++28°89°40″ +3000 ++- +2 arcmin +19 47 29. 28° 28 42 +69 +19#-4653 28°.28 +2500 +19h 46 00 28° 19 04 +19: 48 127 28 19 06 +2000 +Pixels +1500 +1000 +500 +N up, E to left +1.2x1.2 deg +1000 +2000 +3000 +Pixels +15 + +Figure 16. Spectrum of the apparent transient emission-line at 745 nm in a 1-sec exposure (squares). Left: The solid line is the average +of 20 exposures surrounding the one exposure with the enhanced emission. Note the enhanced emission at 745 nm. Right: The 20 +individual spectra are shown as small dots, conveying the scatter in the number of photons in each exposure. The apparent transient +emission at 745 nm indeed stands more intense than the ensemble of individual spectra. But the scatter in the number of photons at each +wavelength shows that the enhanced emission is at the extreme end of that distribution, but not detached from that distribution. Thus +the enhanced emission flagged by the automatic detection algorithm is justified, but not inconsistent with the end of the distribution of +noise. + + +We could not identify a definite M-type star near the coordinates above. The two reddest stars within the +3 arcmin error circle are IRAS 19436+2834 (19 45 40.4 +28 41 55) that has V=10.72, J=5.25, and +K=3.733, and IRAS 19433 +2829 (19 45 24, +28 36 55) that has magnitudes G=13.8, J=7.40, and +K=5.44. Both stars have colors consistent with a mid to late M-type star as observed here. The first of +them is closer to the coordinates measured here. We have no other candidate stars. In any case, as noted, +the apparent emission at 745 nm is most likely mere noise from photon-arrival statistics and seeing +variations. +At Galactic longitude 248 deg, the automatic algorithm identified a similar apparent transient emission in +an M dwarf at RA=8h 01m 26s DEC=-30o 15’ 11”. The star is magnitude, R ~ 10, with the continuum +heavily chopped by TiO absorption, typical for a star of spectral type ~M3. Figure 17 shows, at left, the +spectrum from the 1-sec exposure, and at right, the average spectrum of 20 exposures. There is apparent +enhanced emission at 720 nm that is likely to be due to momentary photon-arrival fluctuations and +improved seeing. + + +600 +500 +口 +Transient +sec exp +口 +400 +6 +Avg. 20 +口 +exposures +300 +00 +口 +口 +吕 +200 +000 +口 +口 +m +口 +口 +6 +100 +口 +酒 +11 +■ +0 +Ir. +750 +800 +850 +006 +Wavelength +[nm]N Photons +200 +300 +40 +5 +600 +100 +500 +5 +0 +800 +nm +850 +6 +900 +16 + +Figure 17. Another example of an M dwarf spectrum that triggered the automatic search algorithm for transient emission, this being at +720 nm where a natural peak in the stellar spectrum occurs. Momentary excellent seeing peaks up the natural peak in the stellar +spectrum. + +4.4 P Cygni +The automated difference-image search for transient monochromatic light found “transient” line emission +at Galactic longitude 76 deg on 2022 Dec 02. The variable line emission appeared in multiple exposures, +suggesting that it was actually constant emission fluctuating due to seeing variations. Figure 18 shows the +entire spectrum of this candidate, revealing that the emission line has a wavelength of 656 nm, consistent +with H-a. Seeing variations no doubt caused the H-a intensity to vary by ~10%, occasionally triggering a +“detection” by the difference-image algorithm. The spectrum also has H-b in emission and the telluric +absorption at the A-band and B-band. There is also emission apparently at 588 nm and 503 nm, likely +from HeI. The star has Vmag ~ 5 and its approximate coordinates are RA = 20h 24 m and DEC = +37d +30’, consistent with the well-known iconic star, P Cygni. + + + +2000 +150( +S +Photons +1000 +N +500 +400 +500 +600 +700 +800 +900 +Wavelength +[nm]3×10 +Photons +2×104 +N +1×10 +400 +500 +600 +700 +800 +900 +Wavelength +[nm] +17 + +Figure 18. A 1-sec exposure revealing variable strong emission line, found by the automated code, that triggered a detailed examination +by eye. The wavelength scale shows the strong emission line is H-a, and the spectrum also contains emission at H-b and HeI (587.6 nm), +and fainter emission lines. Coordinates show this object is probably P Cygni itself, with spectral resolution inadequate to reveal the +blueward absorption, but showing a steep shortward edge of its profile. + +4.5 Be Star or Planetary Nebula +At Galactic longitude 112 the automatic search algorithm identified a transient emission line at RA ~ 23h 24m 51s DEC~+61o +14 18” (within 30”), based on astrometry calibrated by neighboring stars. The extracted spectrum is shown in Figure 19, the +sum of 20 1-s exposures. Strong Ha triggered the difference-image search algorithm due, no doubt, to seeing changes. The +object has a continuum that is blue, similar to stars of spectral type early A or B-type. The Balmer emission lines have a sharp +blueward edge indicative of gas outflow, and the spectrum contains other emission lines common from 104 K gas. Lists of +stars within a 1 arcmin revealed no obvious identifications, with closest being an A2III star BD+60 2536 (V=9.57, B=9.93), +with no spectrum published to check for emission lines. The blue continuum and emission line profiles suggest an early type +star with mass outflow, somewhat reminiscent of KjPn8 (Vazquez, Kingsburgh, and Lopez 1998). In any case, the spectrum is +consistent with an astrophysical explanation, removing it relevant for our purposes. It deserves follow-up spectroscopy. + +α +H +2.0×104 +d +Photons +1.5x10 +an +B +d +an +B +H +1.0×104 +B +Z +5.0×103 +400 +500 +600 +700 +800 +900 +Wavelength +[nm] +18 + +Figure 19. A candidate transient emission candidate at Galactic longitude 112 deg that is simply an astrophysical object with Ha that +varied due to seeing as is common. The spectrum is likely an V ~ 11 mag Be star or planetary nebula (or both), and the triggering +transient emission is at the wavelength of Ha. This blue object at RA ~ 23h 24m 51s DEC~+61o 14 18” (within 30”) remains +unidentified. + +4.6 Summary of Monochromatic Candidates + +The difference-image algorithm performed a search of 124 fields, each 3.2x2.1 deg, along the Galactic Plane in this +objective prism survey. The limiting magnitude for monochromatic emission was approximately Vmag = 13. Added to the +previously observed fields near the Galactic Centre, a total of 973 square degrees were surveyed near the entire Galactic Plane +accessible from latitude 38 deg. Each field was observed with 600 consecutive 1-sec exposures, allowing a difference-image +search for monochromatic objects of interest. The automated difference-image search for transient emission +identified 11 candidate sources of monochromatic emission, some of which were continuously emitting +but fluctuating due to seeing variations. We carefully examined each candidate, including any associated +stellar spectrum. None of the 11 candidates were pulses nor continuous emission of monochromatic light. +Instead, all of them were either astrophysical objects with a strong emission feature in the spectrum that +varied due to seeing changes or aircraft with flashing xenon lights, including one that was nearly +monochromatic. Unidentified aircraft or spacecraft that have unexpected spectral characteristics would +stand out. We found no point-sources of monochromatic emission, pulsing or continuous, that were +plausibly extraterrestrial lasers. + + +5 DETECTION EFFICIENCY: INJECTION AND RECOVERY OF LASER PULSES + +We generated 100 synthetic monochromatic pulses consisting of 2D Gaussians having FWHM ~ 6 pixels, representative of the +actual PSF of our images of the Milky Way Plane. We scaled these synthetic monochromatic pulses to various total numbers +of photons within the entire profile, from 400 to 1000 photons. These synthetic pulses ranged from roughly 0.2x background +to 1.5x the background photons per pixel. We added these synthetic pulses to actual individual images, simulating a pulse + +Sum 20 Exposures +Hα +3×104 +Photons +[ArII],[O] +Nal +Z +1X10 +4 +Hel, +H +Band +Hel +400 +500 +600 +700 +800 +006 +Wavelength +[nm] +19 +duration less than 1 sec. We placed the pulses at random locations within the image, both in between and coincident with +stellar spectra. +For each of these real images with injected monochromatic pulses, we executed the blind difference-image analysis to +determine if it “discovered” the synthetic pulses. We ran 100 cases for each level of pulse intensity. The fraction of injected +pulses detected is shown graphically in Figure 20. Blindly executing the image-difference algorithm described above, we +found the code successfully discovered 50% of the injected pulses that had at least 650 total photons in the profile. It found +none of the pulses containing fewer than 500 photons, and it found 97% of the pulses having more than 900 photons. +Thus, the nominal detection threshold at which 50% of the pulses would be detected is 650 photons total within the +monochromatic pulse. This 650 photon threshold represents the number of photons that must be detected in 1 sec such that +half of such pulses would be detected. The search algorithm has diminishing sensitivity for pulses lasting over the 1 sec +exposure time of each frame. For such cases, some of the adjacent six bookend exposures would contain the emission, +diminishing their contrast with the target image. In particular, for continuous monochromatic emission, ~6500 photons per sec +would be required in order for the 10% variations to reveal itself as a pseudo-pulse. The term “continuous” here refers to a +cadence of pulses that is more frequent than 1 pulse per second. A train of pulses of nanosecond duration and arriving 106 per +second would be detected here only as “continuous” monochromatic emission, requiring seeing variations for detection. + +Figure 20. The fraction of injected monochromatic pulses detected blindly by the difference-image search algorithm as a function of the +number of photons in the monochromatic pulse. Pulses containing 650 photons (total within the PSF) are detected in 50% of trials. +Pulses with >1000 photons are detected in ~100% of the trials. The nominal detection threshold is 650 photons per laser pulse. + + +6. DISCUSSION + + +We searched 2/3 of the Galactic Plane in a swath 2 deg wide for sub-second pulses of monochromatic emission between +wavelengths 380 and 950 nm. The technique was also sensitive to sources of constant monochromatic emission as seeing +causes momentary 10% fluctuations in the acquired photons in a one-second exposure. The goal was to search a domain of +transients, in time and wavelength, that could have been missed in past transient surveys that use exposure times over 30 sec +and usually had modest or no spectroscopic ability to detect unexpected emission lines. For example, searches for planetary + +1.0 +Detected +0.8 +Pulses +0.6 +of +Fraction +0.4 +0.2 +0.0 +400 +600 +800 +1000 +Number of Photons in Laser Pulse +20 +nebulae, HII regions, and flare stars used exposures more than 1 minute and were often confined to the detection of Balmer +lines, especially Ha. + +The detection threshold of 650 photons within a pulse translates into a threshold of photon fluxes entering the telescope. The +650-photon threshold corresponds to a fluence per unit area by using the effective collecting area of the 0.278-m RASA +telescope system, including efficiency between 450 – 800 nm and blockage by the camera at prime focus. We find that the +effective collecting area is Aeff = 0.020 m2 (Marcy, Tellis, and Wishnow 2021,2022). Thus, the detection threshold of 650 +photons implies a fluence threshold of 32500 photons per square meter at the Earth’s surface for monochromatic pulses of +duration less than 1 sec. For a pulse lasting 1 sec, that fluence corresponds to Vmag = 15. For wavelengths below 450 nm and +above 800 nm the quantum efficiency drops below 50% of peak QE (at ~600 nm), thus requiring more than 32500 photons per +square meter for detection. Atmospheric extinction raises this threshold fluence at the top of the Earth’s atmosphere by a few +percent. + +One driver for this new search was the detection of laser beams in the Galaxy. A monochromatic light source lasting a few +nanoseconds, microseconds, or milliseconds would have been detected in one exposure relative to reference exposures, with a +detection threshold of 650 photons in the pulse. We found no pulsed monochromatic sources, nor any unknown continuous +monochromatic sources, between Galactic Longitude -4 to 248 deg, in a swath ~2 deg wide along the Galactic Plane. + +A major consideration in this optical SETI program was to minimize false positives. We engineered the optics, pixel size, and +difference-imaging algorithm thresholds to avoid false positives, such as from cosmic rays, satellite glints, Cherenkov +radiation, or electronics noise. Our entire system, end-to-end, was designed to avoid them, and indeed we found none, after +careful scrutiny of candidates. Without a doubt, the optics, detector, and algorithm could be modified to make the entire +system more sensitive to monochromatic pulses, by allowing some false positives to pass through. But we elected to choose +parameters that avoided the difficult task of identifying them (e.g., Sheikh et al. 2021, Villarroel et al. 2022). + +Some pulse cadences would not have been detected. For example, a duty cycle of one pulse per day, week, or month, implies +that one pulse could arrive while we were not observing. Obviously, arbitrary pulse cadences with low duty cycle bring a +lower detection probability. More observations are warranted to provide greater cadence coverage. This survey effectively +searches for cadences of at least one pulse every 10 minutes, the duration of the observing sequence of a given field. Pulse +cadences more rapid than 1 Hz are effectively “continuous” in time. In such cases, the detection threshold is approximately +10x higher, as only the ~10% fluctuations from seeing variations will cause such continuous sources to vary above the +detection threshold in the difference images. + +We can compute the laser power needed, from interstellar distances, to produce a detectable fluence. We consider a benchmark +laser that is diffraction limited with a 10-meter diameter aperture, and located 100 ly from Earth. Extinction by dust is ~10%. +It emits a beam with an opening angle of ~0.01 arcsec at a wavelength of, say, 500 nm. To produce a photon flux at Earth +having the detection threshold of 32500 photons per square meter, a power of 100 Megawatt is required during the 1 sec pulse. +For a laser launcher located 1000 ly away, a power of 10 Gigawatt is required. Extinction from interstellar dust will increase +this requirement by ~30%. At 1 kpc, the required power is ~90 Gigawatt. Extinction from dust will require another factor of +two in power. The laser beam footprint at Earth would have a diameter of 0.3 to 3 AU, respectively, a fraction of the area of the +inner Solar System. Such small footprints imply that the laser is purposely or fortuitously (or mistakenly) pointed at Earth. +Lasers having apertures smaller than 10 meter can also be detected, but their larger opening beam angle would require more +laser power, increasing inversely as the square of aperture diameter. + +A benchmark receiver might have 100x the diameter of the collector and thus 10,000x higher collecting area. This larger +collector can thus reliably receive 10,000x the bit rate here (Gertz & Marcy 2022). A string of cell towers overcomes this +limitation of bit-rate. + +7. SUMMARY + +More than 5000 stars have been searched for monochromatic laser emission (Reines and Marcy 2002, Tellis and Marcy 2015, +2017, Marcy 2021, and Marcy et al. 2022). Most surveyed stars are Sun-like or smaller with spectral types F, G, K, or M-type. +But several hundred are massive stars of spectral types O, B, A, and early F-type (Tellis, private communication 2022). We have +also searched the focal points of the solar gravitational lens (Marcy et al. 2021). Many searches for optical pulses of sub-second +duration have been performed, covering over half the sky (e.g. Wright et al. 2001, Stone et al. 2005, Howard et al. 2004, Howard +et al. 2007, Maire et al. 2020). No optical laser emission, pulsed, monochromatic, or otherwise, has ever been detected. + +Optical lasers could have been detected previously by “conventional” astrophysics searches, both broad-band and +spectroscopic. Objective prism telescopes 100 years ago (Fleming et al. 1907, Pickering 1912, Cannon & Pickering 1922) +revealed thousands of objects that emit emission lines, such as planetary nebulae, HII regions, T Tauri stars, Be stars, Wolf- +Rayet stars, M dwarf flare stars, and active galactic nuclei, including at high redshift. Such objective prism surveys yield + + +21 +emission-line objects that are then pursued with spectroscopy of modest resolution that is able to reveal non-astrophysical +emission lines if any existed. Even optical searches using mere broadband filters, such as BVRI, would reveal objects whose +emission was confined to one or two bands. A 13th mag source whose emission was confined to one or two emission lines +would exhibit bizarre colors, meriting follow-up spectroscopy (ala Lyman-break galaxies). The emission at non-astrophysical +wavelength would be immediately obvious. Many all-sky surveys reached 18th magnitude, and modern ones reach 21st +magnitude. No object with strange colors turned out to be lasers. In summary no monochromatic sources with non- +astrophysical emission lines, e.g., lasers, were discovered in hundreds of past all-sky surveys, nor in the present survey. + +The present non-detection of lasers, and those from the hundreds of past surveys, does not imply that extraterrestrial +technology is absent in the Milky Way. To be sure, other wavelengths and pulse cadences merit observations. Also, the filling +factor of optical laser beams may be too small (Forgan 2014). A vast parameter space is yet to be surveyed (Wright et al. +2018). However, a large fraction of observable SETI parameter space has already been surveyed by both conventional +astronomy surveys of the entire sky and by explicit searches for extraterrestrial technology (e.g., Wlodarczyk-Sroka, Garrett +& Siemion 2021, Garrett & Siemion 2022). + +Most impressively, SETI parameter space has unintentionally been surveyed by all-sky searches for natural, astrophysical +objects. Those surveys gloriously revealed a multitude of astrophysical objects that emit, unexpectedly, at all wavelengths, +including radio, microwave, extreme UV, x-rays, and gamma rays. Those unexpected discoveries constitute non-detections of +extraterrestrial technology in the same search domains, but not recorded as such. The dearth of beacons of technology leaves +us revealing more of a great SETI desert, especially at the intensely surveyed optical and radio wavelengths. + + +ACKNOWLEDGMENTS +This work benefitted from valuable communications with Beatriz Villarroel, Franklin Antonio, John Gertz, Ben Zuckerman, +Brian Hill, Susan Kegley, Dan Werthimer, Ariana Paul, Roger Bland, Martin Ward, and Paul Horowitz. We thank the team at +Space Laser Awareness for outstanding technical help. + +DATA AVAILABILITY + +This paper is based on raw CMOS sensor images obtained with Space Laser Awareness double objective prism telescopes. The +8864 raw images are 125 MB each, totaling 1.1 TB. 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SPIE, 4273, 173 + +Zhang Y., Liu X.-W. 2003, A&A, 404, 545 + +Zuckerman B., 1985, Acta Astronautica, 12, 127. + + + + +24 +This paper was typeset from Microsoft WORD document prepared by the author. + diff --git a/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/load_file.txt b/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1939f6d547dca75e5852c5ee8c1b4ace71a09c63 --- /dev/null +++ b/8dAzT4oBgHgl3EQfSPuZ/content/tmp_files/load_file.txt @@ -0,0 +1,1029 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf,len=1028 +page_content='1 A Search for Transient, Monochromatic Light from the Galactic Plane Geoffrey W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Marcy1 & Nathaniel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Tellis2 1 Center for Space Laser Awareness, 3388 Petaluma Hill Rd, Santa Rosa, CA, 95404, USA 2RocketCDL Accepted xxx Received xxx ABSTRACT The Galactic Plane was searched for transient, monochromatic light at optical and near-IR wavelengths to detect pulses shorter than 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' An objective-prism Schmidt telescope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='28-meter aperture and a CMOS camera were used to observe 973 square degrees, with 8864 exposures of 1-sec each, within a strip 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 deg wide along the Galactic Plane, from Galactic longitude -4 deg to +248 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' All exposures were analyzed for transient, monochromatic sources using a “difference image” algorithm that yielded 11 candidate sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' All 11 sources were found to be associated with either astrophysical emission-line objects or aircraft with sub-second blinking lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Our survey “rediscovered” many Wolf-Rayet stars, M dwarf flare stars, and planetary nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It also identified an aircraft, of unknown type, that apparently had a nearly monochromatic lamp and a xenon lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This survey would have revealed optical and near-IR pulses having a power of ~180 GW (wavelength dependent) if emitted by a 10-meter aperture laser located 1 kiloparsec away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' These non-detections of laser pulses from the Galactic Plane, including a 10-degree region toward the Galactic Centre, add to the non-detections from more than 5000 nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Indeed, all-sky surveys for emission-line objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', ionized gas, supernovae remnants, and active galactic nuclei) would have revealed lasers of a wide range of average brightness, wavelength, and cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The absence of beacons reveals more of a SETI desert, notably at the intensely surveyed optical and radio wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Key Words: Transients, Extraterrestrial intelligence, Galaxy, Techniques: Spectroscopic 1 INTRODUCTION Surveys of the sky with time resolution have serendipitously revealed unexpected time-variable, dynamical astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Historic examples include eclipsing binary stars, Cepheid and RR Lyrae pulsating stars, flare stars, cataclysmic variables, and supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' More recent examples include previously unknown classes of objects such as pulsars, x-ray binaries, kilonovae, gamma ray bursts, and fast radio bursts (FRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Surveys at radio wavelengths have enjoyed a natural propensity at finding unexpected sub-second transients because of the necessity to record voltage every ~10-6 s or ~10-9 s (at MHz or GHz frequencies), leading to the unanticipated discoveries of pulsars and FRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' These fortuitous successes motivate searching unexplored domains of time and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Very few all-sky surveys at ultraviolet, optical, and IR wavelengths have been done with exposure times shorter than 20 s, and fewer still with exposures shorter than 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Long exposure times cause sub-second flashes to be diluted relative to the background night sky brightness, making them less visibl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The LaserSETI program at the SETI Institute employs millisecond exposures to overcome such dilution (Gillum 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The recent development of astronomy-quality CMOS sensors offers sub- second readout times, small pixels of a few microns size, and large pixel arrays >50 megapixels, providing access to transient phenomena having sub-second time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Here we report a search for monochromatic, sub-second pulses of optical and near IR light, motivated both by the unexplored domain of time and wavelength and by a speculative model of interstellar communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The Milky Way Galaxy may contain spacecraft, communication relay stations, or home star systems that communicate by lasers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', Bracewell 1960,1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Freitas 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Maccone 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Gillon 2014, Hippke 2020, 2021abc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Gertz 2018, 2021, Gertz & Marcy 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Indeed, lasers are already widely used for communication by Earth-orbiting satellites because they offer narrow-beams for privacy, high bit rate, and minimal payload mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Similarly, laser communication, or next-generation coherent quantum communication methods, may be used for interstellar communication (Schwartz and Townes 1961, Zuckerman 1985, Hippke 2018, 2021abc), perhaps using repeater nodes (Gertz & Marcy 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Laser light may be identified in telescopes by its narrow range of wavelengths, hereafter “monochromatic” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', Naderi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2016, Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2014, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2020), regardless of the unknown duration or cadence of the pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Previously, we have 2 searched for monochromatic light from more than 5000 individual stars of all masses, ages, and chemical compositions (O,B,A,F, G, K, and M) using high-resolution optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' These extensive searches yielded no laser detections and no viable candidates (Reines & Marcy 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Tellis & Marcy 2017, Marcy 2021, Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022, and Tellis 2022, private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' These laser searches employed spectra of high resolution, l/Dl > 60000, in the wavelength range l = 3600 to 9500 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The detection threshold of laser power was 50 kW to 10 MW, assuming a diffraction-limited laser emitter consisting of a benchmark 10-meter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We also searched a 10x10 degree field at the Milky Way Centre, albeit at low spectral resolution, yielding no laser detections (Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We also searched the solar gravitational lens focal points for nearby stars (Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' No monochromatic light was found, neither sub-second pulses nor continuous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Searches for narrowband radio waves from technological entities also make use of the “monochromatic” nature of a signal to distinguish it from ordinary astrophysical sources and to promote the candidacy of technological signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Isaacson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2017, Enriquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2017, Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2020, Wlodarczyk-Sroka, Garrett, & Siemion 2020, Sheikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021, Garrett & Simion 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is no guarantee that interstellar communication will be nearly monochromatic or multi-bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' But that characteristic offers a search property that excludes the vast majority of false positives that are either astrophysical or terrestrial noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Without spectroscopic information, other searches for technological signals have been done by hunting for sub-second optical pulses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2007), Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2009), Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2016), Villarroel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' No definitive optical pulses were found, but some candidates emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Next generation searches for sub-second optical pulses are planned (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Here we describe a search for sub- second pulses of monochromatic light along the Milky Way Plane using a special optical system designed to optimize this search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2 OBJECT-PRISM OBSERVATIONS OF THE MILKY WAY PLANE We used the objective prism Schmidt telescope operated by the Center for Space Laser Awareness and described in Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2021, 2022) and at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='spacelaserawareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='org .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In brief, the telescope is a modified Schmidt design with aperture diameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='28 m and a 7-degree wedge prism to produce spectra of low resolution, R~100, of every point within the 2x3 deg field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The CMOS camera at the focal plane contains 9400 x 6600 pixels, each 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='7 microns with a quantum efficiency over 80% between 500 – 800 nm and lower QE extending to 370 nm and 950 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We operated with an exposure time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 sec and dead time of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='01 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The system is optimized to detect monochromatic pulses of optical light (see Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021, 2022) having pulse duration less than 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Monochromatic emission has the shape of a two-dimensional PSF, with a FWHM ~ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 pixels, allowing efficient search algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In contrast, direct hits by particles or gamma rays (cosmic rays) make “dots” sharper than the PSF, allowing discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Another nemesis comes from reflections off satellites (Corbett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021, Nir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021) which exhibit the solar spectrum, obviously not monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The optical design is remarkably similar to the objective prism telescopes of the Harvard Observatory (Pickering 1912;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Fleming 1917), albeit with 50x the QE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We performed multiple performance tests of the QHY600 CMOS camera, finding read noise is ~2 photons (RMS), the dark noise <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 e-/s per pixel, and the response is linear within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='3% over a dynamic range of a few to 56,000 photo-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We operate with modest thermoelectric cooling to -20C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The properties of the QHY600M are comparable to CCDs (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022, Betoule et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022), but offer frame rates up to 30 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We can detect light pulses having sub-second duration with minimal contamination from the background “noise” of stars, galaxies, and sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' From our observing station at Taylor Mountain in California, the sky produces ~40 photons/pixel during 1 sec coming mostly from Santa Rosa city lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 1 shows a sum of 10 images obtained with the objective prism telescope system and QHY600 CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This image is centred at Galactic Longitude 38 deg and Galactic Latitude 0 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The image shows hundreds of stellar spectra oriented vertically, each spanning wavelengths 380 – 950 nm spread over 1200 pixels, with long wavelengths downward and North up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In the 1 sec exposures, the faintest stars have Vmag=13 with signal-to-noise ratios of ~10 per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Stars brighter than Vmag = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 saturate the sensor with >56000 photons/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A monochromatic, spatially unresolved point source would appear as a two-dimensional “dot” with a PSF shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Sub-second monochromatic pulses would appear in only one image as a PSF-shape “dot” within a sequence of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This survey searches for cadences of at least one pulse every 10 minutes, the duration of the observing sequence of a given field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Continuous sources and high cadence sources, >1 Hz, would appear in all images in a sequence of 600 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We judge the PSF by the width of the spectra in the spatial direction that is dominated by both seeing and optical imperfections in the prism, yielding a PSF width of typically 6 to 8 arcseconds corresponding to a FWHM 5 to 6 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The sum of 10 images (1 sec each) from the objective prism system with a field of view of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 deg, 9500 x 6300 pixels, each subtending 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='3 arcsec on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The stellar spectra span wavelengths 370 – 950 nm with longer wavelengths downward and north up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This image is centred at Galactic longitude 38 deg, at RA=19h 00m, DEC=+4o 30’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The stellar spectra come from stars of magnitude 8 – 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Monochromatic emission would appear as a PSF-shape “dot”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 [deg] DEC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 6 4 2 0 2 4 6 RA 一 19hr 00min min 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Spectrophotometry of Vega vs wavelength obtained with the objective prism telescope and CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum establishes the spectrophotometric sensitivity, wavelength scale, and spectral resolution, R~100, of all spectra that have a length 1200 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Prominent absorption lines in Vega and strong atmospheric lines are labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Exposure time was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A spectrum of the planetary nebula NGC7027 with the objective prism system and 5 sec exposure total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It shows emission lines common from ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The wavelength scale is based on a spectrum of Vega, with a zero point set by Ha here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Laser emission is easily distinguished from known astrophysical sources by their pattern of known emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figures 2 and 3 show spectrophotometry of Vega and NGC7027 obtained with the objective prism system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The reduction to one dimensional spectra was accomplished by a simple summing of the photons along the spatial width at each wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum of Vega (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 s exposure) shows the Balmer lines up to H11, along with telluric lines (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The wavelength calibration was done with a 7th order polynomial fit to 14 pixel positions and the corresponding wavelengths in that Vega spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The prism creates a highly nonlinear wavelength dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The resulting Vega spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2 shows that stellar spectra can be classified and that emission-line spectra can be identified, to distinguish them from non-astrophysical NGC7027 10x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 sec nm 2021 July 20 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='7 4×10* Pixel + 3×104 α per n H 9 Photons 5 9 4 2x104 u 732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 nm m [ArIV] uu n 713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 1×10* + 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 6 + 5 Arl H H I H S 400 500 600 00 800 006 Wavelength 1 [nm] 5 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrophotometry of Vega in Figure 2 is given in photons per nm per sec detected with our objective prism system, allowing this spectrophotometry to map magnitude to photons per nm per sec of other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The monotonic decrease in photons detected for wavelengths shortward of 440 nm is due to decreasing quantum efficiency of the CMOS sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectral and spatial resolutions are set by the PSF that has FWHM ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 pixels, dominated by seeing and optical aberrations in the prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum of NGC7027 (Figure 3) shows the usual emission lines from ionized gas at 10000K (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Zhang & Li 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The two [OIII] lines at 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='9 and 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='7 nm are barely resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This modest spectral resolution, 2 to 10 nm from 380 to 950 nm, allows the identification of stars, galaxies, ionized gas, asteroids, aircraft, and satellites, to distinguish them from less common phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Reflected sunlight and glints from orbiting satellites are easily identified by their solar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This allows light pulses from Earth-orbiting satellites and discarded rocket boosters to be immediately distinguished from extraterrestrial laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectroscopic information allows instant discrimination of both astrophysical and terrestrial sources from monochromatic extraterrestrial sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' However, laser pulses from human-made satellites could be indistinguishable from extraterrestrial laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Satellite-born laser pulses that are sufficiently brief to avoid detecting an orbital “streak” on the image, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' less than ~1 arcsecond, can masquerade as extraterrestrial laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 4 shows the RA and DEC of the fields we observed, each field having of angular size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 deg located along the Galactic Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Also shown on Figure 4 is the 14 x 10 deg region we previously observed near the Galactic Centre, as labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The total coverage is 973 square degrees along the Galactic Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Missing is the Galactic Plane south of 34 deg, inaccessible to our observatory in Northern California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The field toward the Galactic Anti-Centre, at RA = 5h 45m 37s DEC=+28 56’, was observed four times, as that direction is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At the Anti-Centre, laser guide stars or communication lasers may be pointed toward us, but actually intended for the Galactic Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each field was observed with 600 consecutive 1-sec exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This survey detects arrival of at least one photon pulse every 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The fields overlap to provide both complete coverage of the region and security against algorithmic or optical poverty at the edges of the field such as from poor background assessment or vignetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The 124 fields, each 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 deg, observed along the Galactic Plane in this objective prism survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Also shown are the previously observed fields near the Galactic Centre, for a total of 973 square degrees surveyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each field was observed 60 40 Galactic Anti-Centre (deg) DEC 20 Galactic Centre 0 5 10 15 20 RA (hr) 6 with 600 consecutive 1-sec exposures, giving time-resolved spectra of all points in the sky to reveal sub-second or continuous monochromatic optical pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 3 THE DIFFERENCE-IMAGE ALGORITHM We search for monochromatic emission that appears as a transient PSF-shape “dot” in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We employ a difference- image technique described in Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2022), similar to the difference algorithm in Vasilyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In brief, the algorithm operates on a set of 600 exposures, each 1 s, for a specific 2x3 deg field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The algorithm takes the average of six “bookend” images (three prior and three subsequent) surrounding a given “target” image and subtracts the average bookend image from the target image to yield the “difference image”, having pixel values near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Residuals are due to Poisson noise of the arrival of photons and to the variations in atmospheric “seeing” from image to image that compromises the subtraction of stellar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We suppress these residuals by performing a 50-pixel boxcar smoothing of the difference image along the direction of dispersion of the spectra, and we subtract that smoothed version from the original difference image (see Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This process subtracts the residual continuum of each star spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Narrow emission lines in the target image that are not in the bookend images will persist in the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This difference-image algorithm yields any monochromatic point sources that were present in each image but not present (or only weakly present) in the average of the six “bookend” images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Examples of this process are shown in Figure 5 of Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (2022), and we used the same algorithm here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each target difference-image (9500x6300 pixels) was examined blindly by this algorithm, yielding PSF-like candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Emission-line sources may be coincident with stellar spectra or they may be located in between them, and the difference-image algorithm suppresses light from both stars and the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The algorithm further demands that the candidate point sources must have a 2D shape consistent with the instantaneous point spread function (PSF), as measured by the spatial profile of the stellar spectra determined by cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For each exposure, the algorithm measures the FWHM of the spatial profile of stellar spectra, commonly 5 to 6 pixels (6 to 8 arcseconds), caused by seeing and optical aberrations in the prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We note that “cosmic-ray” particles that hit the CMOS sensor are immediately rejected, as they affect only a few neighboring pixels, inconsistent with the smooth Gaussian-like shape of the PSF with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 pixel FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The algorithm is designed to detect sub-second monochromatic pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' However, if the cadence of light pulses is more frequent than 1 pulse per second, the image-difference algorithm will be compromised in detecting them because the pulses appear in both the target image and the bookend reference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' If the pulse amplitudes are not constant or if the seeing changes over a period of 7 s, the pulses may still be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Cadences slower than ~1 pulse per second will yield individual frames containing the point source and neighboring ones that do not, making the pulse detectable with the difference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Somewhat surprisingly, sources of monochromatic light that are constant in time are usually detected by the difference-image algorithm despite appearing in both the target and bookend images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The seeing changes on sub-second time scales causes the captured number of photons to vary by more than 10% from image to image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The intensity of emission lines from astrophysical objects such as flare stars, planetary nebulae, and Wolf-Rayet stars varied in apparent brightness by 10% - 20% (RMS) among the 1 s exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The emission lines are detected by the difference-image algorithm as if they were transient sources, even though they are actually steady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, the algorithm actually detects monochromatic point sources no matter if they last less than 1 s or are continuous in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4 DETECTED MONOCHROMATIC CANDIDATES We executed the difference-image algorithm to all 124 fields and their 600 exposures per field along the Galactic Plane region (Figure 4), yielding 11 monochromatic objects of interest, each requiring visual inspection and assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We describe the monochromatic candidates here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 Moving Multiple Flashes: Probably Aircraft On 2022 Jan 24, we obtained 600 exposures, 1 s each, at Galactic longitude 92 deg for which the automated difference-image algorithm gave an alert of several monochromatic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Upon inspection, the object clearly had several components, as seen in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' That figure shows the seven consecutive raw images, each panel showing the full frame image, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 deg, with north to the left and longer wavelengths to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A few hundred stellar spectra are apparent, of V magnitudes 6 to 13, fixed in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The field has coordinates RA=21h 20m 30s +49o 46’, in the northwest region of the sky at an altitude 40 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', and the angular velocity of the object is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='55 deg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 7 An odd-shaped object first appears (barely) in the upper left of the first frame, and it moves down and to the right each successive image during the full 7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Successive locations along a diagonal path, from upper left to lower right, represents time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A sequence of seven consecutive full frames, each a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 s exposure, at Galactic longitude 92 deg observed on 2022 Jan 24 at 2:33 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' An object enters the field of view at upper left and moves down and to the right each second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' During 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 s, several flashes occur having a broad spectrum, 1000 pixels across left-right, and one flash (the brightest) occurs having a smaller range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The diagonal line in each frame represents a light that is shining continuously during the full 7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It is apparently a nearly monochromatic light because it exhibits very little extent in the wavelength direction (left-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A magnified view of the third frame is shown in Figure 6, in which the stellar spectra were subtracted using the previous and next images, leaving only the moving object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figures 5 and 6 show that the object travelled from upper left to lower right during each second, including a continuous diagonal line caused by a light source that stayed on during the entire 7 seconds of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Remarkably, this “diagonal light source” has a wavelength extent, left to right length, that is only ~60 pixels FWHM, corresponding to a wavelength spread of ~22 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This light is nearly monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We do not know its central wavelength, nor how this continuous light was produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figures 5 and 6 also show at least six flashes of light that happened during the 1 second exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each flash caused a long horizontal line showing that the light in the flashes consisted of the full optical wavelengths, 380 – 950 nm, similar to the background stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The horizontal lines are narrow, only ~20 pixels wide along the diagonal (time) direction, corresponding to ~1% of the full diagonal length of travel during 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This shows that each flash lasted ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='010 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectra of all 6 flashes show spectral structure at the far right end (longest, near-IR, wavelengths), consisting of at least to emission bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The magnified view of the laser candidate in Figure 5, the 3rd image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Background stars were subtracted using adjacent images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The diagonal streak is caused by motion of a light source from upper left to lower right and shining during the full one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Its narrow width, left right, shows it was nearly monochromatic, still not understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each of the six flashes exhibits a broad range of wavelengths (left right extent), with emission lines at the longest wavelengths (at the right end of each flash, consistent with xenon lamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The brightest flash (bottom right) is only 400 pixels long, indicating a short wavelength range, but the same emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 8000F 8000 2000 10002000 Column7000 6500 Row Pixel 6000 5500 500 1000 1500 2000 2500 3000 Pixel Column 8 The brightest horizontal line located at bottom right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 6 happened at a time ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='9 sec during the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 s exposure, judging from its location along the diagonal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It contains only 1/3 of the full range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The emission structure at its far right of its spectrum resembles the emission structure at the far right of the other flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Apparently, those wavelengths are at the far red and infrared end of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' That brightest flash at bottom right also has a duration of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='01 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' One other attribute of all of the lights is that they continued to illuminate the telescope during the full 7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This suggests that all the light sources emitted a beam that was broad enough in solid angle to keep the telescope bathed in light during the full 7 s of the object’s motion across 4 deg of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' To accomplish this steady bathing, the beams of light must have been many degrees across at least, and perhaps nearly isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We wonder what combination of light sources can create the multi-component images in Figures 5 and 6, and what type of object they are attached to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A clue comes from spectral features in the full-length spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At the long wavelength end of each spectrum (far right) is clearly some structure, and perhaps broad emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We extracted those spectra from the raw image by simply adding 10 rows along the full length of the spectrum, one of which is shown in the left panel of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum shows two broad, strong emission lines in the near-IR, at wavelengths ~830 nm and ~900 nm, along with some weaker emission lines at 480 nm and ~540 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Laboratory spectra of xenon displays its two strongest lines at 850 and 910 nm, in good agreement with the two strongest seen here in Figure 7 (Povrozin & Barbieri 2016, and https://mmrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='edu/Stark/Xe%20lamp%20spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='pdf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The third strongest emission feature fin the observed object is a pair of closely spaced lines at ~470 nm (see Figure 7) that also agrees with the lines seen in lab xenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum in Figure 7 also has a close pair of lines at 540 nm, which is not in the lab xenon spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This may indicate another gas, besides xenon, in the lamp, such as mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Large airplanes typically have at least eight different external lights, usually composed of xenon gas, each having different color filters and beam directions, some of which flash at intervals near 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The angular speed of the object, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='55 deg s-1 is consistent with that of aircraft by common experience, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', a Moon diameter per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Aircraft xenon lamps often have filters to produce “green” and “red” lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' LED lights are now also being used on aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We wonder if an LED coupled with a narrow band filter could produce the “diagonal” line that was on continuously with its narrow wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' LIDAR seems unlikely, as the lasers usually operate at 1064 nm, or frequency doubled to 530nm, both having narrow wavelength ranges of < 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The observed wavelength range of 22 nm is inconsistent with such lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Alternatively, a xenon lamp that was continuously on, but covered by a narrowband filter, could produce the persistent diagonal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is also the possibility of reflected light, laser or otherwise, off the aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In any case, the most likely explanation for this “monochromatic object of interest” is an aircraft with multiple, flashing lights with different filters, with one lamp shining continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Indeed, the northwest region of the sky is the direction of a small airport, Charles M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Schulz Airport, 18 km away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We don’t know what causes the continuous “diagonal” light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Photons per pixel vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' wavelength of the two bright flashes in rows 6120 and 5580 in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Both spectra show emission lines at 830 and 910 nm, which match the known strongest emission lines in Xenon gas, commonly used in airplane lamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The weak emission lines at 475 nm here also appear in laboratory spectra of Xenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum at right is missing light shortward of 600 nm, undoubtedly caused by a filter that transmits only light longward of 600 nm, making a “red” light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, the lamps are made of xenon gas, as is commonly used on aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' On 30 Nov 2021, observations at Galactic longitude 46 deg revealed a new object of interest, detected by the automated difference-image algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 8 shows a sequence of seven full frames, each 1 second duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' As with the previous candidate (Figures 5, 6, 7), there are several flashes lasting less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='05 sec, several being full spectrum and one being red and bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' All of them contain the strong emission lines in the near-IR indicating xenon lamps again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A light with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×105 Lamp 5 Pixel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5×105 Photons per 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×10§ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×104 400 500 600 002 800 900 1000 Wavelength [nm]6×10 Red Lamp per Pixel 5×105 4×105 Photons j 2×105 1×105 0 400 500 600 700 800 900 1000 Wavelength [nm] 9 continuous spectrum is also apparent, again lit continuously, seen as the faint diagonal streak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The angular speed is again ~1 deg/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We presume this object is also an aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A transient found at Galactic longitude 46, seen entering the field in the third frame and exiting in the 5th frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The full spectrum flashes of light, and diagonal trajectory, are consistent with the xenon lights on an aircraft .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 Wolf Rayet Stars At Galactic longitude 8 deg, the automated difference-image code identified a transient emission line among the 600 1-s exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Extraction of the raw image along the full spectrum revealed other emission lines, shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We accomplished the wavelength calibration by employing, blindly, the calibration of wavelength vs pixel obtained from the spectrum of Vega used in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The zero-point of a wavelength scale was set by using the pixel at the far infrared end of the spectrum in the raw image, deemed to be a wavelength of ~950 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The uncertainty of that zero-point is ~50 nm, due to the gradual dimming at the end of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This approximate wavelength scale allowed the emission lines to be identified, within 50 nm, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Spectrum of a changing emission line at wavelength 585 nm, identified by the automated difference image algorithm, in the field at Galactic longitude 8 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The extracted spectrum shows other emission lines, consistent with a Wolf Rayet star of type WC5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The coordinates from our image (upper right) suggest this is the known Wolf Rayet star, “WR111”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 246 3000F 2600 2000 1500 1000F 600 1000 2000 3000 400 1000 3000 40 1000 2000 3000 400 1000 2000 3000 4自 1000 2000 8000 40 1000 2000 9000 4000 1000 2000 3000 40005000 wu RA=18h08m nm 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='6 DEC=-21 15 581 4000 CIV Hel Photons CIII,CIV 465 nm Hell468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='6nm 3000 P 2000 Z CIII Hell 1000 400 500 600 700 800 006 Wavelength [nm] 10 That approximate wavelength scale showed the similarity of the spectrum with a standard Wolf-Rayet star of type WC5, based on the catalog Wolf-Rayet spectra given at: https://lweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='edu/~pberlind/atlas/htmls/wrstars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='html and on the classification by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (1996) and Crowther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Using the lab wavelengths of identified lines, we refined the zero-point of the wavelength calibration to yield the spectrum shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We performed astrometry of our image, yielding coordinates, RA = 18h 08m and DEC=-21d 15’, at which exists the known Wolf-Rayet star, WR 111=HD 165763 (type WC5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This obviously removes this candidate from further consideration as non-astrophysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A Galactic longitude 74 deg, the automated code revealed another emission line that varied in brightness during the 600 1-sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It met the criteria of a pulsing laser candidate in 28 of the exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 10 shows the full spectrum, revealing the emission line at 465nm, and also several other emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The line at 465 nm appears to vary in intensity because of seeing changes, easily verified by the widths in the spatial direction of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The pattern of emission lines matches that of Wolf-Rayet stars of type WC8, with coordinates approximately RA=20h 15m and DEC = +36d 38’, with a brightness approximately V = 9, with a possible identification as HD 192641, a WC7 Wolf-Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, there is no support for a non-astrophysical interpretation, laser or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' An emission candidate detected automatically by the intensity variations of the emission line at 465nm due to changes in seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=" This is apparently a WC8 Wolf Rayet Star of magnitude V~9, at approximate coordinates, RA = 20h 15m , DEC=+36d 38'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We rule out extraterrestrial technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In the field at Galactic longitude 76 deg, the automated search triggered on another candidate transient emission line, shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum shows it to be a WC7 Wolf-Rayet star, and the coordinates show that it is HD 192641, an 8th mag WC7 Wolf-Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The automated search for transients was triggered by seeing variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 3000 Photons 2000 Z 1000 400 500 600 700 800 900 Wavelength [nm] 11 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The automated search for transient emission lines discovered this candidate, which is clearly a WC7 Wolf Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This spectrum results from adding 20 1 sec exposures to improve the signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Seeing changes caused the most intense emission line at HeI 465 nm line to momentarily brighten, triggering the alert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It is likely HD 192641, a WC7 Wolf Rayet star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='3 M-Type Stars At Galactic longitude 38, our automated code revealed an apparent, changing emission line at a wavelength of 720 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A plot of the full spectrum, shown in Figure 12 (top), shows a spectral energy distribution that is clearly an M dwarf, with the characteristic TiO absorption bands at the red and near-IR wavelengths (Leggett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is a naturally occurring peak at 720 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Examination of a sequence of seven images, shown in Figure 12 (bottom), shows that the momentary improved seeing conspired to yield an apparent increase in the peak intensity, fooling the code into sensing an emerging emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We see no evidence of a PSF-like line, and instead relegate this candidate to the common occurrence of momentarily improved seeing at a wavelength with natural high intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5×104 Photons Z 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0x10° 500 600 700 800 900 Wavelength [nm]1000 800 Photons 600 400 200 0 400 500 600 700 800 006 Wavelength [nm] 12 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A candidate emission line at a wavelength ~738 nm (column 2181) at Galactic Longitude 38 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='. Top panel: the spectrum of the object, clearly an M dwarf (M4 to M5) with an apparent emission line at 738 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Bottom Panel: the seven consecutive raw images zoomed on the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The apparent intensity increase of the emission is just due to momentary improved seeing at a peak of the spectral energy distribution, between TiO absorption bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At Galactic Longitude 72 deg on 2021 Dec 02, the automated code identified a monochromatic brightening at wavelength 827 nm (column 6167), in a star with an M4 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 13 shows the extracted spectrum vs column #, both for the single exposure (left panel) that yielded the candidate and for the sum of 20 exposures (right panel) that gives an average spectrum over 20 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The candidate identified by the difference-image algorithm is located at a persistent peak in the spectrum of the M4 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' On that one exposure there was a momentary enhancement of the intensity that is obviously consistent with the noise level in that single exposure, thus making the apparent transient emission line to be likely noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Indeed, examination of the raw image shown in Figure 14 shows that the enhanced emission was concentrated within 4 pixels, which is inconsistent with the PSF that has FWHM ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, we suggest that this candidate monochromatic emission is simply noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Follow-up spectroscopy of this M star may be warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Unfortunately, the identity of the M star in Figures 13 and 14 remains a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The coordinates from our images are approximately RA= 20h 13m 10s, DEC=+33d 07’ (eq 2000, epoch 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' From our image, its magnitude is Rmag ~11, where two plausible stars reside, HD331958 and TYC 2675-1608-1, neither of which has high quality characterization in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' HD331958 has properties on SIMBAD listed B-V=+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='19 mag, consistent with a K5 star not the M4 spectrum we see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' SIMBAD lists its spectral type as B8, which is inconsistent with both K5 and M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There are multiple inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Its parallax is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='16 mas and proper motion is 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 mas/yr, implying a transverse space velocity of 26 km s-1, which is consistent with a Milky Way disk star and with its measured radial velocity of 48 km s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The Vmag and parallax imply an absolute magnitude of MV = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='62, which is consistent with K dwarf implied by the B-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' However, the spectrum we see is clearly an M4-M5 dwarf from the obvious TiO bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, HD 331958 seems unlikely to be the candidate shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The other nearby catalog star of comparable brightness is TYC 2675-1608-1 (20 13 10, DEC=+33 07) with SIMBAD photometry, V=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='86 and B-V=+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='65 and J-V = +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='28, sufficiently red to be an M4 star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Its listed parallax is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='623 mas, implying a distance of 1600 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' That great distance rules out M dwarf status for a star having Vmag=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' One solution that satisfies the parallax, Vmag, and the observed M4 spectral type is an M supergiant at 1600 pc having MV ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The measured proper motion of 12 mas yr-1 and distance ~1600 pc implies a transverse velocity of ~91 km s-1, much larger than the stated radial velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='12 km s-1 listed on SIMBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Such a mismatch of velocity components raises concerns about a mistake somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In any case, the apparent emission appears to be noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Follow-up spectra are certainly warranted to verify that the apparent emission at 827 nm is indeed merely noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A candidate transient emission line in an M dwarf at Galactic Longitude 72 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At left: The extracted spectrum vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Column #, showing the location of the candidate emission found by the automated code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At right: The sum of 20 exposures, showing that the candidate emission (top) resides where there is persistent emission, but is consistent with the noise in a single spectrum, given 600 exposures to draw from, as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 590 591 592 594 595 598 9850 3B40 皖· 9830 3B20E 1402160 2180 2200 2880 2140 2160 2180 2800 2880 2140 2180 2180 2200 2280 2140 160 2180 2200 2880 2140 2180 2180 2800 2280 2140 2160 2180 2800 2880 8140 8180 8160 2200 2880400 Candidate Emission 300 S Photons 200 100 5950 6000 6050 6100 6150 6200 6250 Column [Pixels]6000 Candidate Emission 5000 S Photons 4000 3000 Z 2000 1000 5950 6000 6050 6100 6150 6200 6250 Column [Pixels] 13 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The raw image of the emission candidate shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The bright pixels near the center are not distributed smoothly over the full vertical length of the PSF in the spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This indicates the brightening of “emission” is due to noise or a cosmic ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At Galactic longitude 64 deg the automated difference-image code identified candidate transient emission located on the spectrum of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 15 shows a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 deg zoom with the spectrum of the star in the NE corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum is short in wavelength, and it exhibits four broad, bright wavelength domains, typical of the molecular bands of M stars with most of the flux longward of 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Identification and astrometry of five stars in the vicinity, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 15, shows that the mid-M-type star is at RA = 19h 45m 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0s and DEC=+28o 39’ 40” (2000), with an uncertainty of 2 arcmin, and it is magnitude R~11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The large astrometric and photometric uncertainty is due to the dispersion by the prism in DEC and to the vignetting in the corner of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The transient emission appears in only one exposure (#588 out of 600) and is located on a broad peak at 745 nm as shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The stellar spectrum indeed exhibits four broad peaks in the spectral energy distribution due to the usual absorption by TiO, CaH, CaOH in mid-M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 16 shows the stellar spectrum between wavelengths 710 and 920 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is very little stellar flux detectable outside that wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The enhanced emission at 745 nm is apparent in the 1-sec exposure (shown as squares) relative to the average spectrum (solid line) from 20 exposures, with 10 taken before and 10 after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In the right panel of Figure 16, the individual 20 spectra are shown that comprised the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The scatter in the number of photons at any given wavelength reveals the noise in the individual 1-sec exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The noise is caused by Poisson fluctuations of photon arrival and seeing variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The dispersion of the number of photons, at each wavelength, among the 20 exposures shows the level of combined noise from Poisson fluctuations and seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The enhanced emission in the one exposure at 745 nm triggered the automated search algorithm (squares), and it is indeed more intense than the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' However, that enhanced intensity has a magnitude that resides at the end of a distribution of noise rather than detached from that distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, the apparent transient emission is likely a result of rare, but expected, noise as seen in the other spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Further, stellar photospheric flux at 745 nm is naturally the most intense region of the spectrum, which allows a slight seeing improvement to concentrate the arriving photons in both the wavelength and spatial directions of the raw image, boosting the peak intensity momentarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We are satisfied that this apparent transient emission is merely noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4196 4190 4185 4180 4175 4170 4165 6150 6160 6170 6180 6190 Pixels 14 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The sum of 20 consecutive 1 sec images, zoomed on a transient emission line candidate at Galactic longitude 64 deg, at upper right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Spectra of stars of brightness Vmag 7 to 14 are visible, with North up, short wavelengths up, and East to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The stellar spectrum with the emission candidate is short because it emits nearly all of its light in the red and infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The coordinates of five identified stars are shown, but the identity of the red star with the candidate emission remains unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The transient emission is shown in Figure 16, and it is consistent with noise fluctuations from seeing changes and Poisson noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 17 19h 46 24± 28° 38 1 Candidate 19h 45㎡ 38" +28°89°40″ 3000 +- 2 arcmin 19 47 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 28° 28 42 69 19#-4653 28°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='28 2500 19h 46 00 28° 19 04 19: 48 127 28 19 06 2000 Pixels 1500 1000 500 N up, E to left 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 deg 1000 2000 3000 Pixels 15 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Spectrum of the apparent transient emission line at 745 nm in a 1 sec exposure (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Left: The solid line is the average of 20 exposures surrounding the one exposure with the enhanced emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Note the enhanced emission at 745 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Right: The 20 individual spectra are shown as small dots, conveying the scatter in the number of photons in each exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The apparent transient emission at 745 nm indeed stands more intense than the ensemble of individual spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' But the scatter in the number of photons at each wavelength shows that the enhanced emission is at the extreme end of that distribution, but not detached from that distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus the enhanced emission flagged by the automatic detection algorithm is justified, but not inconsistent with the end of the distribution of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We could not identify a definite M-type star near the coordinates above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The two reddest stars within the 3 arcmin error circle are IRAS 19436+2834 (19 45 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='4 +28 41 55) that has V=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='72, J=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='25, and K=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='733, and IRAS 19433 +2829 (19 45 24, +28 36 55) that has magnitudes G=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='8, J=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='40, and K=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Both stars have colors consistent with a mid to late M-type star as observed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The first of them is closer to the coordinates measured here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We have no other candidate stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In any case, as noted, the apparent emission at 745 nm is most likely mere noise from photon-arrival statistics and seeing variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At Galactic longitude 248 deg, the automatic algorithm identified a similar apparent transient emission in an M dwarf at RA=8h 01m 26s DEC=-30o 15’ 11”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The star is magnitude, R ~ 10, with the continuum heavily chopped by TiO absorption, typical for a star of spectral type ~M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 17 shows, at left, the spectrum from the 1-sec exposure, and at right, the average spectrum of 20 exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is apparent enhanced emission at 720 nm that is likely to be due to momentary photon-arrival fluctuations and improved seeing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 600 500 口 Transient sec exp 口 400 6 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 20 口 exposures 300 00 口 口 吕 200 000 口 口 m 口 口 6 100 口 酒 11 ■ 0 Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 750 800 850 006 Wavelength [nm]N Photons 200 300 40 5 600 100 500 5 0 800 nm 850 6 900 16 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Another example of an M dwarf spectrum that triggered the automatic search algorithm for transient emission, this being at 720 nm where a natural peak in the stellar spectrum occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Momentary excellent seeing peaks up the natural peak in the stellar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='4 P Cygni The automated difference-image search for transient monochromatic light found “transient” line emission at Galactic longitude 76 deg on 2022 Dec 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The variable line emission appeared in multiple exposures, suggesting that it was actually constant emission fluctuating due to seeing variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 18 shows the entire spectrum of this candidate, revealing that the emission line has a wavelength of 656 nm, consistent with H-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Seeing variations no doubt caused the H-a intensity to vary by ~10%, occasionally triggering a “detection” by the difference-image algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum also has H-b in emission and the telluric absorption at the A-band and B-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' There is also emission apparently at 588 nm and 503 nm, likely from HeI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The star has Vmag ~ 5 and its approximate coordinates are RA = 20h 24 m and DEC = +37d 30’, consistent with the well-known iconic star, P Cygni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2000 150( S Photons 1000 N 500 400 500 600 700 800 900 Wavelength [nm]3×10 Photons 2×104 N 1×10 400 500 600 700 800 900 Wavelength [nm] 17 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A 1 sec exposure revealing variable strong emission line, found by the automated code, that triggered a detailed examination by eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The wavelength scale shows the strong emission line is H a, and the spectrum also contains emission at H b and HeI (587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='6 nm), and fainter emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Coordinates show this object is probably P Cygni itself, with spectral resolution inadequate to reveal the blueward absorption, but showing a steep shortward edge of its profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5 Be Star or Planetary Nebula At Galactic longitude 112 the automatic search algorithm identified a transient emission line at RA ~ 23h 24m 51s DEC~+61o 14 18” (within 30”), based on astrometry calibrated by neighboring stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The extracted spectrum is shown in Figure 19, the sum of 20 1-s exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Strong Ha triggered the difference-image search algorithm due, no doubt, to seeing changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The object has a continuum that is blue, similar to stars of spectral type early A or B-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The Balmer emission lines have a sharp blueward edge indicative of gas outflow, and the spectrum contains other emission lines common from 104 K gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Lists of stars within a 1 arcmin revealed no obvious identifications, with closest being an A2III star BD+60 2536 (V=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='57, B=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='93), with no spectrum published to check for emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The blue continuum and emission line profiles suggest an early type star with mass outflow, somewhat reminiscent of KjPn8 (Vazquez, Kingsburgh, and Lopez 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In any case, the spectrum is consistent with an astrophysical explanation, removing it relevant for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It deserves follow-up spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' α H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×104 d Photons 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5x10 an B d an B H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×104 B Z 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0×103 400 500 600 700 800 900 Wavelength [nm] 18 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A candidate transient emission candidate at Galactic longitude 112 deg that is simply an astrophysical object with Ha that varied due to seeing as is common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The spectrum is likely an V ~ 11 mag Be star or planetary nebula (or both), and the triggering transient emission is at the wavelength of Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This blue object at RA ~ 23h 24m 51s DEC~+61o 14 18” (within 30”) remains unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='6 Summary of Monochromatic Candidates The difference-image algorithm performed a search of 124 fields, each 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 deg, along the Galactic Plane in this objective prism survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The limiting magnitude for monochromatic emission was approximately Vmag = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Added to the previously observed fields near the Galactic Centre, a total of 973 square degrees were surveyed near the entire Galactic Plane accessible from latitude 38 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Each field was observed with 600 consecutive 1-sec exposures, allowing a difference-image search for monochromatic objects of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The automated difference-image search for transient emission identified 11 candidate sources of monochromatic emission, some of which were continuously emitting but fluctuating due to seeing variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We carefully examined each candidate, including any associated stellar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' None of the 11 candidates were pulses nor continuous emission of monochromatic light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Instead, all of them were either astrophysical objects with a strong emission feature in the spectrum that varied due to seeing changes or aircraft with flashing xenon lights, including one that was nearly monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Unidentified aircraft or spacecraft that have unexpected spectral characteristics would stand out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We found no point-sources of monochromatic emission, pulsing or continuous, that were plausibly extraterrestrial lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 5 DETECTION EFFICIENCY: INJECTION AND RECOVERY OF LASER PULSES We generated 100 synthetic monochromatic pulses consisting of 2D Gaussians having FWHM ~ 6 pixels, representative of the actual PSF of our images of the Milky Way Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We scaled these synthetic monochromatic pulses to various total numbers of photons within the entire profile, from 400 to 1000 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' These synthetic pulses ranged from roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2x background to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='5x the background photons per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We added these synthetic pulses to actual individual images, simulating a pulse Sum 20 Exposures Hα 3×104 Photons [ArII],[O] Nal Z 1X10 4 Hel, H Band Hel 400 500 600 700 800 006 Wavelength [nm] 19 duration less than 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We placed the pulses at random locations within the image, both in between and coincident with stellar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For each of these real images with injected monochromatic pulses, we executed the blind difference-image analysis to determine if it “discovered” the synthetic pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We ran 100 cases for each level of pulse intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The fraction of injected pulses detected is shown graphically in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Blindly executing the image-difference algorithm described above, we found the code successfully discovered 50% of the injected pulses that had at least 650 total photons in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It found none of the pulses containing fewer than 500 photons, and it found 97% of the pulses having more than 900 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, the nominal detection threshold at which 50% of the pulses would be detected is 650 photons total within the monochromatic pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This 650 photon threshold represents the number of photons that must be detected in 1 sec such that half of such pulses would be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The search algorithm has diminishing sensitivity for pulses lasting over the 1 sec exposure time of each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For such cases, some of the adjacent six bookend exposures would contain the emission, diminishing their contrast with the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In particular, for continuous monochromatic emission, ~6500 photons per sec would be required in order for the 10% variations to reveal itself as a pseudo-pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The term “continuous” here refers to a cadence of pulses that is more frequent than 1 pulse per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A train of pulses of nanosecond duration and arriving 106 per second would be detected here only as “continuous” monochromatic emission, requiring seeing variations for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The fraction of injected monochromatic pulses detected blindly by the difference image search algorithm as a function of the number of photons in the monochromatic pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Pulses containing 650 photons (total within the PSF) are detected in 50% of trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Pulses with >1000 photons are detected in ~100% of the trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The nominal detection threshold is 650 photons per laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' DISCUSSION We searched 2/3 of the Galactic Plane in a swath 2 deg wide for sub-second pulses of monochromatic emission between wavelengths 380 and 950 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The technique was also sensitive to sources of constant monochromatic emission as seeing causes momentary 10% fluctuations in the acquired photons in a one-second exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The goal was to search a domain of transients, in time and wavelength, that could have been missed in past transient surveys that use exposure times over 30 sec and usually had modest or no spectroscopic ability to detect unexpected emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For example, searches for planetary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 Detected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='8 Pulses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='6 of Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='0 400 600 800 1000 Number of Photons in Laser Pulse 20 nebulae, HII regions, and flare stars used exposures more than 1 minute and were often confined to the detection of Balmer lines, especially Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The detection threshold of 650 photons within a pulse translates into a threshold of photon fluxes entering the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The 650-photon threshold corresponds to a fluence per unit area by using the effective collecting area of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='278-m RASA telescope system, including efficiency between 450 – 800 nm and blockage by the camera at prime focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We find that the effective collecting area is Aeff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='020 m2 (Marcy, Tellis, and Wishnow 2021,2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Thus, the detection threshold of 650 photons implies a fluence threshold of 32500 photons per square meter at the Earth’s surface for monochromatic pulses of duration less than 1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For a pulse lasting 1 sec, that fluence corresponds to Vmag = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For wavelengths below 450 nm and above 800 nm the quantum efficiency drops below 50% of peak QE (at ~600 nm), thus requiring more than 32500 photons per square meter for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Atmospheric extinction raises this threshold fluence at the top of the Earth’s atmosphere by a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' One driver for this new search was the detection of laser beams in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A monochromatic light source lasting a few nanoseconds, microseconds, or milliseconds would have been detected in one exposure relative to reference exposures, with a detection threshold of 650 photons in the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We found no pulsed monochromatic sources, nor any unknown continuous monochromatic sources, between Galactic Longitude -4 to 248 deg, in a swath ~2 deg wide along the Galactic Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A major consideration in this optical SETI program was to minimize false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We engineered the optics, pixel size, and difference-imaging algorithm thresholds to avoid false positives, such as from cosmic rays, satellite glints, Cherenkov radiation, or electronics noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Our entire system, end-to-end, was designed to avoid them, and indeed we found none, after careful scrutiny of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Without a doubt, the optics, detector, and algorithm could be modified to make the entire system more sensitive to monochromatic pulses, by allowing some false positives to pass through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' But we elected to choose parameters that avoided the difficult task of identifying them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', Sheikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021, Villarroel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Some pulse cadences would not have been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For example, a duty cycle of one pulse per day, week, or month, implies that one pulse could arrive while we were not observing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Obviously, arbitrary pulse cadences with low duty cycle bring a lower detection probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' More observations are warranted to provide greater cadence coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This survey effectively searches for cadences of at least one pulse every 10 minutes, the duration of the observing sequence of a given field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Pulse cadences more rapid than 1 Hz are effectively “continuous” in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In such cases, the detection threshold is approximately 10x higher, as only the ~10% fluctuations from seeing variations will cause such continuous sources to vary above the detection threshold in the difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We can compute the laser power needed, from interstellar distances, to produce a detectable fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We consider a benchmark laser that is diffraction limited with a 10-meter diameter aperture, and located 100 ly from Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Extinction by dust is ~10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' It emits a beam with an opening angle of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='01 arcsec at a wavelength of, say, 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' To produce a photon flux at Earth having the detection threshold of 32500 photons per square meter, a power of 100 Megawatt is required during the 1 sec pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' For a laser launcher located 1000 ly away, a power of 10 Gigawatt is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Extinction from interstellar dust will increase this requirement by ~30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' At 1 kpc, the required power is ~90 Gigawatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Extinction from dust will require another factor of two in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The laser beam footprint at Earth would have a diameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='3 to 3 AU, respectively, a fraction of the area of the inner Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Such small footprints imply that the laser is purposely or fortuitously (or mistakenly) pointed at Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Lasers having apertures smaller than 10 meter can also be detected, but their larger opening beam angle would require more laser power, increasing inversely as the square of aperture diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A benchmark receiver might have 100x the diameter of the collector and thus 10,000x higher collecting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' This larger collector can thus reliably receive 10,000x the bit rate here (Gertz & Marcy 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A string of cell towers overcomes this limitation of bit-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' SUMMARY More than 5000 stars have been searched for monochromatic laser emission (Reines and Marcy 2002, Tellis and Marcy 2015, 2017, Marcy 2021, and Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Most surveyed stars are Sun-like or smaller with spectral types F, G, K, or M-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' But several hundred are massive stars of spectral types O, B, A, and early F-type (Tellis, private communication 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We have also searched the focal points of the solar gravitational lens (Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Many searches for optical pulses of sub-second duration have been performed, covering over half the sky (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2001, Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2005, Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2004, Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2007, Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' No optical laser emission, pulsed, monochromatic, or otherwise, has ever been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Optical lasers could have been detected previously by “conventional” astrophysics searches, both broad-band and spectroscopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Objective prism telescopes 100 years ago (Fleming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 1907, Pickering 1912, Cannon & Pickering 1922) revealed thousands of objects that emit emission lines, such as planetary nebulae, HII regions, T Tauri stars, Be stars, Wolf- Rayet stars, M dwarf flare stars, and active galactic nuclei, including at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Such objective prism surveys yield 21 emission-line objects that are then pursued with spectroscopy of modest resolution that is able to reveal non-astrophysical emission lines if any existed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Even optical searches using mere broadband filters, such as BVRI, would reveal objects whose emission was confined to one or two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A 13th mag source whose emission was confined to one or two emission lines would exhibit bizarre colors, meriting follow-up spectroscopy (ala Lyman-break galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The emission at non-astrophysical wavelength would be immediately obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Many all-sky surveys reached 18th magnitude, and modern ones reach 21st magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' No object with strange colors turned out to be lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' In summary no monochromatic sources with non- astrophysical emission lines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', lasers, were discovered in hundreds of past all-sky surveys, nor in the present survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The present non-detection of lasers, and those from the hundreds of past surveys, does not imply that extraterrestrial technology is absent in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' To be sure, other wavelengths and pulse cadences merit observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Also, the filling factor of optical laser beams may be too small (Forgan 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' A vast parameter space is yet to be surveyed (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' However, a large fraction of observable SETI parameter space has already been surveyed by both conventional astronomy surveys of the entire sky and by explicit searches for extraterrestrial technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=', Wlodarczyk-Sroka, Garrett & Siemion 2021, Garrett & Siemion 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Most impressively, SETI parameter space has unintentionally been surveyed by all-sky searches for natural, astrophysical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Those surveys gloriously revealed a multitude of astrophysical objects that emit, unexpectedly, at all wavelengths, including radio, microwave, extreme UV, x-rays, and gamma rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' Those unexpected discoveries constitute non-detections of extraterrestrial technology in the same search domains, but not recorded as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The dearth of beacons of technology leaves us revealing more of a great SETI desert, especially at the intensely surveyed optical and radio wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work benefitted from valuable communications with Beatriz Villarroel, Franklin Antonio, John Gertz, Ben Zuckerman, Brian Hill, Susan Kegley, Dan Werthimer, Ariana Paul, Roger Bland, Martin Ward, and Paul Horowitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' We thank the team at Space Laser Awareness for outstanding technical help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' DATA AVAILABILITY This paper is based on raw CMOS sensor images obtained with Space Laser Awareness double objective prism telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' The 8864 raw images are 125 MB each, totaling 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content='1 TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} +page_content=' They are located on a peripheral disk that is not 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAzT4oBgHgl3EQfSPuZ/content/2301.01230v1.pdf'} diff --git a/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/2301.01250v1.pdf.txt b/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/2301.01250v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55bef9dbb6dfd19eb7f54f8d7f89d4a1adb7dcc5 --- /dev/null +++ b/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/2301.01250v1.pdf.txt @@ -0,0 +1,2386 @@ +Highlights +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Maxime Chaveroche,Franck Davoine,Véronique Cherfaoui +• Provides a way to learn an efficient decentralized communication policy between autonomous vehicles +• Proposes a new generative model that learns to build state representations for RL through prediction and reconstruction +• Proposes a reward function with interpretable parameters to adjust the trade-off between information gain and volume +• With our experiment parameters, achieved 25% gain in relevant information, with only 5% of the total queryable volume +arXiv:2301.01250v1 [cs.LG] 12 Dec 2022 + +Decentralized cooperative perception for autonomous vehicles: +Learning to value the unknown⋆ +Maxime Chaverochea,∗, Franck Davoinea and Véronique Cherfaouia +aAlliance Sorbonne Université, Université de technologie de Compiègne, CNRS, Heudiasyc, CS 60319 - 60203 Compiègne Cedex, , France +A R T I C L E I N F O +Keywords: +cooperative perception +decentralized +V2V +communication +efficiency +filtering +prediction +model-based +DRL +Deep Learning +Reinforcement Learning +A B S T R A C T +Recently, we have been witnesses of accidents involving autonomous vehicles and their lack of +sufficient information. One way to tackle this issue is to benefit from the perception of different view +points, namely cooperative perception. We propose here a decentralized collaboration, i.e. peer-to- +peer, in which the agents are active in their quest for full perception by asking for specific areas +in their surroundings on which they would like to know more. Ultimately, we want to optimize a +trade-off between the maximization of knowledge about moving objects and the minimization of +the total volume of information received from others, to limit communication costs and message +processing time. For this, we propose a way to learn a communication policy that reverses the usual +communication paradigm by only requesting from other vehicles what is unknown to the ego-vehicle, +instead of filtering on the sender side. We tested three different generative models to be taken as base +for a Deep Reinforcement Learning (DRL) algorithm, and compared them to a broadcasting policy and +a policy randomly selecting areas. More precisely, we slightly modified a state-of-the-art generative +model named Temporal Difference VAE (TD-VAE) to make it sequential. We named this variant +Sequential TD-VAE (STD-VAE). We also proposed Locally Predictable VAE (LP-VAE), inspired by +STD-VAE, designed to enhance its prediction capabilities. We showed that LP-VAE produced better +belief states for prediction than STD-VAE, both as a standalone model and in the context of DRL. +The last model we tested was a simple state-less model (Convolutional VAE). Experiments were +conducted in the driving simulator CARLA, with vehicles exchanging parts of semantic grid maps. +Policies learned based on LP-VAE featured the best trade-off, as long as future rewards were taken into +account. Our best models reached on average a gain of 25% of the total complementary information, +while only requesting about 5% of the ego-vehicle’s perceptual field. We also provided interpretable +hyperparameters controlling the reward function, which makes this trade-off adjustable (e.g. allowing +greater communication costs). +1. Introduction +Recently, we have been witnesses of accidents involving +autonomous vehicles and their lack of sufficient information +at the right time. One way to tackle this issue is to benefit +from the perception of different viewpoints, namely collab- +orative perception. While setting a multitude of sensors in +the road infrastructure could be imagined, this would require +a lot of investments and limit its usage to some areas in +the world. Instead, we focus on the exchange of information +between vehicles about their common environment, where +they are the only sources available. +These communications can simply be centralized by a +server that would gather all information from all vehicles +to process it and re-distribute it to all, as suggested in +[1]. However, this still consists of Vehicle-to-Infrastructure +(V2I) communications, which implies (1) an infrastructure +cost and the impossibility to share information with other +⋆This work was carried out and co-funded in the framework of the +Labex MS2T and the Hauts-de-France region of France. It was supported by +the French Government, through the program “Investments for the future” +managed by the National Agency for Research (Reference ANR-11-IDEX- +0004-02). +∗Corresponding author +maxime.chaveroche@gmail.com (M. Chaveroche); +franck.davoine@hds.utc.fr (F. Davoine); veronique.cherfaoui@hds.utc.fr +(V. Cherfaoui) +ORCID(s): 0000-0002-0834-4022 (M. Chaveroche); +0000-0002-8587-6997 (F. Davoine); 0000-0003-2064-9838 (V. Cherfaoui) +agents when there is no server available nearby. It also +features the disadvantage of (2) making the agents broadcast +their entire perception, which can be heavy on the means of +communication and computation and give rise to delays. +In contrast, the decentralized Vehicle-to-Vehicle (V2V) +approach [2, 3, 4, 5, 6] does not require any extra infrastruc- +ture to work, i.e. does not implies (1). In this setting, agents +directly exchange pieces of information between them. It +also comes with new problems such as data incest and +lower computation capabilities. We will ignore them here +as we already tackled the issue of avoiding data incest +using Dempster-Shafer Theory (DST) [7] in spite of low +computation capabilities with two conference papers [8, 9] +and a journal paper [10]. But V2V communications bring +a potentially heavier communication burden as well, due to +redundancies. In fact, (2) is worse in this setting than in the +centralized one if agents are passive, meaning if they simply +broadcast their perception for the others to know, without +filtering it beforehand. Nevertheless, this decentralized ap- +proach offers the possibility to make the agents active in +their quest for full perception, i.e. making the agents ask for +specific areas in their surroundings on which they would like +to know more, instead of always broadcasting everything. +This is impossible in the centralized setting, as the server +decides and thus needs to gather all perceptions beforehand. +Here, we propose such a system, where each agent builds +its own local top-down semantic grid and sends specific +Chaveroche et al.: Preprint submitted to Elsevier +Page 1 of 19 + +aDecentralized cooperative perception for autonomous vehicles: Learning to value the unknown +requests to others in the form of bounding boxes described +in the global reference frame. We choose local grid maps for +their ability to map an agent’s knowledge and to deduce its +uncertainties in space. +2. Related Works +Since not all uncertain areas are relevant, Active Ex- +ploration [11, 12] is not enough; a truly efficient collabo- +ration policy requires some understanding of the scenery +[13], extracted from the spatial arrangement of grid cells +and their classes. What could lie in the shadows and how +to best discover it? If a pedestrian is heading towards an +occluded area, we expect the agent to request for this area, +as a tracking system. If the agent has no idea of what +could be in the unknown, maybe it could ask for some +key points to understand the layout of the environment. If +an area on the road is near a crowd of people or in the +continuity of a pedestrian crossing, ask for it as some unseen- +before pedestrians could be crossing, etc. More generally, +we would like the agent to know as much as possible about +moving objects in its vicinity, while avoiding to request too +much information from others. This represents a complex +bounding box selection policy to be learned from pixels. +Given the long-lasting successes of Deep Learning in +such ordeals, it seems natural to consider neural networks +for our problem. But, while it is theoretically possible (but +practically challenging) to learn our policy in an end-to- +end fashion with Model-free Deep Reinforcement Learning +(DRL), we choose to first learn a deep generative model +to pre-process our inputs. Indeed, training deep neural net- +works is easier, faster and more stable when the loss on the +output is in the form of a well-justified derivable function, +which is hard to achieve with reward signals from a RL +environment. Building this generative model also allows for +more control and insights on what is learned, and reduces +the size of the neural networks that are supposed to be +trained through model-free DRL. As demonstrated in World +Models [14], learning a policy on top of a model can even be +achieved with simple heuristics such as Evolution Strategies +(ES), with performances equivalent to RL algorithms. +Our model needs to be generative, for inference in un- +known areas. In addition, we want it to be predictive, in order +to make it understand latent dynamics, anticipating disap- +pearances or inferring hidden road users from the behavior of +visible ones. Doing so, it could even eventually compensate +for communication latencies. Such a model would be useful +in itself for other tasks as well, e.g. autonomous driving. +Several existing works [15, 16, 17, 18, 19] employed +generative models with convolutional networks in a U-Net +architecture in order to augment instantaneous individual +grid maps. Some used deterministic networks such as Gen- +erative Adversarial Networks (GAN). Others tried to incor- +porate stochasticity with Monte Carlo Dropout or simply +using a Variational Auto-Encoder (VAE). Most used occu- +pancy grids as input, but some chose semantic grid maps +or DOGMa (occupancy grid with velocities). These inputs +were either expressed in a static global reference frame or +given to a system that had no prediction capability. Doing +so, it appears that none of these approaches really modeled +the long-term dynamics of the environment that would be +necessary to learn our desired policy. On the other hand, +a kind of recurrent generative model inspired by the VAE, +namely Temporal Difference VAE (TD-VAE) [20], was de- +signed with the specific intent of being taken as base for +a reinforcement learning algorithm. It puts an emphasis on +the learning of belief states for long-term predictions, which +are important for the development of complex strategies. +It has been proven in [21] that explicitly predicting future +states enhances data-efficiency in a number of RL tasks, +though they train their model jointly with the policy and +do not use the loss defined in [20]. Appealed by the the- +oretical justifications of TD-VAE, its decoupling regarding +specific RL tasks (which simplifies the search for good +RL hyperparameters) and its demonstrated ability to predict +plausible sequences of images in a 3D world at different time +horizons and from a variable number of observations, we +have implemented and adapted this TD-VAE to our problem. +However, correcting some of its weaknesses regarding its +actual prediction capability, we finally proposed our own +model, called Locally Predictable VAE (LP-VAE). To learn +our communication policy based on this model, we chose the +widely used Proximal Policy Optimization (PPO) algorithm +[22], which is a fairly stable and simple policy-gradient +based DRL algorithm with few hyperparameters. +Closely related to our goal, other works try to address the +problem of efficiently communicating between autonomous +vehicles. In [23], they used a joint Perception and Prediction +(P&P) model that transforms sensor data into learned fea- +tures to broadcast to other vehicles. This model also fuses +received features with local ones and tries to predict the +trajectory of nearby communicating vehicles. This informa- +tion compression is also present in our work in the form +of a Convolutional VAE preprocessing each observation +grid. We go one step further in communication efficiency +as our system does not broadcast every piece of informa- +tion, but chooses instead which one it wishes to receive. +Sending learned features also forces them to make another +neural network learn to spatially and temporally transform +all pieces of information received from the vehicular net- +work. Even the fusion operation is done by making a neural +network learn how to fuse two learned features, without any +guarantee on the result. Instead, here we rely on top-down +semantic grids, which are simple discretizations of the space +around the ego-vehicle. Doing so, we can transform the +content of our transmissions using linear transformations. +Furthermore, our system keeps its integrity by only fusing +probability distributions. +In [24], they used Deep Reinforcement Learning to select +only a portion of the perceptive field of an autonomous +vehicle to send to others. However, this information filtering +is done on the sender side, contrary to our approach that +filters on the receiver side. Doing so, their approach still +Chaveroche et al.: Preprint submitted to Elsevier +Page 2 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Figure 1: Illustration of our application. CARLA provides a semantic segmentation corresponding to a camera attached to the +ego-vehicle hood, as well as its corresponding depth (images taken from [26]). This gives us enough information to create a +semantic 3D point cloud, i.e. to scatter all pixels in space according to their depth and image coordinates (and the camera +deformation). From it, we project these pixels back into a 2D plane (i.e. a grid), but from a top-down point of view (and without +camera deformations). In parallel, we get the ego-vehicle motion since the previous time step in order to update a perception +memory containing 2D points from previous time steps. We add the current semantic grid to this memory and give the resulting +augmented grid to our learned world model (STD-VAE or LP-VAE), along with the ego-motion and driving policy commands. In +turn, this model tries to guess what is hidden in occluded areas and provides a belief state about latent dynamics. These outputs +are then given to a DRL algorithm that chooses a grid area to request to the world. This area is extracted at the next time +step from a grid generated by a camera above the ego-vehicle. Finally, this information is fused at the next time step with the +ego-vehicle perception. +consists in broadcasting pieces of information, regardless of +the actual needs of others. +The same can be stated for [25], where they describe +a V2V cooperative perception system in which vehicles +exchange object detections. They try to reduce redundancies +by estimating the value of a piece of information for a +potential receiver. The value here is the novelty, i.e. the +probability that the potential receiver is not aware of some +object of interest. +Section 3 formally introduces our communication prob- +lem, justifying the use of a preprocessing generative model. +Section 4 formalizes the aforementioned generative model, +introducing TD-VAE and LP-VAE. Section 5 presents our +deep networks implementing these models. Then, section 6 +evaluates and compares the performance of different ver- +sions of our models and policy learnings. Finally, we con- +clude this article with section 7. +3. Problem formulation +We formulate our communication problem as a Markov +Decision Process (MDP). Fig. 1 gives an overview of it, +working with the driving simulator CARLA [26] for our +experiments. +3.1. State space +We assume the existence of a driving policy from which +we only know the actions taken at each time step: ego- +vehicle controls (acceleration and steering angle, each rang- +ing in [−1, 1]) and global direction (average of the next +10 equally-spaced points the planner set to visit in meters +relative to the ego-vehicle’s reference). This driving policy +influences the road environment in which the ego-vehicle +is moving. This is not the case with the communication +environment that we consider in this MDP. Each observation +is a tuple (퐺푡, 퐶푡, 푉푡), where 퐺푡 is an ego-centered semantic +grid, 퐶푡 represents the actions taken by the driving policy +at a given instant 푡 (which influence 퐺푡+1) and 푉푡 is the +motion of the ego-vehicle between 푡−1 and 푡. Each semantic +grid 퐺푡 is a top-down 6-channels pseudo-Bayesian mass grid +corresponding to the five classes of the frame of discernment +Ω = {pedestrian, car, road lines, road, other}. The class car +actually contains any type of vehicle, even bikes. The class +road lines contains any road marking: road lines, arrows, +painted stop signs, etc. The class other contains the rest of +the static objects perceivable by the agent, such as vegeta- +tion, sidewalks, buildings, etc. The last channel represents +ignorance, i.e. the mass put on Ω. This means that 퐺푡 ≥ 0 +and, for any cell index 푖 of 퐺푡, we have ∑6 +푘=1 퐺푡[푖][푘] = 1. +Chaveroche et al.: Preprint submitted to Elsevier +Page 3 of 19 + +Semantic segmentation +CARLA +Semantic 3D point +cloud +Top-down semantic 2D arid +(Correcting +Depth camera +Perception +camera +deformations +Ego Motion +memory +as for the +frontal view) +Driving controls +Learned World +Planned general +Belief state +model +direction +(dynamics) +DRL +Bounding +algorithm +box a +(PPO) +Top-down semantic 2D grid +Inferred top-down semantic 2D gridDecentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Figure 2: Left: Illustration of an instance of top-down semantic +grid 퐺푡 corresponding to a partial observation 푥푡 in our model. +Red is for pedestrians, blue is for cars, yellow is for road +lines, purple is for road, white is for other and black is for +ignorance. The displayed class is the one with the greatest +mass. The intensity of its color depends on its mass: the +closer to 0, the darker. Notice all the occlusions due to walls +or other road users, in addition to the limited distance of +perception of the ego-vehicle. Right: Instance of top-down +semantic grid corresponding to a complete observation 푦푡 in +our model. Actually, this view is obtained with a facing ground +camera above the ego-vehicle. Doing so, it contains itself some +occlusions due to trees, poles, buildings, etc. Thus, it is rather +a hint about the true 푦푡. This view can also be obtained by +the fusion of multiple view points, from autonomous vehicles +or infrastructure sensors. +These cells are distributed as a matrix (grid) of 80 rows and +120 columns, i.e. 퐺푡 is analog to a 80 × 120 × 6 image of +values in [0, 1]. See Fig. 2 for a visualization of this semantic +grid. +These observations constitute a very large and complex +space which would be hard to transform into exploitable +neural network features without a derivable loss function. +Thus, we will first build a generative model of the driving en- +vironment (implicitly including the agent’s driving policy). +Besides, learning this model beforehand will give us more +control on the information flow that should be considered by +the communication policy. Therefore, the state space of our +MDP is made of learned features from this generative model. +Several versions of this generative model are proposed in +Section 4. +3.2. Action space +Our MDP has 4 continuous actions that each ranges in +[0, 1], defining a bounding box in the local grid 퐺푡 of the +ego-vehicle at time 푡: width, height, column and row. This +bounding box is supposed to represent an area in the ego- +vehicle’s future surroundings. +3.3. Transition function +Transitions from a state-action pair to a new state depend +also on the driving environment, i.e. CARLA. First, this +environment generates a new partial grid 퐺푡+1 and other +observations already described. The bounding box described +by the action given at time 푡 is then translated into an area of +퐺푡+1 filled with complete information. Fig. 3 illustrates this +process. +In addition, a visual memory mechanism, specific to +our MDP, makes perceptions persist for a few time steps, +Figure 3: Illustration of our decision process: 1) Based on +what is known at time 푡, select a bounding box where there +is high uncertainty and high probability to discover road users. +2) Send this request in global coordinates to the vehicular +network (which may consists of both infrastructure sensors +and other autonomous vehicles). 3) At time 푡 + 1, we expect +some vehicles to transmit their perception of this area. In our +implementation, complete perceptions are simply obtained by a +camera above the ego-vehicle since we focus on the selection of +bounding boxes, i.e. 1). 4) The transmitted partial perception +is fused with the one of the ego-vehicle at time 푡 + 1. +discounted a little more every time. This implements short- +term memory, so that we only consider as unknown what +has not been perceived in a long time (or never). This also +has the effect of giving consequences to past actions, since +bounding boxes in the same area will have close to no +potential information gain for a few time steps. +3.4. Rewards +Finally, let us define a reward function for our MDP. Let +푟푡 be a reward density, defined for each cell 푖 of 퐺푡+1 as: +푟푡(푖) = −휂.푟min + 푆[푖]. +5 +∑ +푘=1 +푟obj[푘]. max +( +0, +퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘] +)푤 +(1) +where 푤 ∈ ℝ+∗, 휂 ∈ [0, 1] and ̃퐺푡+1 is the grid before fusion +with the grid 퐺푀 +푡+1 corresponding to 푀푡+1. The quantity 푟obj +is a nonnegative reward per object pixel (only null for the +static class other, i.e. 푟obj[5] = 0) such that 푟obj[푘] ≥ 푟obj[푘+ +1]. Indeed, pedestrian are the smallest identifiable objects +among our classes and so must have the highest reward +per pixel. The quantity 푟min is equal to the least positive +reward per pixel, i.e. 푟min = 푟obj[4]. It is used to discourage +the selection of uninteresting cells. The coefficient 휂 that +multiplies it represents the minimum informational gain that +is needed to consider this cell worth to be requested. For +some value of 휂, this minimum gain applies to the class with +the least reward, while it becomes virtually more and more +forgiving as the class has a greater reward per cell. Moreover, +notice that max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘]) ∈ [0, 1], which +Chaveroche et al.: Preprint submitted to Elsevier +Page 4 of 19 + +Transmission: +at t +Request to +vehicular +network at t +2Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Figure 4: Heatmap illustrating our spatial filter 푆 for 훼 = 0.5, +훽퐹 = 0.8, 훽퐿 = 1 and 휁 = 0.01. Deep blue is 0, while deep +red is 1, which means that the reward in a cell located in a +blue region will be 0, no matter what is inside. The center of +the ego-vehicle is in the middle of the first row starting from +bottom. +implies that max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘])푤 ∈ [0, 1]. +This means that 푤 only alters the significance of some gain +in mass: for 푤 ∈ (0, 1), max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘]) +will be greater than for 푤 = 1, while for 푤 ∈ (1, +∞), +max(0, 퐺푡+1[푖][푘]− ̃퐺푡+1[푖][푘]) will be less. In other words, +if 푤 ∈ (1, +∞), then the gain will have to be more important +to have an impact on 푟푡(푖). Finally, 푆 represents a spatial +filter to account for the fact that we are not equally interested +everywhere in discovering road users. For example, a road +user very far ahead is not as valuable an information as a +road user just around the corner. We defined a forward filter +푆퐹 and a lateral filter 푆퐿, such that 푆 = 푆퐹 .푆퐿. We set +푆퐹 [푖] = 1 − +[ 훽퐹 +1 − 훼 . max +( +0, +퐹(푖) +max(퐹) − 훼 +)] +where 훼 ∈ [0, 1) and 훽퐹 ∈ [0, 1]. The quantity 퐹(푖) is the +forward distance (number of rows from the row in which +the center of the ego-vehicle is) corresponding to cell 푖. The +greater the parameter 훽퐹 , the less the farest cells are valued. +The greater the parameter 훼, the farer from the ego-vehicle +the decrease in value starts. +The second filter is defined as +푆퐿[푖] = 1 − 훽퐿 +휁 . max (0, 휁 − |cos (arctan2 (퐿(푖), 퐹(푖)))| ) +where 휁 ∈ (0, 1]. The quantity 퐿(푖) is the lateral distance +(number of columns from the column in which the center +of the ego-vehicle is) corresponding to cell 푖. This filter de- +scribes a cone in front of the ego-vehicle (and symmetrically +at the back of it) in which the cells are the most valued. The +greater the parameter 휁, the narrower this cone. The greater +the parameter 훽퐿 is, the less the cells outside the cone (i.e. +on the sides of the ego-vehicle) are valued. Fig. 4 provides a +visualization of 푆. +The reward associated with some action 푎푡 is defined as +푅푡(푎푡) = −퐾.(1 − 휂).푟min+ +∑ +푖∈퐼(푎푡) +푟푡(푖), +(2) +where 퐾 is the minimum number of interesting cells that +must be entirely discovered in order to make the request +worthwhile, 퐼(푎) = [푣(푎), 푣(푎) + ℎ(푎)] × [푢(푎), 푢(푎) + 푤(푎)] +and 푢(푎), 푣(푎), 푤(푎), ℎ(푎) are respectively the column index, +row index, width and height indicated by some action 푎. +3.4.1. Grid fusion +In order to produce 퐺푡 from ̃퐺푡 and the grid 퐺푀 +푡 +cor- +responding to 푀푡 in Eq. (1), we need to define a fusion +procedure. As each cell 푖 in both ̃퐺푡 and 퐺푀 +푡 +is a mass +function, we know that: +퐺푡[푖][6] = ̃퐺푡[푖][6] . 퐺푀 +푡 [푖][6], +where 6 is the channel corresponding to the mass on Ω. +Furthermore, we can get the contour functions of these +pseudo-Bayesian mass functions simply by adding the mass +on Ω to the mass on each of our 5 classes. Then, a sim- +ple pointwise multiplication of these two contour functions +produces the contour function corresponding to 퐺푡. This +also implies a mass on ∅, which is caused by conflicting +pieces of evidence between the two mass functions. Since +we are not interested in this level of conflict, we choose +to renormalize masses as in Dempster’s combination rule +[27]. Unlike Dempster’s rule however, we only distribute this +conflict on singletons 퐺푡[푖][1 ∶ 5] and keep the true value +퐺푡[푖][6], as the distinction between ignorance and conflict +is crucial to our communication policy. Algorithm 1 details +this procedure. +Algorithm 1: Fusion procedure for two pseudo- +Bayesian mass functions 푚1 and 푚2. +Input: Two pseudo-Bayesian mass functions 푚1, +푚2 +Output: The fused mass function 푚12 +푁 ← len(푚1); +푚12[푁] ← 푚1[푁] . 푚2[푁]; +푚12[1 ∶ 푁−1] ← (푚1[1 ∶ 푁−1]+푚1[푁]) . (푚2[1 ∶ +푁 − 1] + 푚2[푁]) − 푚12[푁]; +푠 ← sum(푚12[1 ∶ 푁 − 1]); +if 푠 > 0 then +푚12[1 ∶ 푁 − 1] ← (1 − 푚12[푁]) . 푚12[1∶푁−1] +푠 +; +Return 푚12; +4. Models +In this section, we will present several versions of the +generative model mentioned in section 3.1, namely STD- +VAE and LP-VAE. In the end, this generative model will +provide us with learned features describing the state of the +environment related to the MDP presented in section 3, in +order to reduce the size of the network optimized through +DRL and to control what is kept in the information flow. We +will start by formalizing in section 4.1 a draft of this model +that ignores the actions the agent takes at each time step. +Chaveroche et al.: Preprint submitted to Elsevier +Page 5 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Then, we will briefly introduce in section 4.2 the original +TD-VAE [20]. Following that, we will propose in section 4.3 +our sequential variant of TD-VAE, i.e. STD-VAE. Inspired +by this model, we will then propose LP-VAE in section +4.4. Finally, section 4.5 will demonstrate with LP-VAE how +to modify this generative model to incorporate the actions +chosen by the agent. +4.1. Action-independent modeling +As a vehicle clearly cannot access the complete state +of its surroundings through its sole perception, we can +model our problem as a Partially Observable Discrete-Time +Markov Chain (PO-DTMC), where 푋푡 and 푍푡 denote ran- +dom variables representing respectively a partial observation +and a latent state at time 푡. However, we consider that 푍푡 +and 푋푡 are in different spaces, the latent space describing +the whole environment and containing information about +object dynamics and trajectories allowing for predictions. +More precisely, 푋푡 corresponds to the sole perception of the +ego-vehicle at time 푡, without memory of the past. We also +introduce a third random variable 푌푡 which represents the +spatially complete observation corresponding to 푍푡 in the +space of 푋푡. In other words, 푋푡 is a partial observation of 푌푡 +which is itself a partial observation of 푍푡. +So, let 휃 be a set containing the parameters of a genera- +tive model that projects a latent state 푍푡 onto the observation +space as (푋푡, 푌푡). We choose to implement this generative +model as a deep neural network and we set the following +Gaussian distributions as constraints, for numerical stability +and simplicity: +• 푍푖 ∼  (0, 퐼푑) +• 푝푍푖+1|푍푖(⋅|푧푡; 휃) =  (휇푧(푧푡; 휃), 휎2 +푧(푧푡; 휃).퐼푑) +• 푝푌푖|푍푖(⋅|푧푡; 휃) =  (휇푦(푧푡; 휃), 훼푦.퐼|푋푡|) +• 푝푋푖|푌푖,푍푖(⋅|푦푡, 푧푡; 휃) =  (휇푥(푦푡, 푧푡; 휃), 훼푥.퐼|푋푡|) +where 휇푧, 휎푧, 휇푥 and 휇푦 are all deep neural networks taking +their parameters in 휃, where 푑 is an arbitrary number of +dimensions for 푍푡, where 푧푡 is a realization of 푍푡 for some +푡 ∈ [1, 푇 ] and where 훼⋅ ∈ [ 1 +2휋 , +∞). This last constraint +implies that the generative model recreates independently +each dimension of 푋푡 from a latent state 푧푡 with the same +fixed precision. Moreover, the PO-DTMC formulation im- +plies that each pair of observations (푋푡, 푌푡) is only dependent +on 푍푡, i.e. +푝푋,푌 |푍 (푥, 푦 | 푧; 휃) = +푇 +∏ +푡=1 +푝푋푖,푌푖|푍푖(푥푡, 푦푡 | 푧푡; 휃), +and that the Markovian property holds in latent space, i.e. +푝푍(푧; 휃) = 푝푍푖(푧1). +푇 +∏ +푡=2 +푝푍푖+1|푍푖(푧푡 | 푧푡−1; 휃). +Fig. 5 provides the Bayesian network corresponding to +our model. +푁 +휃 +푌1 +푌2 +푌푇 −1 +푌푇 +푋1 +푋2 +푋푇 −1 +푋푇 +푍1 +푍푇 −1 +... +푍2 +푍푇 +Figure 5: Bayesian network of our generative model of parame- +ters in 휃. We have 푁 replications of this model, corresponding +to the 푁 sequences of length 푇 in our dataset. The parameter +set 휃 influences the inference of all variables in the model for +the 푁 sequences we have. +Thus, based on a dataset of 푁 independent sequences of +partial and complete observations 퐷 = (푥1∶푇 , 푦1∶푇 )1∶푁, we +want to optimize the parameters 휃 so that the probability that +the model generates the sequences of 퐷 is maximal under its +constraints. In other words, we want to find the parameters 휃 +that maximize 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷; 휃), which is the same as +finding 휃 maximizing log 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷; 휃). We have: +log 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷; 휃) = +∑ +(푥,푦)∈퐷 +log 푝푋,푌 (푥, 푦; 휃) +where +푝푋,푌 (푥, 푦; 휃) += ∫ 푝푋,푌 |푍(푥, 푦 | 푧; 휃) . 푝푍(푧; 휃) 푑푧 += ∫ ⋯ ∫ 푝푍푖(푧1). +푇 +∏ +푡=1 +푝푋푖,푌푖|푍푖(푥푡, 푦푡 | 푧푡; 휃) +. +푇 +∏ +푡=2 +푝푍푖+1|푍푖(푧푡 | 푧푡−1; 휃) +푇 +∏ +푡=1 +푑푧푡 +which is intractable, due to the fact that 휇푧, 휎푧, 휇푥 and 휇푦 +are multi-layers neural networks with nonlinearities. This +intractability is amplified by the fact that we work with +sequences of 푇 non-independent continuous latent states, +which implies a multiple integral over ℝ푇 ×푑. This means that +we cannot evaluate or differentiate the marginal likelihood +푝푋,푌 (푥, 푦; 휃). For the same reasons, the posterior distribution +푝푍|푋,푌 (⋅| 푥, 푦; 휃) = +푝푋,푌 |푍(푥, 푦| ⋅ ; 휃).푝푍(⋅ ; 휃) +푝푋,푌 (푥, 푦; 휃) +, +is intractable, which implies that methods based on the +posterior distribution such as the Expectation-Maximization +(EM) algorithm cannot be employed either. So, let us adopt +the Variational Bayesian (VB) approach by introducing a +variational distribution dependent on a parameter set 휙 to +approximate 푝푍|푋,푌 (⋅| 푥, 푦; 휃). But, more than just a mathe- +matical trick, we want this variational distribution to actually +be a recognition model such that it is able to infer latent states +only given past partial observations, in order to infer 푦 and +to be able to generate plausible next observations. +Chaveroche et al.: Preprint submitted to Elsevier +Page 6 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +4.2. TD-VAE model +TD-VAE [20] is a variant of the original VAE [28] +for temporal sequences which features the particularity to +separate belief states from latent states. A belief state 푏푡 +is a statistics describing 푥1∶푡 such that 푝푍푡|푋1∶푡(⋅|푥1∶푡; 휃) ≈ +푝푍푡|퐵푡(⋅|푏푡; 휃). The end goal motivating this distinction, aside +theoretical accuracy, is to learn a model able to determin- +istically aggregate observations by updating a statistics 푏푡 +that contains enough information to infer some latent state +푧푡, avoiding the accumulation of estimation errors on 푧1∶푡−1. +Since 푧푡 alone allows for predictions of next latent states, +푏푡 constitutes a belief on plausible latent dynamics that is +simply updated with each new observation. This feature is +important for model-based RL. +In [20], they chose additionally to make their model +provide jumpy predictions, i.e. directly predicting a latent +state 푧푡+훿 from some 푧푡 where 훿 is not precisely known, in +order to abstract latent dynamics for the benefit of computa- +tional efficiency. Formally, they seek to optimize 휃 so that it +maximizes the expression +피 +훿∼[훿푖,훿푠] +[ +피 +푡∼[1,푇 −훿] +[ +log 푝푋푡+훿|퐵푡 +(푥푡+훿|푏푡; 휃)]] +, +(3) +where [푎,푏] is the uniform distribution on the interval [푎, 푏] +and 퐵푡 += RNN(푋푡, 퐵푡−1; 휙). This cannot be optimized +directly, as showed in the previous section. However, we can +maximize a lower bound of this expression by introducing a +variational distribution. +Let 푄푡,훿(휙) = 푞푍푡,푍푡+훿|퐵푡,퐵푡+훿(⋅|푏푡, 푏푡+훿; 휙) be this varia- +tional distribution, dependent on a parameter set 휙, such that +푞푍푡,푍푡+훿|퐵푡,퐵푡+훿 +(⋅|푏푡, 푏푡+훿; 휙) ≈ 푝푍푡,푍푡+훿|퐵푡,푋푡+훿 +(⋅|푏푡, 푥푡+훿; 휃) +where it is important to notice that +푝푍푡,푍푡+훿|퐵푡,푋푡+훿 +(⋅|푏푡, 푥푡+훿; 휃) = +푝푋푡+훿,푍푡,푍푡+훿|퐵푡 +(푥푡+훿, ⋅|푏푡; 휃) +푝푋푡+훿|퐵푡 +(푥푡+훿|푏푡; 휃) += +푃푡,훿(휃) +푝푋푡+훿|퐵푡 +(푥푡+훿|푏푡; 휃). +To find the optimal parameters 휙 that minimize its ap- +proximation error, we can optimize 휙 so that it minimizes +through gradient descent the following average Kullback- +Leibler (KL) divergence: +피 +훿∼[훿푖,훿푠] +[ +피 +푡∼[1,푇 −훿] +[ +퐷퐾퐿 +( +푄푡,훿(휙) |||| +|||| +푃푡,훿(휃) +푝푋푖+훿|퐵푖 +(푥푡+훿|푏푡; 휃) +)]] +, +This cannot be optimized directly either. Yet, it can be shown +that we can equivalently minimize this divergence, while +also maximizing a lower bound of (3), by minimizing the +following loss w.r.t. 휙 and 휃: +TD-VAE(푥; 휃, 휙) += +피 +훿∼[훿푖,훿푠] +[ +피 +푡∼[1,푇 −훿] +[퐷퐾퐿 +(푄푡,훿(휙) || 푃푡,훿(휃))] ] +where +퐷퐾퐿 +(푄푡,훿(휙) || 푃푡,훿(휃)) += +피 +푍푡,푍푡+훿∼푄푡,훿(휙) +[ +log 푞푍푖|퐵푖 +(푧푡+훿|푏푡+훿; 휙) ++ log 푞푍푡|퐵푡,퐵푡+훿,푍푡+훿 +(푧푡|푏푡, 푏푡+훿, 푧푡+훿; 휙) +− log 푝푍푖|퐵푖 +(푍푡|푏푡; 휃) − log 푝푍+훿|푍 +(푍푡+훿|푍푡; 휃) +− log 푝푋푖|푍푖 +(푥푡+훿|푍푡+훿; 휃)] +. +In complement, the authors of [20] had to make the strong +assumption that 푝푍푖|퐵푖 +(⋅|푏푡; 휃) = 푞푍푖|퐵푖 +(⋅|푏푡; 휙) for any +휃, 휙. They also set 푝푍+훿|푍 +(⋅|푧푡; 휃) as a multivariate normal +distribution with diagonal covariance matrix, corresponding +to the distribution of latent states at any instants in [푡 + +훿푖, 푡 + 훿푠]. This is in contradiction with our sequential latent +model 푝푍푖+1|푍푖 +(⋅|푧푡; 휃), which is itself a multivariate normal +distribution with diagonal covariance matrix. In this regard, +푝푍+훿|푍 +(⋅|푧푡; 휃) can be seen as a rough approximation. +This abstraction of latent dynamics may be useful in +some cases where precision is not needed and the variability +of observations 푥푡∶푡+훿 gathered in a moment can be summa- +rized in latent space by smooth transitions between states +corresponding to dataset samples. However, we argue that +models of complex environments, in which the observation +space is combinatorially extremely large and in which mul- +tiple agents interact with each other, require precise learning +signals to understand latent dynamics and so to generalize +well outside the training set. More importantly, TD-VAE +cannot consider the actions taken by the observing agent +between 푡 and 푡 + 훿. Yet, learning the link between actions +and observations is central in RL. +4.3. Our Sequential variant STD-VAE of the +TD-VAE model +The authors of [20] also proposed a sequential version +of their model. Its corresponding Bayesian network is given +in Fig. 6. They chose to train its parameters as a particular +case of the jumpy one, simply taking 훿 = 1. Yet, this +would only maximize a lower bound of the probability to ob- +serve 푥푡+1 after 푏푡, i.e. +피 +푡∼[1,푇 −1] +[ +log 푝푋푡+1|퐵푡 +(푥푡+1|푏푡; 휃)] +, +instead of the whole future sequence 푥푡+1∶푇 after 푏푡, i.e. +피 +푡∼[1,푇 −1] +[ +log 푝푋푡+1∶푇 |퐵푡 +(푥푡+1∶푇 |푏푡; 휃)] +. +From a practical point of view, this would prove to be +computationally heavy if done multiple times per sequence +and would not learn from the accumulation of prediction +errors: particularly in a stochastic network such as TD-VAE +and with a time step small enough, the network will tend +to optimize weights such that the predicted next state looks +almost identical to the initial state. It is only by chaining +these predictions that their errors become significant. Thus, +we choose a slightly different variational distribution. Let +Chaveroche et al.: Preprint submitted to Elsevier +Page 7 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +푁 +퐵1 +퐵2 +... +퐵푇 −1 +퐵푇 +푋1 +푋2 +푋푇 −1 +푋푇 +푍1 +푍푇 −1 +... +푍2 +푍푇 +Figure 6: Bayesian networks corresponding to STD-VAE. Solid +lines represent the Bayesian network of our generative model +(without 푌푡) of parameters in 휃. Dashed lines represent the +Bayesian network of the recognition model of parameters in +휙 proposed by TD-VAE. Parameter dependencies are not +represented for the sake of clarity. Only 퐵푡 is not directly +influenced by 휃, while only variables at the end of a dashed +arrow are influenced by 휙. We have 푁 replications of this +model, corresponding to the 푁 sequences of length 푇 in our +dataset. +푄푡(휙) = 푞푍푡∶푇 |퐵푡∶푇 +(⋅|푏푡∶푇 ; 휙) be this variational distribu- +tion, dependent on a parameter set 휙, such that +푞푍푡∶푇 |퐵푡∶푇 +(⋅|푏푡∶푇 ; 휙) ≈ 푝푍푡∶푇 |퐵푡,푋푡+1∶푇 +(⋅|푏푡, 푥푡+1∶푇 ; 휃) +where it is important to notice that +푝푍푡∶푇 |퐵푡,푋푡+1∶푇 +(⋅|푏푡, 푥푡+1∶푇 ; 휃) += +푝푋푡+1∶푇 ,푍푡∶푇 |퐵푡 +(푥푡+1∶푇 , ⋅|푏푡; 휃) +푝푋푡+1∶푇 |퐵푡 +(푥푡+1∶푇 |푏푡; 휃) += +푃푡(휃) +푝푋푡+1∶푇 |퐵푡 +(푥푡+1∶푇 |푏푡; 휃). +To find the optimal parameters 휙 that minimize its ap- +proximation error, we can optimize 휙 so that it minimizes +through gradient descent the following average Kullback- +Leibler (KL) divergence: +피 +푡∼[1,푇 −1] +[ +퐷퐾퐿 +( +푄푡(휙) |||| +|||| +푃푡(휃) +푝푋푡+1∶푇 |퐵푡 +(푥푡+1∶푇 |푏푡; 휃) +)] +, +It can be shown that we can equivalently minimize this +divergence, while also maximizing a lower bound of +피 +푡∼[1,푇 −1] +[ +log 푝푋푡+1∶푇 |퐵푡 +(푥푡+1∶푇 |푏푡; 휃)] +, +by minimizing the following loss w.r.t. 휙 and 휃: +STD-VAE(푥; 휃, 휙) = +피 +푡∼[1,푇 −1] +[퐷퐾퐿 +(푄푡(휙) || 푃푡(휃))] +where +퐷퐾퐿 +(푄푡(휙) || 푃푡(휃)) += +피 +푍푡∶푇 ∼푄푡(휙) +[ +log 푞푍푖|퐵푖 +(푍푇 |푏푇 ; 휙) ++ +푇 −1 +∑ +푘=푡 +log 푞푍푖|퐵푖,푍푖+1 +(푍푘|푏푘, 푍푘+1; 휙) +− log 푝푍푖|퐵푖 +(푍푡|푏푡; 휃) − +푇∑ +푘=푡+1 +log 푝푍푖+1|푍푖 +(푍푘|푍푘−1; 휃) +− +푇∑ +푘=푡 +log 푝푋푖|푍푖 +(푥푘|푍푘; 휃) +] +(4) +Fig. 7 visually explains the process of evaluating (4), +which is very similar to the original TD-VAE. The belief +network aggregates observations such that each belief 푏푡 is +assumed to be a sufficient statistics for 푥1∶푡. The smoothing +network, knowing what the final latent state 푧푇 is, given +observations 푥1∶푇 , infers what should have been latent states +푧푡∶푇 −1. This gives us two different distributions for the +inference of 푧푡: one given only observations 푥1∶푡, and the +other given all observations 푥1∶푇 . In the learning phase, we +measure the divergence between these two distributions as a +loss to prompt correct dynamics recognition and consistency +in the belief network. Then, the Markovian transition model +infers the next state from the current one. We infer the +Gaussian parameters of the next state for each latent state +inferred by the smoothing network and measure as loss the +divergence between the distribution inferred by the smooth- +ing network and the one inferred by the transition model. +Finally, for each latent state 푧푘 sampled from the smoothing +network, we infer the Gaussian parameters describing the +observation 푥푘 with the decoding network and compute the +negative log-likelihood of 푥푘 given these parameters as loss. +However, our preliminary experiments on this model +with a dataset acquired in CARLA [26] revealed very poor +prediction quality when 푧푡 is sampled from 푞푍푡|퐵푡(⋅|푏푡; 휙), +while providing very good predictions when 푧푡 is sampled +from 푞푍푡|퐵(⋅|푏푡∶푇 ; 휙), i.e. from the smoothing network. In +fact, this seems obvious considering that the prediction +part of this model is trained with the latent states sampled +from the variational distribution 푞푍푡∶푇 |퐵푡∶푇 +(⋅|푏푡∶푇 ; 휙) and +not 푞푍푡∶푇 |퐵푡 +(⋅|푏푡; 휙). This is what motivates the introduction +in the next section of a local predictability constraint, allow- +ing us to train our model on samples from 푞푍푡∶푇 |퐵푡 +(⋅|푏푡; 휙). +This will also allow us to keep the idea of predicting distant +latent states from current observations while avoiding the +strong assumption that 푝푍|퐵 +(⋅|푏푡; 휃) = 푞푍|퐵 +(⋅|푏푡; 휙). +4.4. Our Locally Predictable VAE (LP-VAE) +model +First, we put a local predictability constraint for the +model to be able to predict multiple time steps into the +future: +푝푍|푋1∶푡(⋅| 푥1∶푡; 휃) ≈ 푝푍|푋,푌 (⋅| 푥, 푦; 휃) +(5) +for any instant 푡 ≥ 푡min. This means that there must be some +instant 푡min such that the partial observations 푥1∶푡min are suffi- +cient to recognize the latent dynamics of the whole sequence, +i.e. such that all observations 푦1∶푇 and all subsequent partial +Chaveroche et al.: Preprint submitted to Elsevier +Page 8 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +푏푡 +푏푡+1 +... +푏푇 −1 +푏푇 +푥푡 +푥푡+1 +푥푇 −1 +푥푇 +푧푡 +푧푇 +푧푇 −1 +... +푧푡+1 +푍푡 +푍푇 +푍푇 −1 +... +푍푡+1 +푋푡+1 +푋푇 −1 +푋푇 +... +푏푡−1 +Figure 7: Illustration of the forward computations allowing for +the evaluation of the STD-VAE loss (4). A diamond indicates +a deterministically inferred variable. A rectangle indicates the +deterministic inference of distribution parameters. A circle +indicates the deterministic inference of distribution parameters +and a sample from this distribution. The blue network is the +belief network. The red network is the smoothing network. The +black network is the Markovian transition model. The brown +network is the decoding network. +observations 푥푡min+1∶푇 bring negligible additional informa- +tion in the recognition of these latent dynamics. Notice that +푝푍|푋,푌 (⋅| 푥, 푦; 휃) = +푝푋,푌 ,푍(푥, 푦, ⋅ ; 휃) +푝푋,푌 (푥, 푦; 휃) += +푃 (휃) +푝푋,푌 (푥, 푦; 휃), +and let us note 푃푡(휃) = 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃). To enforce Eq. (5), +we want to minimize the average KL divergence +피 +푡∼ [푡min, 푇 −1] +[ +퐷퐾퐿 +( +푃푡(휃) |||| +|||| +푃(휃) +푝푋,푌 (푥, 푦; 휃) +)] += log 푝푋,푌 (푥, 푦; 휃) + +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푃푡(휃) || 푃(휃))] , +which we cannot minimize directly, due to the intractability +of 푝푋,푌 (푥, 푦; 휃) and 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃). However, we have: +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푃푡(휃) || 푃(휃))] += − log 푝푋,푌 (푥, 푦; 휃) ++ +피 +푡∼ [푡min, 푇 −1] +[ +퐷퐾퐿 +( +푃푡(휃) |||| +|||| +푃 (휃) +푝푋,푌 (푥, 푦; 휃) +)] +≥ − log 푝푋,푌 (푥, 푦; 휃), +(6) +since the KL divergence is always nonnegative for two +probability distributions. So, by optimizing 휃 to minimize +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푃푡(휃) || 푃(휃))], we maximize a lower +bound of 푝푋,푌 (푥, 푦; 휃), which is our primary goal. Thus, we +can simply introduce a variational distribution to approxi- +mate 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃) as long as we simultaneously mini- +mize the aforementioned KL divergence. Such a variational +distribution corresponds to a recognition model that tries to +predict the next latent states in addition to recognizing the +current and past ones, which is more useful than one that +would directly approximate 푝푍|푋,푌 (⋅| 푥, 푦; 휃). +Notice that: +푝푍|푋1∶푡(푧| 푥1∶푡; 휃) += 푝푍|푋(푧푡| 푥1∶푡; 휃) . 푝푍|푍,푋(푧1∶푡−1|푧푡, 푥1∶푡; 휃) +. 푝푍|푍,푋(푧푡+1∶푇 |푧1∶푡, 푥1∶푡; 휃) += 푝푍|푋(푧푡| 푥1∶푡; 휃) . +푡−1 +∏ +푘=1 +푝푍|푍,푋(푧푘|푧푘+1, 푥1∶푘; 휃) +. +푇 +∏ +푘=푡+1 +푝푍|푍(푧푘|푧푘−1; 휃), +(7) +omitting variable indices in distribution indices for the sake +of clarity. Based on this decomposition, let us introduce +two variational distributions 푄1 +푡 (휙) = 푞푍푡|푋1∶푡(⋅|푥1∶푡; 휙) and +푄2 +푡 (휙) = 푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1; 휙) taking their parameters +in the parameter set 휙 such that: +푞푍푡|푋1∶푡(⋅|푥1∶푡; 휙) ≈ 푝푍푡|푋1∶푡(⋅|푥1∶푡; 휃) +푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1; 휙) ≈ 푝푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1; 휃). +We assume that both 푝푍푡|푋1∶푡(⋅|푥1∶푡; 휃) and +푝푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1; 휃) have an approximate Gaussian +form with an approximately diagonal covariance matrix, i.e. +푄1 +푡 (휙) =  (휇푏(푥1∶푡; 휙), 휎푏(푥1∶푡; 휙).퐼푑) +푄2 +푡 (휙) =  (휇푠(푥1∶푡, 푧푡+1; 휙), 휎푠(푥1∶푡, 푧푡+1; 휙).퐼푑), +where 휇푏, 휎푏, 휇푠 and 휎푠 are deep neural networks taking their +parameters in the parameter set 휙. Taking back Eq. (7), we +get: +푝푍|푋1∶푡(푧| 푥1∶푡; 휃) +≈ 푞푍|푋(푧푡| 푥1∶푡; 휙) . +푡−1 +∏ +푘=1 +푞푍|푍,푋(푧푘|푧푘+1, 푥1∶푘; 휙) +. +푇 +∏ +푘=푡+1 +푝푍|푍(푧푘|푧푘−1; 휃) += 푞푍|푋(푧1∶푡|푥1∶푡; 휙) . 푝푍|푍(푧푡+1∶푇 | 푧푡; 휃) += 푞푍|푋1∶푡(푧| 푥1∶푡; 휃, 휙) = 푄푡(휃, 휙), +which means that posing our two variational distributions +푄1 +푡 (휙) and 푄2 +푡 (휙) is equivalent to posing the variational +distribution 푄푡(휃, 휙) ≈ 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃). +Therefore, we want to optimize 휙 and 휃 to minimize +피 +푡∼ [푡min, 푇 −1] +[ +퐷퐾퐿 +( +푄푡(휃, 휙) |||| +|||| +푃(휃) +푝푋,푌 (푥, 푦; 휃) +)] +while optimizing 휙 to minimize +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃푡(휃))] . +Chaveroche et al.: Preprint submitted to Elsevier +Page 9 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Actually, to achieve both these objectives, we only need to +minimize +LP-VAE(푥, 푦; 휃, 휙) = +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃))] +(8) +w.r.t. both 휙 and 휃. See Appendix A for more details. De- +veloping the KL divergence of Eq. (8) to make our recurrent +distributions appear, we finally obtain: +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|퐵1∶푡(푍1∶푡|푏1∶푡; 휙) ++ log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 | 푍푡; 휃) +] +− +피 +푍∼푄푡(휃,휙) +[ +log 푝푍1∶푡(푍1∶푡 ; 휃) ++ log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 |푍푡 ; 휃) ++ log 푝푋,푌 |푍(푥, 푦|푍 ; 휃) +] += 퐷퐾퐿 +( +푞푍1∶푡|퐵1∶푡(⋅|푏1∶푡; 휙) || 푝푍1∶푡(⋅ ; 휃) +) +− +피 +푍∼푄푡(휃,휙) +[log 푝푋,푌 |푍(푥, 푦|푍; 휃)] +(9) +which leads to +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍푖|퐵푖 +(푍푡|푏푡; 휙) ++ +푡−1 +∑ +푘=1 +log 푞푍푖|퐵푖,푍푖+1 +(푍푘|푏푘, 푍푘+1; 휙) +− log 푝푍푖 +(푍1; 휃) − +푡∑ +푘=2 +log 푝푍푖+1|푍푖 +(푍푘|푍푘−1; 휃) +− +푇∑ +푘=1 +log 푝푋푖,푌푖|푍푖 +(푥푘, 푦푘|푍푘; 휃) +] +(10) +Fig. 8 illustrates the process of evaluating (10). We can +easily give an interpretation to this loss: we can identify +two global objectives in Eq. (9) that are reminiscent of the +original VAE [28] in terms of interpretation: the 퐷퐾퐿 term +is an encoder loss for the recognition model of parameters +휙, while the second term is a decoder loss for the generative +model of parameters 휃. It can be viewed as a precision loss +(second term) optimized against a regularization (first term) +to prevent from overfitting. +We can even go deeper in interpretation to highlight what +differs from the original VAE. Contrary to the original VAE, +our model generates a sequence of observations instead of +an isolated one. Doing so, we have a Markovian transition +model that predicts a latent state from the previous one with +its own set of parameters separated from the decoder ones. +Therefore, it seems natural to have a third loss term for +prediction. We can make it appear by splitting the second +term of Eq. (9), i.e.: +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += 퐷퐾퐿 +( +푞푍1∶푡|퐵1∶푡(⋅|푏1∶푡; 휙) || 푝푍1∶푡(⋅ ; 휃) +) +− +피 +푍∼푄푡(휃,휙) +[ +log 푝(푋,푌 )1∶푡|푍1∶푡((푥, 푦)1∶푡|푍1∶푡; 휃) +] +− +피 +푍∼푄푡(휃,휙) +[ +log 푝(푋,푌 )푡+1∶푇 |푍푡+1∶푇 ((푥, 푦)푡+1∶푇 |푍푡+1∶푇 ; 휃) +] +The first term is an encoder loss. The second term is a +decoder loss. The third term is a prediction loss. This pre- +diction loss can also be viewed as a loss optimized against +a regularization since the 퐷퐾퐿 term affects the inference of +푍푡 by the recognition model from which the next latent states +are predicted. +4.5. LP-VAE with actions +The models we described up to this point represents the +environment evolving around the observing agent. However, +our agent also acts on this environment and influences the +observations gathered to train our model. Thus, we need to +modify it in order to integrate this subtlety. +Let 퐴푡 be the action applied at time 푡 on perceptions. This +action describes a mask on the information contained in 푌푡. +This partial information is then transmitted to the observing +agent, influencing 푋푡. It has no influence on the environment +evolving around the agent, only on its perception of it. This +means that 푌푡 and 푍푡 are not affected by 퐴푡. Moreover, +we will now consider that the random variable 푋푡 is the +ego-vehicle perception at time 푡, eventually augmented with +information from 푌푡, in accordance with 퐴푡, and combined +with the discounted memory of the previous partial obser- +vations 푋1∶푡−1. Fig. 9 provides the corresponding Bayesian +network. +We set the following constraints: +• 푍푖 ∼  (0, 퐼푑) +• 푝푍푖+1|푍푖(⋅|푧푡; 휃) =  (휇푧(푧푡; 휃), 휎2 +푧(푧푡; 휃).퐼푑) +• 푝푌푖|푍푖(⋅|푧푡; 휃) =  (휇푦(푧푡; 휃), 훼푦.퐼|푋푡|) +• 푝푋푖|푋푖−1,푌푡,푍푡,퐴푡(⋅|푥푡−1, 푦푡, 푧푡, 푎푡; 휃) +=  (휇푥(푥푡−1, 푦푡, 푧푡, 푎푡; 휃), 훼푥.퐼|푋푡|) +where all parameters 휇⋅ and 휎⋅ are deep neural networks +taking their parameters in 휃, and 훼⋅ ∈ [ 1 +2휋 , +∞). +Our dataset 퐷 is composed of 푁 independent sequences +of partial and complete observations with a randomly chosen +bounding box 퐴푡, i.e. 퐷 = (푥1∶푇 , 푦1∶푇 , 푎2∶푇 )1∶푁. Fortu- +nately, Eq. (7) still holds in this new model. Moreover, we +know that the environment does not depend on the actions +퐴2∶푇 taken on its perception of it and that the actions only +mask regions of 푌푡 while not altering the remaining. Finally, +since 푋푡 contains the information transmitted from 푌푡 in +accordance with 퐴푡, the actions 퐴2∶푡 do not bring any infor- +mation for the inference of the latent states 푍1∶푡. Given the +Bayesian network in Fig. 9, the actions 퐴 without knowing +Chaveroche et al.: Preprint submitted to Elsevier +Page 10 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +푏1 +푏2 +... +푏푡−1 +푏푡 +푥1 +푥2 +푥푡−1 +푥푡 +푧1 +푧푡−1 +... +푧2 +푧푡 +푧푇 +푧푇 −1 +... +푧푡+1 +푍2 +푍푡−1 +... +푍푡 +푋푌1 +푋푌2 +푋푌푡−1 +푋푌푡 +푋푌푡+1 +푋푌푇 −1 +푋푌푇 +... +... +Figure 8: Illustration of the forward computations allowing for the evaluation of the LP-VAE loss. A diamond indicates a +deterministically inferred variable. A rectangle indicates the deterministic inference of distribution parameters. A circle indicates +the deterministic inference of distribution parameters and a sample from this distribution. The blue network is the belief network. +The red network is the smoothing network. The black network is the Markovian transition model. The brown network is the +decoding network. +푁 +휃 +푌1 +푌2 +푌푇 −1 +푌푇 +퐴2 +퐴푇 −1 +퐴푇 +푋1 +푋2 +... +푋푇 −1 +푋푇 +푍1 +푍푇 −1 +... +푍2 +푍푇 +Figure 9: Bayesian network of our generative model of parameters in 휃. We have 푁 replications of this model, corresponding to +the 푁 sequences of length 푇 in our dataset. The parameter set 휃 influences the inference of all variables in the model for the 푁 +sequences we have. +푋푡+1∶푇 do not bring any information for the inference of the +latent states 푍푡+1∶푇 either. We have: +푝푍|푋1∶푡,퐴(⋅| 푥1∶푡, 푎; 휃) = 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃) +Thus, we keep the LP-VAE variational distributions +푄1 +푡 (휙) = 푞푍푡|푋1∶푡(⋅|푥1∶푡; 휙), +푄2 +푡 (휙) = 푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1; 휙), +푄푡(휃, 휙) ≈ 푝푍|푋1∶푡(⋅| 푥1∶푡; 휃). +Then, for our local predictability constraint (See Eq. (5)), we +consider 푝푍|푋,푌 ,퐴(⋅| 푥, 푦, 푎; 휃) instead of 푝푍|푋,푌 (⋅| 푥, 푦; 휃). +Notice that +푝푍|푋,푌 ,퐴(⋅| 푥, 푦, 푎; 휃) = +푝푋,푌 ,푍|퐴(푥, 푦, ⋅ |푎; 휃) +푝푋,푌 |퐴(푥, 푦 |푎; 휃) += +푃(휃) +푝푋,푌 |퐴(푥, 푦 |푎; 휃) +We take as loss function LP-VAE(푥, 푦 |푎; 휃, 휙) instead of +LP-VAE(푥, 푦; 휃, 휙), where +LP-VAE(푥, 푦 |푎; 휃, 휙) += +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃))] +(11) +This loss maximizes a lower bound of +푝푋,푌 |퐴(푥, 푦 |푎; 휃). +Developing the KL divergence of Eq. (11) in accordance +with our new model, we get: +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|퐵1∶푡 +(푍1∶푡|푏1∶푡; 휙) ++ log 푝푍푡+1∶푇 |푍푡 +(푍푡+1∶푇 |푧푡; 휃) − log 푝푍1∶푡 +(푍1∶푡; 휃) +− log 푝푍푡+1∶푇 |푍푡 +(푍푡+1∶푇 |푧푡; 휃) − log 푝푌 |푍 (푦 |푍; 휃) +Chaveroche et al.: Preprint submitted to Elsevier +Page 11 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +− log 푝푋|푌 ,푍,퐴 (푥 |푦, 푍, 푎; 휃) +] += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|퐵1∶푡 +(푍1∶푡|푏1∶푡; 휙) +− log 푝푍1∶푡 +(푍1∶푡; 휃) − log 푝푌 |푍 (푦 |푍; 휃) +− log 푝푋|푌 ,푍,퐴 (푥 |푦, 푍, 푎; 휃) +] +which leads to +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍푖|퐵푖 +(푍푡|푏푡; 휙) − log 푝푍푖 +(푍1; 휃) ++ +푡−1 +∑ +푘=1 +log 푞푍푖|퐵푖,푍푖+1 +(푍푘|푏푘, 푍푘+1; 휙) +− +푡∑ +푘=2 +log 푝푍푖+1|푍푖 +(푍푘|푍푘−1; 휃) − +푇∑ +푘=1 +log 푝푌푖|푍푖 +(푦푘|푍푘; 휃) +− +푇∑ +푘=2 +log 푝푋푖|푋푖−1,푌푖,푍푖,퐴푖 +(푥푘|푥푘−1, 푦푘, 푍푘, 푎푘; 휃) +− log 푝푋1|푌1,푍1 +(푥1|푦1, 푍1; 휃) +] +(12) +In practice however, we will neglect the term +− log 푝푋1|푌1,푍1 +(푥1|푦1, 푍1; 휃) for several reasons. First, it +avoids to optimize parameters that would only be used in +the learning phase, while not corresponding to an important +component (the complete observation 푦1 being already con- +sidered and containing 푥1). But maybe more importantly, +since 푋푡 keeps a memory of past observations in this for- +mulation of the LP-VAE, 푥1 may also contain information +on actions preceding 푎2∶푇 that should be given as well if +푥1 is actually not the start of an episode of interactions +in the environment. Not generating 푥1 allows us to start +the inference of latent states at any point of the episode, +independently from the previous actions and observations +that produced 푥1. This means that we can re-use different +subsequences of the same training sequence in the learning +phase, without having to make sure that 푥1 do not contain +information related to past observations and actions. +5. Implementation as neural networks +5.1. Belief state computation +The grids 퐺푡 introduced in section 3.1 are not directly +taken as input of our LP-VAE. Beforehand, we train a Con- +volutional VAE (CVAE) to learn a compressed, essentialized +representation of these observations in which spatial features +have been extracted. This CVAE is itself separated into 4 +independent parts in order to preserve the semantics of these +features: a CVAE for the pedestrian channel, another for the +car channel, another for static elements (road lines, road, +other) and a last one for the ignorance. The projection of +퐺푡 into the latent space of this Convolutional VAE is the +푋푡 taken by our LP-VAE. Then, we feed 푋푡, 푋푡−1 and the +ego-motion 푉푡 to a Multilayer perceptron (MLP) in order to +extract features about the motion of road users around the +ego-vehicle. The output of this MLP serves as input to a +Recurrent Neural Network (RNN) composed of Long Short- +Term Memory (LSTM) cells to form and update a belief over +the dynamics of other road users. The concatenation of the +hidden state of this RNN with 푋푡 and the driving controls 퐶푡 +represents the belief state 퐵푡 at time 푡. Fig. 10 visually sums +up this procedure. +5.2. Inference of Gaussian parameters +In [20], they proposed to use what they called D maps1 to +infer the Gaussian parameters of any of the distributions over +the latent state 푧푡. It is a part of a LSTM cell (new features +multiplied by the input gate), as indicated in Fig. 11, where +the output is passed to two fully connected (FC) layers in +parallel without activation function, one to determine 휇푧푡 and +the other to determine log(휎푧푡). Yet, in our sequential setting, +this D map becomes a truly recurrent unit, chaining itself +multiple times from 푡1 to 1 in the smoothing network and +from 푡1 to 푡2 in the prediction network. As for any recurrent +network, this poses the issue of vanishing gradients. Fur- +thermore, it lacks the semantics of a transition model: some +components could disappear from the frame (forget gate) and +some other could become visible or simply move from their +initial state (input gate, followed by an addition to the initial +components). These are exactly the transformations applied +to the cell state of a LSTM cell. Thus, using the cell state +of a LSTM cell as latent state mean 휇푧푡 as in Fig. 11, where +ℎ = 푧푡+1 and input = 푏푡, solves both the vanishing gradient +issue and the lack of model semantics. Giving ℎ as both +hidden and cell states also has the effect of implementing +peephole connections [29], giving the cell state some control +over the input, forget and output gates (the three sigmoïd +layers), which better captures sporadic events. In addition, +uncertainty should be encoded within the latent state to be +self-sufficient for a transition model. This encourages the +computation of the standard deviation 휎푧푡 from 휇푧푡 with +some filtering gate (output gate), which is exactly what a +LSTM cell does to output a quantity based on its cell state. +Similarly, we use this LSTM cell in the prediction network +for 푝푍푖+1|푍푖(⋅|푧푡; 휃), where ℎ = 푧푡 and input = ∅. For the +belief network, we keep this D map as there is no propagation +in time. +5.3. Decoding +So far, we determined the networks outputting distribu- +tion parameters describing the latent states 푍 used in the +evaluation of LP-VAE, both for the generative model and +the recognition model. It remains to propose the decoding +network that is part of the generative model and produces +푋 and 푌 . Given the conditional distributions appearing in +LP-VAE, we need a decoder inferring 푌푡 from 푍푡 and another +one inferring 푋푡 from 푋푡−1, 푌푡, 퐴푡 and 푍푡. +1In [20], they used a 16-layer model where the information transits +from layer to layer through the states of a LSTM, possibly in place of this D +map, in their DeepMind Lab experiment. Note however that it is recurrent +through layers, not time. This is different from what is proposed here. +Chaveroche et al.: Preprint submitted to Elsevier +Page 12 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +CNN +CNN +CNN +CNN +[pedestrian, car, road line, road, other, Ω] +6-channels mass grid at 푡 +[pedestrian] +[car] +[roal line, road, other] +[Ω] +푥푡 +푥푡−1 +푥푡 +푣푡 +MLP +ℎ푡−1 +LSTM +ℎ푡 +푐푡 +푏푡 +ℎ푡 +Belief state computation +Figure 10: Illustration of the process of computing the observation 푋푡 and the belief state 퐵푡 from 퐺푡, 푋푡−1, 푉푡 and 퐶푡. Four +independent Convolutional VAEs are trained to learn a sufficient representation of pedestrian, car, {road lines, road, other} and +ignorance. These encodings form 푋푡. A Multilayer perceptron (MLP) tries to learn features about the motion of road users around +the ego-vehicle. The output of this MLP serves as input to a Recurrent Neural Network (RNN) composed of Long Short-Term +Memory (LSTM) cells to form and update a belief over the dynamics of other road users. The concatenation of the hidden state +of this RNN with 푋푡 and the driving controls 퐶푡 represents the belief state 퐵푡 at time 푡. +LSTM cell +D map +ℎ +input +× ++ +휎 +FC +휎 +FC +× +tanh +tanh +× +FC +휎 +FC +FC +휇푧푡 +log(휎푧푡) +Figure 11: Proposed replacement for D maps. The FC rectan- +gles indicate a single Fully Connected layer. Circles indicate +point-wise operations, where 휎 is the sigmoïd activation +function. +However, since 푋푡 and 푌푡 are not given in the original +space but in a learned compressed one, extracting features +from 푌푡 according to the bounding box 퐴푡 is not directly +possible. One has to decode 푌푡, extract features according +to 퐴푡, decode 푋푡 and then fuse it with the leaked features +from 푌푡. For the sake of efficiency, we will learn to directly +extract these features that we denote by the random variable +푀푡 in the learned compressed space and to fuse them with +푋푡. Thus, in parallel to LP-VAE, we minimize an extra loss +term +− +푇∑ +푘=2 +log 푝푀푖|퐴푖,푌푖 +(푚푘|푎푘, 푦푘; 휃) , +푧푡 +푥푡−1 +휎 +× ++ +tanh +휎 +× +tanh +휎 +× +푦푡 +푎푡 +휎 +× ++ +tanh +휎 +× +푚푡 +휎 +× ++ +tanh +휎 +× +푥푡 +Decoder +Figure 12: Illustration of our decoding architecture. The +decoder block infers 푥푡 the partial observation, 푦푡 the spatially +complete observation and 푚푡 the masked 푦푡 (as dictated by +the bounding box 푎푡). It takes as inputs a latent state 푧푡, a +previous partial observation 푥푡−1 and a bounding box 푎푡. A +rectangle indicates a fully connected layer, while the symbol at +its center indicates the activation function applied to its output +(휎 for sigmoid, tanh for hyperbolic tangent and nothing for +the identity function). Each updating network is composed of +a forget gate (first 휎) and a D map, i.e. input features (tanh), +an input gate (last 휎) and a fully connected layer. +where 푚푡 corresponds to 푦푡 masked in accordance with 푎푡 +and compressed by the same CVAE as for 푦푡. Note that our +dataset becomes 퐷 = (푥1∶푇 , 푦1∶푇 , 푚2∶푇 , 푎2∶푇 )1∶푁. +Chaveroche et al.: Preprint submitted to Elsevier +Page 13 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +CNN +CNN +CNN +CNN +[pedestrian, car, road line, road, other, Ω] +6-channels mass grid at 푡 +[pedestrian] +[car] +[roal line, road, other] +[Ω] +푥푡 or 푦푡 or 푚푡 +Grid decoder +Figure 13: Illustration of the decoding of 푋푡 or 푌푡 or 푀푡 by the decoder of the CVAE that gave 푋푡 to get back into the observation +space. The CNN blocks are Transposed CNNs. +We choose to infer 푌푡 from 푍푡 through a D map as intro- +duced in section 5.2. All other inferences are done through an +updating module that is inspired by the updating of a LSTM +cell state. The masking of 푌푡 is orchestrated by 퐴푡, producing +푀푡 by filtering. Finally, 푋푡−1 is updated in two steps. The +first update is assumed to change its reference frame and to +determine which parts of 푌푡 are visible to the ego-vehicle. +This implicitly produces the 푋푡 corresponding to the null +action, i.e. the action that consists in doing nothing. We +consider this transformation deterministic, given 푦푡 and 푧푡. +The second update transmits the excerpt 푀푡 from 푌푡 to this +prior perception, producing the actual 푋푡 influenced by 퐴푡. +Fig. 12 depicts these networks. In addition, Fig. 13 illustrates +the decoding of 푋푡 by the decoder of the Convolutional VAE. +6. Experiments +6.1. Data acquisition & RL Environment +To conduct our experiments, we chose to work with the +open-source driving simulator CARLA [26]. Our semantic +grids 퐺푡 are computed online from a frontal 320 × 480 +depth camera with FOV of 135◦ and its corresponding pixel- +wise semantic classification. These simulated sensors are +attached to a simulated vehicle autonomously wandering in +a city with other vehicles, bikes and pedestrians (see Fig. 2). +More precisely, 퐺푡 is obtained by counting the number of +occurrences of each class in each possible configuration of +4 × 4 consecutive pixels. All classes corresponding to static +objects are merged into the class other. Then, in each cell of +the resulting 80 × 120 × 5 grid, these numbers are divided +by 16 and we add a channel representing ignorance (i.e. Ω) +to store the quantity needed to make the sum on all channels +equal to 1. We also discount the resulting mass functions by a +factor of 0.01 to simulate noise, i.e. all masses are multiplied +by 0.99 and 0.01 is added to the mass on ignorance. Finally, +thanks to the depth and information about the camera, we +create a 3D point cloud of this frontal perception. Thus, to +get the 2D grid 퐺푡, we ignore points higher than 2.5 meters +and we take the highest of the remaining ones (if more than +one point at the same ground coordinates). For this reason, +it sometimes happens that the ground under a vehicle is +perceived, but not its top, leading to road cells surrounded +by car cells, as can be observed in Fig. 2 Left. An important +road elevation may also conflict with the threshold of 2.5 +meters. This view can be obtained by a LIDAR and a 3D +semantic classifier [30] as well. +Our top-down semantic grids corresponding to complete +observations 푦푡 in our model are obtained with a facing +ground camera above the ego-vehicle. Doing so, it contains +itself some occlusions due to trees, poles, buildings, etc. +Thus, it is rather a hint about the true 푦푡. This grid can also be +obtained by the fusion of multiple view points, from a fleet +of autonomous vehicles or infrastructure sensors, which can +be acquired in the real world. A drone may be able to acquire +this information as well. In any case, this ground truth grid +is in fact itself uncertain and so is computed as 퐺푡 with an +ignorance channel. +We created a dataset composed of 1560 sequences of +50 timesteps (5 seconds) each, where each perception is +80×120×6. There are 30 runs in each of four cities available +in CARLA, including small towns, big towns and fast lanes. +Each run is 35 seconds long and a sequence is recorded every +2.5 seconds, leading to 13 sequences per run, hence the size +of our dataset. This dataset provides the grids corresponding +to 푋푡 and 푌푡 in the action-independent model of section 4.1. +To provide the grids corresponding to 푋푡 as defined in +the full model of section 4.5, we created a second dataset +from the first one by choosing random regions of 푌푡 to be +given to 푋푡. We also added a visual memory that keeps a +buffer of grid cells, transforms their coordinates according +to the given motion of the ego-vehicle, discounts their mass +functions to account for information ageing and fuses them +Chaveroche et al.: Preprint submitted to Elsevier +Page 14 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Binary classification per class +Mass +P +C +RL +R +O +Ω +score +LP-VAE +20.5% +68.5% +28.7% +84.3% +77.8% +49.5% +68.3% +STD-VAE +33.7% +72.7% +30.7% +85.9% +80.6% +46.2% +68.8% +Table 1 +Mass score and binary classification accurracy per class. P indicates the pedestrian channel, C the car channel, RL the road lines +channel, R the road channel, O the other channel and Ω the complete out-of-sight channel. It is clear that STD-VAE outperforms +LP-VAE for simple grid completion, though the total mass score is not so different. +with the current perception grid, resulting in this 푋푡. In +fact, the first dataset combined with our visual memory +and our fusion procedure of Algorithm 1 for ̃퐺푡 and 퐺푀 +푡 +constitutes the environment in which our agent will learn a +communication policy. +6.2. Models +During training, we give between 8 and 10 timesteps of +observations (i.e. between 0.8 and 1 second) and it is asked +to predict between 5 and 10 timesteps ahead, i.e. between 0.5 +and 1 second. We use the Mean Squared Error (MSE) loss +function to compute the Gaussian negative log likelihoods of +observing the grids corresponding to 푥푡 and 푚푡 given latent +states. Indeed, this is analog to taking 훼 = 1 +2 and ignoring the +constant term log +(√ +2휋훼 +) +. For the negative log likelihoods +on the grid corresponding to 푦푡, we binarize it by taking the +class with maximum mass and use a cross-entropy loss. To +account for the fact that the instances of 푌푡 in our dataset are +not perfect, we simply do a pointwise multiplication between +this loss and the complement to 1 of its ignorance channel +(last channel). That way, if 푦푡 does not have any information +about a cell, no loss on 푦푡 is actually back-propagated. +Furthermore, we weight this cross-entropy loss differently +from one channel to another to account for class imbalance. +We used the weight vector [100, 10, 1, 0.2, 0.1, 1]. Indeed, on +average, there are far less cells containing pedestrians than +cells containing the road or any other static class. Doing so, +without weights, the network would consider pedestrian as +noise and neglect them. +In the following, we compare STD-VAE and LP-VAE for +complete grid inference and prediction. +6.2.1. Grid completion +In this experiment, we use the decoder network de- +scribed in Fig. 12 on the current latent state 푍푡 inferred +from 퐵푡 to retrieve 푌푡. Then, we use the network described +in Fig. 13 to transform 푌푡 into the complete mass grid +퐺푌 . To compare STD-VAE and LP-VAE, we employed two +metrics: binary classification accuracy per class and a mass +score. Our Mass score metric is computed as the mean of +퐺푌 +푡 . ̂ +퐺푌 푡 over all cells in the grid, where 퐺푌 +푡 is the true +binary complete grid classification and ̂ +퐺푌 푡 is a mass grid +inferred by some model. Since 퐺푌 +푡 is binary, it acts as an +indicator function for the correct class and the mass score +represents the mean mass given to the right class by the +model generating ̂ +퐺푌 푡. Results are showed in Table 1. +6.2.2. Prediction +In this experiment, we compare prediction accuracy +between LP-VAE and STD-VAE. For this, we study mass +variations on the super-class {road, road line}, i.e. the sum +of the road and road line grid channels. Indeed, this super- +class represents the road layout. Its absence in a cell indicates +either road users or the other class. Thus, its mass variations +accounts for the dynamics of the whole scene, independently +of classification accuracy. +In practice, for each model, we infer a prediction se- +quence of 10 complete grids ̂푦1∶10 (i.e. 1 second in the +future), based on 10 observations (i.e. the past second). +From it, we compute the corresponding sequence of 9 grid +variations ̂푦′ +푡 = ̂푦푡+1 − ̂푦푡. We execute the same process +with the true complete grids, which produces grids 푦′ +1∶9 of +values ranging in {−1, 0, 1}. We test separately the accu- +racy on positive and negative changes. For the former, we +do a pointwise multiplication between the true complete +positive grids max(0, 푦′ +1∶9) and the inferred positive ones +max(0, ̂푦′ +1∶9). For the latter, we do a pointwise multiplication +between the true complete negative grids max(0, −푦′ +1∶9) and +the inferred negative ones max(0, − ̂푦′ +1∶9). We then sum all +cells of each grid in the sequence, over 4992 sequences, i.e. +49 920 inferred grids and compare it to the separate sums of +positive and negative true changes. Results are displayed in +the first two columns of Table 2. +However, note that this binary mask can be quite hard to +match, as both the exact location of these changes and their +amplitude must be correct. To alleviate this constraint, we +repeat this test with blurring filters applied to each grid of +푦′ +1∶9. The resulting grids, noted ̃푦′ +1∶9, are then renormalized +so that ∑ max(0, 푦′ +1∶9). max(0, ̃푦′ +1∶9) = ∑ max(0, 푦′ +1∶9) and +∑ max(0, −푦′ +1∶9). max(0, − ̃푦′ +1∶9) = ∑ max(0, −푦′ +1∶9). This +allows for slight misplacements of cells in predicted grids. +We repeated this test twice with Gaussian filters, with ker- +nels 5x5 and 11x11. These experiments correspond to the +last 4 columns of Table 2. Our LP-VAE outperforms STD- +VAE in every of these tests, no matter how hard the constraint +on change location is. This means that the predicted changes +of LP-VAE are not just better located, but also better shaped +than the ones of STD-VAE, as expected by design. Fig. 14 +illustrates this experiment. +Chaveroche et al.: Preprint submitted to Elsevier +Page 15 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +True 푦′ +No blur +Gaussian blur 5x5 +Gaussian blur 11x11 ++ +- ++ +- ++ +- +LP-VAE ̂푦′ +6.81% +6.94% +14.41% +14.61% +23.81% +24.36% +STD-VAE ̂푦′ +2.10% +2.37% +4.89% +5.41% +8.66% +9.52% +Table 2 +Prediction accurracies between STD-VAE and LP-VAE. As expected, LP-VAE significantly outperforms STD-VAE on predictions. +(a) +(b) +Figure 14: (a) Left column: partial grid 퐺푡 corresponding to +푋푡. Right column: complete grid 퐺푌 +푡 corresponding to 푌푡. Top +row: true classification grids. Bottom row: classification grids +predicted by LP-VAE from 푋 alone, 4 time steps in the future. +(b) Prediction dynamics. Black represents the absence of +variation, white some mass change in the cells of the road and +road line channels of the grid in (a). Left column corresponds +to the true variations, blurred by a 11x11 Gaussian filter. The +central column corresponds to the prediction dynamics of STD- +VAE, multiplied by the ones of the first column. Same for the +right column but for LP-VAE. The first row represents positive +changes, while the second row represents negative ones. +6.3. Policy learning +Here, we finally compare different policies learned with +PPO, with and without model to test the benefits of using +belief states in our case. Each policy is the best found among +iterations of training with 3000 transitions amounting to 500 +000 time steps in total. We used a batch size of 60, with +10 epochs on each transition dataset, with a learning rate of +0.0003 and an entropy coefficient of 0.01. We also made the +time horizon vary, i.e. we made the hyperparameter 훾 vary +from 0 to 0.7, in order to see if a medium/long term strategy +performs better. +The network learned with PPO has two parts: one for +inferring the Value of a state, representing the mean of all +potential future rewards, and one for inferring the best action +from this same state, representing the policy. Each of these +networks is composed of two fully connected hidden layers +of 128 and 64 neurons. +Different communication behaviors can be obtained by +adjusting reward parameters. In particular, increasing 퐾 in +Eq. 2 will make requests bigger, increasing 푤 in Eq. 1 will +make requests more focused on completely unknown areas, +increasing 휂 will make requests more focused on pedestrians +and cars, less rewarding in general and so less frequent. We +chose the following values: 휂 = 0.3, 퐾 = 36 and 푤 = 2. +We also added a penalty of -15 for no cooperation at all (i.e. +choice of a bounding box with no pixel in it, which means +no transmission cost either) to force the agent to play the +game. Moreover, approximating the top-down dimensions of +cars and pedestrians, we took the following reward densities +per squared meter: 푟푚 +obj = [540∕(0.7 ∗ 1.6), 540∕(3 ∗ +1.8), 20, 20, 0]. Then, we converted them into rewards per +squared cell by multiplying them by our grid resolution. +More precisely, we set our cameras in CARLA so that the +height corresponds to 40 meters. Thus, our reward densities +per squared cell are 푟obj = ( 40 +80)2.푟푚 +obj. Our final rewards are +obtained by normalizing 푟obj to [0, 1] by dividing it by its +maximum. For the spatial filter, we used the parameters of +Fig. 4, i.e. 훼 = 0.5, 훽퐹 = 0.8, 훽퐿 = 1 and 휁 = 0.01. +In order to evaluate and compare the performance of +different policy learning schemes, we take as metrics the +mean request size and the mean informational gain over all +time steps of a test set with same size and characteristics +as the training set described Section 6.1. We applied these +metrics to 3 class groups: pedestrians (P), cars (C) and +{road lines, road} (R). In these conditions, we compared 3 +schemes: PPO on top of the LP-VAE belief state 퐵푡, PPO +on top of the STD-VAE belief state 퐵푡 and PPO on top of +푋푡 alone (i.e. only the features extracted from the current +mass grid 퐺푡 by a Convolutional VAE). Each of them has +been trained with 훾 += +0 (i.e. only immediate rewards +matter), 훾 = 0.35 and 훾 = 0.7, to see if we could benefit +from medium/long term strategies. We also compare these +policies with a simple random policy that has a 50% chance +of making a request and chooses uniformly random size and +Chaveroche et al.: Preprint submitted to Elsevier +Page 16 of 19 + +0 +0 +0 +50 +50 +50 +0 +50 +100 +0 +50 +100 +0 +50 +100 +0 +0 +0 +50 +50 +50 +0 +50 +100 +0 +50 +100 +0 +50 +100Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown +Information gain +Request +P +C +R +size +Random +26.2% +22% +22.9% +13% +LP-VAE 퐵푡 +22% +27.6% +26.5% +6% +훾 = 0 +STD-VAE 퐵푡 +19.9% +26.5% +24.4% +5% +푋푡 alone +21.7% +29.2% +27.6% +6% +LP-VAE 퐵푡 +20.6% +25.7% +24.8% +6% +훾 = 0.35 +STD-VAE 퐵푡 +18.2% +22.8% +23.3% +5% +푋푡 alone +17.8% +23.4% +22.3% +5% +LP-VAE 퐵푡 +15.7% +18.2% +19.5% +5% +훾 = 0.7 +STD-VAE 퐵푡 +13.6% +16.3% +17.2% +4% +푋푡 alone +14.3% +17.8% +18.6% +4% +Table 3 +Learned communication policy performances relatively to a broadcasting policy. The information gain is a mean percentage +representing the mass actually gained after request, over the total mass that can be gained, at each time step. +position of bounding box when it does. Table 3 presents our +results, in percentage relatively to the maximal information +gain and request size possible inherent to a broadcasting +policy. +All of our learned policies only ask for about 5% of the +space around the ego-vehicle, while receiving about 25% of +the relevant information the agent lacks. Requiring about +2.5 times more information from the vehicular network +for about the same relevant information gain or lower, the +random policy is vastly less efficient. It only outperforms the +others for pedestrians, which is consistent with the highly +random behavior of pedestrians in CARLA. However, PPO ++ 푋푡 alone and 훾 = 0 (i.e. greedy policy) is the policy +that performs best overall. Surprisingly enough, taking into +account future rewards actually harms performance in our +case. A lower discounting factor in the memory module (i.e. +observations that are kept longer in memory) would proba- +bly make policies perform best with 훾 > 0. Furthermore, +note that LP-VAE always performs better than the other +learned policies when 훾 > 0. This is consistent with the +fact that LP-VAE has better prediction capabilities and thus +provides useful information in its belief state for predicting +future rewards. +7. Conclusions +In this paper, we tried to elaborate an efficient peer-to- +peer communication policy for collaborative perception. For +this, we made agents learn what could be hidden in their +blind spots through a generative sequence model that we +proposed, named Locally Predictable VAE (LP-VAE). We +compared its performance with another generative sequence +model for RL applications called TD-VAE that we slightly +adapted to our problem by making it both jumpy and se- +quential, referring to it as STD-VAE. We demonstrated that +LP-VAE produces better predictions than STD-VAE, which +translated into better performance for policies learned on +top of its belief state. However, we discovered in the end +that our best communication policy was a greedy one, i.e. +one that does not need prediction capabilities. Combined +with the fact that we augmented each observation with +the discounted memories of past observations, it followed +that only a state-less Convolutional VAE was needed for +this greedy policy. Overall, our best learned policies only +require about 5% of the space around the ego-vehicle, while +gaining about 25% of the relevant information the agent +lacks. Thus, we proved that learning to value the unknown is +much more efficient than employing a broadcasting policy. +It is also more efficient than blindly asking for random +areas around the ego-vehicle since it requires about 13% of +the total information, while gaining less than 25% of the +relevant information the agent lacks. In addition, we defined +interpretable hyperparameters shaping the reward function +corresponding to our problem. This makes it possible to +obtain various communication policies, with different trade- +offs between request size and information gain, as well as +different class valuations, spatial priorities and valuation of +ignorance (i.e. more or less emphasis on total ignorance). For +future works, it would be interesting to compare LP-VAE +and STD-VAE in RL tasks where future rewards are more +important. Also, we would like to test our communication +policies in a truly multi-agent context, where the agent +would need to take into account the availability of nearby +communicating vehicles, and with real sensor data. +A. LP-VAE loss +A.1. Minimization of 퐷퐾퐿 +(푄푡(휃, 휙) || 푃푡(휃)) +Proof. Indeed, we have, for some instant 푡: +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) + log 푝푍푡+1∶푇 |푍푡(⋅| ⋅ ; 휃) +− log 푝푍1∶푡(푍1∶푡; 휃) − log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 |푍푡; 휃) +− log 푝푋,푌 |푍(푥, 푦|푍; 휃)] += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) +− log 푝푍1∶푡(푍1∶푡; 휃) +− log 푝푋|푍(푥|푍; 휃) − log 푝푌 |푋,푍(푦|푥, 푍; 휃)] +Chaveroche et al.: Preprint submitted to Elsevier +Page 17 of 19 + +Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) +− log 푝푍1∶푡(푍1∶푡; 휃) − log 푝푋1∶푡|푍1∶푡(푥1∶푡|푍1∶푡; 휃) +− log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ; 휃) +− log 푝푌 |푋,푍(푦|푥, 푍; 휃)] += +피 +푍∼푄푡(휃,휙) +[ +log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) +− log 푝푋1∶푡,푍1∶푡(푋1∶푡, 푍1∶푡; 휃) +− log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ; 휃) +− log 푝푌 |푋,푍(푦|푥, 푍; 휃)] +Suppose that both 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ; 휃) and +푝푌 |푋,푍(푦|푥, 푍; 휃) range in [0, 1]. This can be easily verified +if they can be written as a factorization of probability +density functions that each ranges in [0, 1], e.g. Gaus- +sian distributions with diagonal covariance matrices where +each term of the diagonal is in [ 1 +2휋 , +∞). Then, both +− log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ; 휃) and +− log 푝푌 |푋,푍(푦|푥, 푍; 휃) are nonnegative, i.e. +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) +≥ 퐷퐾퐿 +( +푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ; 휃) +) +. +Thus, by minimizing 퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)), we minimize +an upper bound of +퐷퐾퐿 +( +푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ; 휃) +) +. +Furthermore, since we have +퐷퐾퐿 +( +푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ; 휃) +) += 퐷퐾퐿 +( +푞푍1∶푡|푋1∶푡(⋅| 푥1∶푡; 휙) || 푝푍1∶푡|푋1∶푡(⋅| 푥1∶푡; 휃) +) +− log 푝푋1∶푡(푥1∶푡; 휃) += 퐷퐾퐿 +(푄푡(휃, 휙) || 푃푡(휃)) − log 푝푋1∶푡(푥1∶푡; 휃), +we know that by optimizing 휙 to minimize +퐷퐾퐿 +( +푞푍1∶푡|푋1∶푡(⋅|푥1∶푡; 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ; 휃) +) +, we mini- +mize 퐷퐾퐿 +(푄푡(휃, 휙) || 푃푡(휃)). To sum up, minimizing +퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃)) w.r.t. 휙 minimizes an upper bound +of 퐷퐾퐿 +(푄푡(휃, 휙) || 푃푡(휃)). +■ +A.2. Maximization of 푝푋,푌 (푥, 푦; 휃) +Proof. Replacing 푃푡(휃) by 푄푡(휃, 휙) in Eq. (6), we get: +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃))] += − log 푝푋,푌 (푥, 푦; 휃) ++ +피 +푡∼ [푡min, 푇 −1] +[ +퐷퐾퐿 +( +푄푡(휃, 휙) |||| +|||| +푃 (휃) +푝푋,푌 (푥, 푦; 휃) +)] +≥ − log 푝푋,푌 (푥, 푦; 휃) +Therefore, by optimizing 휙 to minimize +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃))], we minimize +피 +푡∼ [푡min, 푇 −1] +[ +퐷퐾퐿 +( +푄푡(휃, 휙) +|||| +|||| +푃(휃) +푝푋,푌 (푥,푦;휃) +)] +, and by opti- +mizing 휃 to minimize +피 +푡∼ [푡min, 푇 −1] +[퐷퐾퐿 +(푄푡(휃, 휙) || 푃(휃))], +we maximize a lower bound of 푝푋,푌 (푥, 푦; 휃). +■ +References +[1] Q. Chen, X. Ma, S. Tang, J. Guo, Q. Yang, and S. Fu, “F-cooper: +Feature based cooperative perception for autonomous vehicle edge +computing system using 3D point clouds,” in Proceedings of the 4th +ACM/IEEE Symposium on Edge Computing, pp. 88–100, 2019. +[2] S. W. Kim, B. Qin, Z. 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Schmidhuber, “Recurrent nets that time and count,” +in Proceedings of the IEEE-INNS-ENNS International Joint Con- +ference on Neural Networks. IJCNN 2000. Neural Computing: New +Challenges and Perspectives for the New Millennium, vol. 3, pp. 189– +194, IEEE, 2000. +[30] Y. Li, L. Ma, Z. Zhong, F. Liu, M. A. Chapman, D. Cao, and J. Li, +“Deep learning for LiDAR point clouds in autonomous driving: a re- +view,” IEEE Transactions on Neural Networks and Learning Systems, +2020. +CRediT authorship contribution statement +Maxime Chaveroche: Conceptualization, Formal anal- +ysis, Investigation, Methodology, Software, Data Curation, +Validation, Visualization, Writing - Original Draft, Writing +- Review & Editing. Franck Davoine: Supervision, Writing +- Review & Editing. Véronique Cherfaoui: Supervision, +Writing - Review & Editing. +Chaveroche et al.: Preprint submitted to Elsevier +Page 19 of 19 + diff --git a/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/load_file.txt b/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f59ebbf0833a76527b15c9f437220eb381abbdc --- /dev/null +++ b/BdAzT4oBgHgl3EQfTfx4/content/tmp_files/load_file.txt @@ -0,0 +1,1376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf,len=1375 +page_content='Highlights Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Maxime Chaveroche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Franck Davoine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Véronique Cherfaoui Provides a way to learn an efficient decentralized communication policy between autonomous vehicles Proposes a new generative model that learns to build state representations for RL through prediction and reconstruction Proposes a reward function with interpretable parameters to adjust the trade-off between information gain and volume With our experiment parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' achieved 25% gain in relevant information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' with only 5% of the total queryable volume arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01250v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='LG] 12 Dec 2022 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown⋆ Maxime Chaverochea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Franck Davoinea and Véronique Cherfaouia aAlliance Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Université de technologie de Compiègne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Heudiasyc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' CS 60319 - 60203 Compiègne Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' France A R T I C L E I N F O Keywords: cooperative perception decentralized V2V communication efficiency filtering prediction model-based DRL Deep Learning Reinforcement Learning A B S T R A C T Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' we have been witnesses of accidents involving autonomous vehicles and their lack of sufficient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' One way to tackle this issue is to benefit from the perception of different view points, namely cooperative perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We propose here a decentralized collaboration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' peer-to- peer, in which the agents are active in their quest for full perception by asking for specific areas in their surroundings on which they would like to know more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Ultimately, we want to optimize a trade-off between the maximization of knowledge about moving objects and the minimization of the total volume of information received from others, to limit communication costs and message processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For this, we propose a way to learn a communication policy that reverses the usual communication paradigm by only requesting from other vehicles what is unknown to the ego-vehicle, instead of filtering on the sender side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We tested three different generative models to be taken as base for a Deep Reinforcement Learning (DRL) algorithm, and compared them to a broadcasting policy and a policy randomly selecting areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More precisely, we slightly modified a state-of-the-art generative model named Temporal Difference VAE (TD-VAE) to make it sequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We named this variant Sequential TD-VAE (STD-VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also proposed Locally Predictable VAE (LP-VAE), inspired by STD-VAE, designed to enhance its prediction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We showed that LP-VAE produced better belief states for prediction than STD-VAE, both as a standalone model and in the context of DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The last model we tested was a simple state-less model (Convolutional VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Experiments were conducted in the driving simulator CARLA, with vehicles exchanging parts of semantic grid maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Policies learned based on LP-VAE featured the best trade-off, as long as future rewards were taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our best models reached on average a gain of 25% of the total complementary information, while only requesting about 5% of the ego-vehicle’s perceptual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also provided interpretable hyperparameters controlling the reward function, which makes this trade-off adjustable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' allowing greater communication costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Introduction Recently, we have been witnesses of accidents involving autonomous vehicles and their lack of sufficient information at the right time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' One way to tackle this issue is to benefit from the perception of different viewpoints, namely collab- orative perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' While setting a multitude of sensors in the road infrastructure could be imagined, this would require a lot of investments and limit its usage to some areas in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Instead, we focus on the exchange of information between vehicles about their common environment, where they are the only sources available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These communications can simply be centralized by a server that would gather all information from all vehicles to process it and re-distribute it to all, as suggested in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, this still consists of Vehicle-to-Infrastructure (V2I) communications, which implies (1) an infrastructure cost and the impossibility to share information with other ⋆This work was carried out and co-funded in the framework of the Labex MS2T and the Hauts-de-France region of France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It was supported by the French Government, through the program “Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX- 0004-02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' ∗Corresponding author maxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='chaveroche@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='com (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' franck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='davoine@hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='utc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='fr (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Davoine);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' veronique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='cherfaoui@hds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='utc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='fr (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Cherfaoui) ORCID(s): 0000-0002-0834-4022 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 0000-0002-8587-6997 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Davoine);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 0000-0003-2064-9838 (V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Cherfaoui) agents when there is no server available nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It also features the disadvantage of (2) making the agents broadcast their entire perception, which can be heavy on the means of communication and computation and give rise to delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In contrast, the decentralized Vehicle-to-Vehicle (V2V) approach [2, 3, 4, 5, 6] does not require any extra infrastruc- ture to work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' does not implies (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In this setting, agents directly exchange pieces of information between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It also comes with new problems such as data incest and lower computation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We will ignore them here as we already tackled the issue of avoiding data incest using Dempster-Shafer Theory (DST) [7] in spite of low computation capabilities with two conference papers [8, 9] and a journal paper [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' But V2V communications bring a potentially heavier communication burden as well, due to redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In fact, (2) is worse in this setting than in the centralized one if agents are passive, meaning if they simply broadcast their perception for the others to know, without filtering it beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Nevertheless, this decentralized ap- proach offers the possibility to make the agents active in their quest for full perception, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' making the agents ask for specific areas in their surroundings on which they would like to know more, instead of always broadcasting everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is impossible in the centralized setting, as the server decides and thus needs to gather all perceptions beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Here, we propose such a system, where each agent builds its own local top-down semantic grid and sends specific Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 19 aDecentralized cooperative perception for autonomous vehicles: Learning to value the unknown requests to others in the form of bounding boxes described in the global reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We choose local grid maps for their ability to map an agent’s knowledge and to deduce its uncertainties in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Related Works Since not all uncertain areas are relevant, Active Ex- ploration [11, 12] is not enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' a truly efficient collabo- ration policy requires some understanding of the scenery [13], extracted from the spatial arrangement of grid cells and their classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' What could lie in the shadows and how to best discover it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' If a pedestrian is heading towards an occluded area, we expect the agent to request for this area, as a tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' If the agent has no idea of what could be in the unknown, maybe it could ask for some key points to understand the layout of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' If an area on the road is near a crowd of people or in the continuity of a pedestrian crossing, ask for it as some unseen- before pedestrians could be crossing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More generally, we would like the agent to know as much as possible about moving objects in its vicinity, while avoiding to request too much information from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This represents a complex bounding box selection policy to be learned from pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Given the long-lasting successes of Deep Learning in such ordeals, it seems natural to consider neural networks for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' But, while it is theoretically possible (but practically challenging) to learn our policy in an end-to- end fashion with Model-free Deep Reinforcement Learning (DRL), we choose to first learn a deep generative model to pre-process our inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, training deep neural net- works is easier, faster and more stable when the loss on the output is in the form of a well-justified derivable function, which is hard to achieve with reward signals from a RL environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Building this generative model also allows for more control and insights on what is learned, and reduces the size of the neural networks that are supposed to be trained through model-free DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' As demonstrated in World Models [14], learning a policy on top of a model can even be achieved with simple heuristics such as Evolution Strategies (ES), with performances equivalent to RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our model needs to be generative, for inference in un- known areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In addition, we want it to be predictive, in order to make it understand latent dynamics, anticipating disap- pearances or inferring hidden road users from the behavior of visible ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, it could even eventually compensate for communication latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Such a model would be useful in itself for other tasks as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Several existing works [15, 16, 17, 18, 19] employed generative models with convolutional networks in a U-Net architecture in order to augment instantaneous individual grid maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Some used deterministic networks such as Gen- erative Adversarial Networks (GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Others tried to incor- porate stochasticity with Monte Carlo Dropout or simply using a Variational Auto-Encoder (VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Most used occu- pancy grids as input, but some chose semantic grid maps or DOGMa (occupancy grid with velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These inputs were either expressed in a static global reference frame or given to a system that had no prediction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, it appears that none of these approaches really modeled the long-term dynamics of the environment that would be necessary to learn our desired policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' On the other hand, a kind of recurrent generative model inspired by the VAE, namely Temporal Difference VAE (TD-VAE) [20], was de- signed with the specific intent of being taken as base for a reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It puts an emphasis on the learning of belief states for long-term predictions, which are important for the development of complex strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It has been proven in [21] that explicitly predicting future states enhances data-efficiency in a number of RL tasks, though they train their model jointly with the policy and do not use the loss defined in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Appealed by the the- oretical justifications of TD-VAE, its decoupling regarding specific RL tasks (which simplifies the search for good RL hyperparameters) and its demonstrated ability to predict plausible sequences of images in a 3D world at different time horizons and from a variable number of observations, we have implemented and adapted this TD-VAE to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, correcting some of its weaknesses regarding its actual prediction capability, we finally proposed our own model, called Locally Predictable VAE (LP-VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To learn our communication policy based on this model, we chose the widely used Proximal Policy Optimization (PPO) algorithm [22], which is a fairly stable and simple policy-gradient based DRL algorithm with few hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Closely related to our goal, other works try to address the problem of efficiently communicating between autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In [23], they used a joint Perception and Prediction (P&P) model that transforms sensor data into learned fea- tures to broadcast to other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This model also fuses received features with local ones and tries to predict the trajectory of nearby communicating vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This informa- tion compression is also present in our work in the form of a Convolutional VAE preprocessing each observation grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We go one step further in communication efficiency as our system does not broadcast every piece of informa- tion, but chooses instead which one it wishes to receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Sending learned features also forces them to make another neural network learn to spatially and temporally transform all pieces of information received from the vehicular net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Even the fusion operation is done by making a neural network learn how to fuse two learned features, without any guarantee on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Instead, here we rely on top-down semantic grids, which are simple discretizations of the space around the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, we can transform the content of our transmissions using linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Furthermore, our system keeps its integrity by only fusing probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In [24], they used Deep Reinforcement Learning to select only a portion of the perceptive field of an autonomous vehicle to send to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, this information filtering is done on the sender side, contrary to our approach that filters on the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, their approach still Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Figure 1: Illustration of our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' CARLA provides a semantic segmentation corresponding to a camera attached to the ego-vehicle hood, as well as its corresponding depth (images taken from [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This gives us enough information to create a semantic 3D point cloud, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' to scatter all pixels in space according to their depth and image coordinates (and the camera deformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' From it, we project these pixels back into a 2D plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' a grid), but from a top-down point of view (and without camera deformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In parallel, we get the ego-vehicle motion since the previous time step in order to update a perception memory containing 2D points from previous time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We add the current semantic grid to this memory and give the resulting augmented grid to our learned world model (STD-VAE or LP-VAE), along with the ego-motion and driving policy commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In turn, this model tries to guess what is hidden in occluded areas and provides a belief state about latent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These outputs are then given to a DRL algorithm that chooses a grid area to request to the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This area is extracted at the next time step from a grid generated by a camera above the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, this information is fused at the next time step with the ego-vehicle perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' consists in broadcasting pieces of information, regardless of the actual needs of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The same can be stated for [25], where they describe a V2V cooperative perception system in which vehicles exchange object detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' They try to reduce redundancies by estimating the value of a piece of information for a potential receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The value here is the novelty, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' the probability that the potential receiver is not aware of some object of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Section 3 formally introduces our communication prob- lem, justifying the use of a preprocessing generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Section 4 formalizes the aforementioned generative model, introducing TD-VAE and LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Section 5 presents our deep networks implementing these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, section 6 evaluates and compares the performance of different ver- sions of our models and policy learnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, we con- clude this article with section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Problem formulation We formulate our communication problem as a Markov Decision Process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1 gives an overview of it, working with the driving simulator CARLA [26] for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' State space We assume the existence of a driving policy from which we only know the actions taken at each time step: ego- vehicle controls (acceleration and steering angle, each rang- ing in [−1, 1]) and global direction (average of the next 10 equally-spaced points the planner set to visit in meters relative to the ego-vehicle’s reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This driving policy influences the road environment in which the ego-vehicle is moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is not the case with the communication environment that we consider in this MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each observation is a tuple (퐺푡, 퐶푡, 푉푡), where 퐺푡 is an ego-centered semantic grid, 퐶푡 represents the actions taken by the driving policy at a given instant 푡 (which influence 퐺푡+1) and 푉푡 is the motion of the ego-vehicle between 푡−1 and 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each semantic grid 퐺푡 is a top-down 6-channels pseudo-Bayesian mass grid corresponding to the five classes of the frame of discernment Ω = {pedestrian, car, road lines, road, other}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The class car actually contains any type of vehicle, even bikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The class road lines contains any road marking: road lines, arrows, painted stop signs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The class other contains the rest of the static objects perceivable by the agent, such as vegeta- tion, sidewalks, buildings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The last channel represents ignorance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' the mass put on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that 퐺푡 ≥ 0 and, for any cell index 푖 of 퐺푡, we have ∑6 푘=1 퐺푡[푖][푘] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Page 3 of 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Semantic segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='CARLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Semantic 3D point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Top-down semantic 2D arid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='(Correcting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Depth camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Perception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='deformations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Ego Motion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='as for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='frontal view) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Driving controls ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Learned World ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Planned general ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Belief state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='direction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='(dynamics) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='DRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Bounding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='box a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='(PPO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Top-down semantic 2D grid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Inferred top-down semantic 2D gridDecentralized cooperative perception for autonomous vehicles: Learning to value the unknown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='Figure 2: Left: Illustration of an instance of top-down semantic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='grid 퐺푡 corresponding to a partial observation 푥푡 in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Red is for pedestrians, blue is for cars, yellow is for road lines, purple is for road, white is for other and black is for ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The displayed class is the one with the greatest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The intensity of its color depends on its mass: the closer to 0, the darker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Notice all the occlusions due to walls or other road users, in addition to the limited distance of perception of the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Right: Instance of top-down semantic grid corresponding to a complete observation 푦푡 in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Actually, this view is obtained with a facing ground camera above the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, it contains itself some occlusions due to trees, poles, buildings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, it is rather a hint about the true 푦푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This view can also be obtained by the fusion of multiple view points, from autonomous vehicles or infrastructure sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These cells are distributed as a matrix (grid) of 80 rows and 120 columns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 퐺푡 is analog to a 80 × 120 × 6 image of values in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2 for a visualization of this semantic grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These observations constitute a very large and complex space which would be hard to transform into exploitable neural network features without a derivable loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, we will first build a generative model of the driving en- vironment (implicitly including the agent’s driving policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Besides, learning this model beforehand will give us more control on the information flow that should be considered by the communication policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Therefore, the state space of our MDP is made of learned features from this generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Several versions of this generative model are proposed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Action space Our MDP has 4 continuous actions that each ranges in [0, 1], defining a bounding box in the local grid 퐺푡 of the ego-vehicle at time 푡: width, height, column and row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This bounding box is supposed to represent an area in the ego- vehicle’s future surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Transition function Transitions from a state-action pair to a new state depend also on the driving environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' First, this environment generates a new partial grid 퐺푡+1 and other observations already described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The bounding box described by the action given at time 푡 is then translated into an area of 퐺푡+1 filled with complete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3 illustrates this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In addition, a visual memory mechanism, specific to our MDP, makes perceptions persist for a few time steps, Figure 3: Illustration of our decision process: 1) Based on what is known at time 푡, select a bounding box where there is high uncertainty and high probability to discover road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2) Send this request in global coordinates to the vehicular network (which may consists of both infrastructure sensors and other autonomous vehicles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3) At time 푡 + 1, we expect some vehicles to transmit their perception of this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In our implementation, complete perceptions are simply obtained by a camera above the ego-vehicle since we focus on the selection of bounding boxes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4) The transmitted partial perception is fused with the one of the ego-vehicle at time 푡 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' discounted a little more every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This implements short- term memory, so that we only consider as unknown what has not been perceived in a long time (or never).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This also has the effect of giving consequences to past actions, since bounding boxes in the same area will have close to no potential information gain for a few time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Rewards Finally, let us define a reward function for our MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Let 푟푡 be a reward density, defined for each cell 푖 of 퐺푡+1 as: 푟푡(푖) = −휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='푟min + 푆[푖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 5 ∑ 푘=1 푟obj[푘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' max ( 0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘] )푤 (1) where 푤 ∈ ℝ+∗, 휂 ∈ [0, 1] and ̃퐺푡+1 is the grid before fusion with the grid 퐺푀 푡+1 corresponding to 푀푡+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The quantity 푟obj is a nonnegative reward per object pixel (only null for the static class other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푟obj[5] = 0) such that 푟obj[푘] ≥ 푟obj[푘+ 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, pedestrian are the smallest identifiable objects among our classes and so must have the highest reward per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The quantity 푟min is equal to the least positive reward per pixel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푟min = 푟obj[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It is used to discourage the selection of uninteresting cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The coefficient 휂 that multiplies it represents the minimum informational gain that is needed to consider this cell worth to be requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For some value of 휂, this minimum gain applies to the class with the least reward, while it becomes virtually more and more forgiving as the class has a greater reward per cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Moreover, notice that max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘]) ∈ [0, 1], which Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 19 Transmission: at t Request to vehicular network at t 2Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Figure 4: Heatmap illustrating our spatial filter 푆 for 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5, 훽퐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8, 훽퐿 = 1 and 휁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Deep blue is 0, while deep red is 1, which means that the reward in a cell located in a blue region will be 0, no matter what is inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The center of the ego-vehicle is in the middle of the first row starting from bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' implies that max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘])푤 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that 푤 only alters the significance of some gain in mass: for 푤 ∈ (0, 1), max(0, 퐺푡+1[푖][푘] − ̃퐺푡+1[푖][푘]) will be greater than for 푤 = 1, while for 푤 ∈ (1, +∞), max(0, 퐺푡+1[푖][푘]− ̃퐺푡+1[푖][푘]) will be less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In other words, if 푤 ∈ (1, +∞), then the gain will have to be more important to have an impact on 푟푡(푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, 푆 represents a spatial filter to account for the fact that we are not equally interested everywhere in discovering road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For example, a road user very far ahead is not as valuable an information as a road user just around the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We defined a forward filter 푆퐹 and a lateral filter 푆퐿, such that 푆 = 푆퐹 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='푆퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We set 푆퐹 [푖] = 1 − [ 훽퐹 1 − 훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' max ( 0, 퐹(푖) max(퐹) − 훼 )] where 훼 ∈ [0, 1) and 훽퐹 ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The quantity 퐹(푖) is the forward distance (number of rows from the row in which the center of the ego-vehicle is) corresponding to cell 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The greater the parameter 훽퐹 , the less the farest cells are valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The greater the parameter 훼, the farer from the ego-vehicle the decrease in value starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The second filter is defined as 푆퐿[푖] = 1 − 훽퐿 휁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' max (0, 휁 − |cos (arctan2 (퐿(푖), 퐹(푖)))| ) where 휁 ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The quantity 퐿(푖) is the lateral distance (number of columns from the column in which the center of the ego-vehicle is) corresponding to cell 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This filter de- scribes a cone in front of the ego-vehicle (and symmetrically at the back of it) in which the cells are the most valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The greater the parameter 휁, the narrower this cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The greater the parameter 훽퐿 is, the less the cells outside the cone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' on the sides of the ego-vehicle) are valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4 provides a visualization of 푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The reward associated with some action 푎푡 is defined as 푅푡(푎푡) = −퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (1 − 휂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='푟min+ ∑ 푖∈퐼(푎푡) 푟푡(푖), (2) where 퐾 is the minimum number of interesting cells that must be entirely discovered in order to make the request worthwhile, 퐼(푎) = [푣(푎), 푣(푎) + ℎ(푎)] × [푢(푎), 푢(푎) + 푤(푎)] and 푢(푎), 푣(푎), 푤(푎), ℎ(푎) are respectively the column index, row index, width and height indicated by some action 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Grid fusion In order to produce 퐺푡 from ̃퐺푡 and the grid 퐺푀 푡 cor- responding to 푀푡 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (1), we need to define a fusion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' As each cell 푖 in both ̃퐺푡 and 퐺푀 푡 is a mass function, we know that: 퐺푡[푖][6] = ̃퐺푡[푖][6] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 퐺푀 푡 [푖][6], where 6 is the channel corresponding to the mass on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Furthermore, we can get the contour functions of these pseudo-Bayesian mass functions simply by adding the mass on Ω to the mass on each of our 5 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, a sim- ple pointwise multiplication of these two contour functions produces the contour function corresponding to 퐺푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This also implies a mass on ∅, which is caused by conflicting pieces of evidence between the two mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Since we are not interested in this level of conflict, we choose to renormalize masses as in Dempster’s combination rule [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Unlike Dempster’s rule however, we only distribute this conflict on singletons 퐺푡[푖][1 ∶ 5] and keep the true value 퐺푡[푖][6], as the distinction between ignorance and conflict is crucial to our communication policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Algorithm 1 details this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Algorithm 1: Fusion procedure for two pseudo- Bayesian mass functions 푚1 and 푚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Input: Two pseudo-Bayesian mass functions 푚1, 푚2 Output: The fused mass function 푚12 푁 ← len(푚1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푚12[푁] ← 푚1[푁] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푚2[푁];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푚12[1 ∶ 푁−1] ← (푚1[1 ∶ 푁−1]+푚1[푁]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (푚2[1 ∶ 푁 − 1] + 푚2[푁]) − 푚12[푁];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푠 ← sum(푚12[1 ∶ 푁 − 1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' if 푠 > 0 then 푚12[1 ∶ 푁 − 1] ← (1 − 푚12[푁]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푚12[1∶푁−1] 푠 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Return 푚12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Models In this section, we will present several versions of the generative model mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1, namely STD- VAE and LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In the end, this generative model will provide us with learned features describing the state of the environment related to the MDP presented in section 3, in order to reduce the size of the network optimized through DRL and to control what is kept in the information flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We will start by formalizing in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1 a draft of this model that ignores the actions the agent takes at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Then, we will briefly introduce in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2 the original TD-VAE [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Following that, we will propose in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3 our sequential variant of TD-VAE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' STD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Inspired by this model, we will then propose LP-VAE in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 will demonstrate with LP-VAE how to modify this generative model to incorporate the actions chosen by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Action-independent modeling As a vehicle clearly cannot access the complete state of its surroundings through its sole perception, we can model our problem as a Partially Observable Discrete-Time Markov Chain (PO-DTMC), where 푋푡 and 푍푡 denote ran- dom variables representing respectively a partial observation and a latent state at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, we consider that 푍푡 and 푋푡 are in different spaces, the latent space describing the whole environment and containing information about object dynamics and trajectories allowing for predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More precisely, 푋푡 corresponds to the sole perception of the ego-vehicle at time 푡, without memory of the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also introduce a third random variable 푌푡 which represents the spatially complete observation corresponding to 푍푡 in the space of 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In other words, 푋푡 is a partial observation of 푌푡 which is itself a partial observation of 푍푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' So, let 휃 be a set containing the parameters of a genera- tive model that projects a latent state 푍푡 onto the observation space as (푋푡, 푌푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We choose to implement this generative model as a deep neural network and we set the following Gaussian distributions as constraints, for numerical stability and simplicity: 푍푖 ∼ \ue23a (0, 퐼푑) 푝푍푖+1|푍푖(⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푧(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 휎2 푧(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼푑) 푝푌푖|푍푖(⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푦(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 훼푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼|푋푡|) 푝푋푖|푌푖,푍푖(⋅|푦푡, 푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푥(푦푡, 푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 훼푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼|푋푡|) where 휇푧, 휎푧, 휇푥 and 휇푦 are all deep neural networks taking their parameters in 휃, where 푑 is an arbitrary number of dimensions for 푍푡, where 푧푡 is a realization of 푍푡 for some 푡 ∈ [1, 푇 ] and where 훼⋅ ∈ [ 1 2휋 , +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This last constraint implies that the generative model recreates independently each dimension of 푋푡 from a latent state 푧푡 with the same fixed precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Moreover, the PO-DTMC formulation im- plies that each pair of observations (푋푡, 푌푡) is only dependent on 푍푡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푋,푌 |푍 (푥, 푦 | 푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푇 ∏ 푡=1 푝푋푖,푌푖|푍푖(푥푡, 푦푡 | 푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), and that the Markovian property holds in latent space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푍(푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푍푖(푧1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푇 ∏ 푡=2 푝푍푖+1|푍푖(푧푡 | 푧푡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 5 provides the Bayesian network corresponding to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푁 휃 푌1 푌2 푌푇 −1 푌푇 푋1 푋2 푋푇 −1 푋푇 푍1 푍푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푍2 푍푇 Figure 5: Bayesian network of our generative model of parame- ters in 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We have 푁 replications of this model, corresponding to the 푁 sequences of length 푇 in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The parameter set 휃 influences the inference of all variables in the model for the 푁 sequences we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, based on a dataset of 푁 independent sequences of partial and complete observations 퐷 = (푥1∶푇 , 푦1∶푇 )1∶푁, we want to optimize the parameters 휃 so that the probability that the model generates the sequences of 퐷 is maximal under its constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In other words, we want to find the parameters 휃 that maximize 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), which is the same as finding 휃 maximizing log 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We have: log 푝(푋,푌 )(1),…,(푋,푌 )(푁)(퐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = ∑ (푥,푦)∈퐷 log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) where 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = ∫ 푝푋,푌 |푍(푥, 푦 | 푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푍(푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푑푧 = ∫ ⋯ ∫ 푝푍푖(푧1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푇 ∏ 푡=1 푝푋푖,푌푖|푍푖(푥푡, 푦푡 | 푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푇 ∏ 푡=2 푝푍푖+1|푍푖(푧푡 | 푧푡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푇 ∏ 푡=1 푑푧푡 which is intractable, due to the fact that 휇푧, 휎푧, 휇푥 and 휇푦 are multi-layers neural networks with nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This intractability is amplified by the fact that we work with sequences of 푇 non-independent continuous latent states, which implies a multiple integral over ℝ푇 ×푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that we cannot evaluate or differentiate the marginal likelihood 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the same reasons, the posterior distribution 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푋,푌 |푍(푥, 푦| ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='푝푍(⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) , is intractable, which implies that methods based on the posterior distribution such as the Expectation-Maximization (EM) algorithm cannot be employed either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' So, let us adopt the Variational Bayesian (VB) approach by introducing a variational distribution dependent on a parameter set 휙 to approximate 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' But, more than just a mathe- matical trick, we want this variational distribution to actually be a recognition model such that it is able to infer latent states only given past partial observations, in order to infer 푦 and to be able to generate plausible next observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' TD-VAE model TD-VAE [20] is a variant of the original VAE [28] for temporal sequences which features the particularity to separate belief states from latent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A belief state 푏푡 is a statistics describing 푥1∶푡 such that 푝푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ≈ 푝푍푡|퐵푡(⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The end goal motivating this distinction, aside theoretical accuracy, is to learn a model able to determin- istically aggregate observations by updating a statistics 푏푡 that contains enough information to infer some latent state 푧푡, avoiding the accumulation of estimation errors on 푧1∶푡−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Since 푧푡 alone allows for predictions of next latent states, 푏푡 constitutes a belief on plausible latent dynamics that is simply updated with each new observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This feature is important for model-based RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In [20], they chose additionally to make their model provide jumpy predictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' directly predicting a latent state 푧푡+훿 from some 푧푡 where 훿 is not precisely known, in order to abstract latent dynamics for the benefit of computa- tional efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Formally, they seek to optimize 휃 so that it maximizes the expression 피 훿∼\ue241[훿푖,훿푠] [ 피 푡∼\ue241[1,푇 −훿] [ log 푝푋푡+훿|퐵푡 (푥푡+훿|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)]] , (3) where \ue241[푎,푏] is the uniform distribution on the interval [푎, 푏] and 퐵푡 = RNN(푋푡, 퐵푡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This cannot be optimized directly, as showed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, we can maximize a lower bound of this expression by introducing a variational distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Let 푄푡,훿(휙) = 푞푍푡,푍푡+훿|퐵푡,퐵푡+훿(⋅|푏푡, 푏푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) be this varia- tional distribution, dependent on a parameter set 휙, such that 푞푍푡,푍푡+훿|퐵푡,퐵푡+훿 (⋅|푏푡, 푏푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) ≈ 푝푍푡,푍푡+훿|퐵푡,푋푡+훿 (⋅|푏푡, 푥푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) where it is important to notice that 푝푍푡,푍푡+훿|퐵푡,푋푡+훿 (⋅|푏푡, 푥푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푋푡+훿,푍푡,푍푡+훿|퐵푡 (푥푡+훿, ⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푝푋푡+훿|퐵푡 (푥푡+훿|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푃푡,훿(휃) 푝푋푡+훿|퐵푡 (푥푡+훿|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To find the optimal parameters 휙 that minimize its ap- proximation error, we can optimize 휙 so that it minimizes through gradient descent the following average Kullback- Leibler (KL) divergence: 피 훿∼\ue241[훿푖,훿푠] [ 피 푡∼\ue241[1,푇 −훿] [ 퐷퐾퐿 ( 푄푡,훿(휙) |||| |||| 푃푡,훿(휃) 푝푋푖+훿|퐵푖 (푥푡+훿|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )]] , This cannot be optimized directly either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Yet, it can be shown that we can equivalently minimize this divergence, while also maximizing a lower bound of (3), by minimizing the following loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙 and 휃: \ue238TD-VAE(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) = 피 훿∼\ue241[훿푖,훿푠] [ 피 푡∼\ue241[1,푇 −훿] [퐷퐾퐿 (푄푡,훿(휙) || 푃푡,훿(휃))] ] where 퐷퐾퐿 (푄푡,훿(휙) || 푃푡,훿(휃)) = 피 푍푡,푍푡+훿∼푄푡,훿(휙) [ log 푞푍푖|퐵푖 (푧푡+훿|푏푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + log 푞푍푡|퐵푡,퐵푡+훿,푍푡+훿 (푧푡|푏푡, 푏푡+훿, 푧푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍푖|퐵푖 (푍푡|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푍+훿|푍 (푍푡+훿|푍푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋푖|푍푖 (푥푡+훿|푍푡+훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In complement, the authors of [20] had to make the strong assumption that 푝푍푖|퐵푖 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푞푍푖|퐵푖 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) for any 휃, 휙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' They also set 푝푍+훿|푍 (⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) as a multivariate normal distribution with diagonal covariance matrix, corresponding to the distribution of latent states at any instants in [푡 + 훿푖, 푡 + 훿푠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is in contradiction with our sequential latent model 푝푍푖+1|푍푖 (⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), which is itself a multivariate normal distribution with diagonal covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In this regard, 푝푍+훿|푍 (⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) can be seen as a rough approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This abstraction of latent dynamics may be useful in some cases where precision is not needed and the variability of observations 푥푡∶푡+훿 gathered in a moment can be summa- rized in latent space by smooth transitions between states corresponding to dataset samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, we argue that models of complex environments, in which the observation space is combinatorially extremely large and in which mul- tiple agents interact with each other, require precise learning signals to understand latent dynamics and so to generalize well outside the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More importantly, TD-VAE cannot consider the actions taken by the observing agent between 푡 and 푡 + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Yet, learning the link between actions and observations is central in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our Sequential variant STD-VAE of the TD-VAE model The authors of [20] also proposed a sequential version of their model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Its corresponding Bayesian network is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' They chose to train its parameters as a particular case of the jumpy one, simply taking 훿 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Yet, this would only maximize a lower bound of the probability to ob- serve 푥푡+1 after 푏푡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 피 푡∼\ue241[1,푇 −1] [ log 푝푋푡+1|퐵푡 (푥푡+1|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] , instead of the whole future sequence 푥푡+1∶푇 after 푏푡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 피 푡∼\ue241[1,푇 −1] [ log 푝푋푡+1∶푇 |퐵푡 (푥푡+1∶푇 |푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' From a practical point of view, this would prove to be computationally heavy if done multiple times per sequence and would not learn from the accumulation of prediction errors: particularly in a stochastic network such as TD-VAE and with a time step small enough, the network will tend to optimize weights such that the predicted next state looks almost identical to the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It is only by chaining these predictions that their errors become significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, we choose a slightly different variational distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Let Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 7 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown 푁 퐵1 퐵2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 퐵푇 −1 퐵푇 푋1 푋2 푋푇 −1 푋푇 푍1 푍푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푍2 푍푇 Figure 6: Bayesian networks corresponding to STD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Solid lines represent the Bayesian network of our generative model (without 푌푡) of parameters in 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Dashed lines represent the Bayesian network of the recognition model of parameters in 휙 proposed by TD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Parameter dependencies are not represented for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Only 퐵푡 is not directly influenced by 휃, while only variables at the end of a dashed arrow are influenced by 휙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We have 푁 replications of this model, corresponding to the 푁 sequences of length 푇 in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푄푡(휙) = 푞푍푡∶푇 |퐵푡∶푇 (⋅|푏푡∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) be this variational distribu- tion, dependent on a parameter set 휙, such that 푞푍푡∶푇 |퐵푡∶푇 (⋅|푏푡∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) ≈ 푝푍푡∶푇 |퐵푡,푋푡+1∶푇 (⋅|푏푡, 푥푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) where it is important to notice that 푝푍푡∶푇 |퐵푡,푋푡+1∶푇 (⋅|푏푡, 푥푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푋푡+1∶푇 ,푍푡∶푇 |퐵푡 (푥푡+1∶푇 , ⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푝푋푡+1∶푇 |퐵푡 (푥푡+1∶푇 |푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푃푡(휃) 푝푋푡+1∶푇 |퐵푡 (푥푡+1∶푇 |푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To find the optimal parameters 휙 that minimize its ap- proximation error, we can optimize 휙 so that it minimizes through gradient descent the following average Kullback- Leibler (KL) divergence: 피 푡∼\ue241[1,푇 −1] [ 퐷퐾퐿 ( 푄푡(휙) |||| |||| 푃푡(휃) 푝푋푡+1∶푇 |퐵푡 (푥푡+1∶푇 |푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )] , It can be shown that we can equivalently minimize this divergence, while also maximizing a lower bound of 피 푡∼\ue241[1,푇 −1] [ log 푝푋푡+1∶푇 |퐵푡 (푥푡+1∶푇 |푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] , by minimizing the following loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙 and 휃: \ue238STD-VAE(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) = 피 푡∼\ue241[1,푇 −1] [퐷퐾퐿 (푄푡(휙) || 푃푡(휃))] where 퐷퐾퐿 (푄푡(휙) || 푃푡(휃)) = 피 푍푡∶푇 ∼푄푡(휙) [ log 푞푍푖|퐵푖 (푍푇 |푏푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + 푇 −1 ∑ 푘=푡 log 푞푍푖|퐵푖,푍푖+1 (푍푘|푏푘, 푍푘+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍푖|퐵푖 (푍푡|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푇∑ 푘=푡+1 log 푝푍푖+1|푍푖 (푍푘|푍푘−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푇∑ 푘=푡 log 푝푋푖|푍푖 (푥푘|푍푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] (4) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 7 visually explains the process of evaluating (4), which is very similar to the original TD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The belief network aggregates observations such that each belief 푏푡 is assumed to be a sufficient statistics for 푥1∶푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The smoothing network, knowing what the final latent state 푧푇 is, given observations 푥1∶푇 , infers what should have been latent states 푧푡∶푇 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This gives us two different distributions for the inference of 푧푡: one given only observations 푥1∶푡, and the other given all observations 푥1∶푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In the learning phase, we measure the divergence between these two distributions as a loss to prompt correct dynamics recognition and consistency in the belief network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, the Markovian transition model infers the next state from the current one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We infer the Gaussian parameters of the next state for each latent state inferred by the smoothing network and measure as loss the divergence between the distribution inferred by the smooth- ing network and the one inferred by the transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, for each latent state 푧푘 sampled from the smoothing network, we infer the Gaussian parameters describing the observation 푥푘 with the decoding network and compute the negative log-likelihood of 푥푘 given these parameters as loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, our preliminary experiments on this model with a dataset acquired in CARLA [26] revealed very poor prediction quality when 푧푡 is sampled from 푞푍푡|퐵푡(⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), while providing very good predictions when 푧푡 is sampled from 푞푍푡|퐵(⋅|푏푡∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' from the smoothing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In fact, this seems obvious considering that the prediction part of this model is trained with the latent states sampled from the variational distribution 푞푍푡∶푇 |퐵푡∶푇 (⋅|푏푡∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) and not 푞푍푡∶푇 |퐵푡 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is what motivates the introduction in the next section of a local predictability constraint, allow- ing us to train our model on samples from 푞푍푡∶푇 |퐵푡 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This will also allow us to keep the idea of predicting distant latent states from current observations while avoiding the strong assumption that 푝푍|퐵 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푞푍|퐵 (⋅|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our Locally Predictable VAE (LP-VAE) model First, we put a local predictability constraint for the model to be able to predict multiple time steps into the future: 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ≈ 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) (5) for any instant 푡 ≥ 푡min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that there must be some instant 푡min such that the partial observations 푥1∶푡min are suffi- cient to recognize the latent dynamics of the whole sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' such that all observations 푦1∶푇 and all subsequent partial Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 8 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown 푏푡 푏푡+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푏푇 −1 푏푇 푥푡 푥푡+1 푥푇 −1 푥푇 푧푡 푧푇 푧푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푧푡+1 푍푡 푍푇 푍푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푍푡+1 푋푡+1 푋푇 −1 푋푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푏푡−1 Figure 7: Illustration of the forward computations allowing for the evaluation of the STD-VAE loss (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A diamond indicates a deterministically inferred variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A rectangle indicates the deterministic inference of distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A circle indicates the deterministic inference of distribution parameters and a sample from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The blue network is the belief network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The red network is the smoothing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The black network is the Markovian transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The brown network is the decoding network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' observations 푥푡min+1∶푇 bring negligible additional informa- tion in the recognition of these latent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Notice that 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푋,푌 ,푍(푥, 푦, ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푃 (휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), and let us note 푃푡(휃) = 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To enforce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (5), we want to minimize the average KL divergence 피 푡∼ \ue241[푡min, 푇 −1] [ 퐷퐾퐿 ( 푃푡(휃) |||| |||| 푃(휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )] = log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푃푡(휃) || 푃(휃))] , which we cannot minimize directly, due to the intractability of 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) and 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, we have: 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푃푡(휃) || 푃(휃))] = − log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + 피 푡∼ \ue241[푡min, 푇 −1] [ 퐷퐾퐿 ( 푃푡(휃) |||| |||| 푃 (휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )] ≥ − log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), (6) since the KL divergence is always nonnegative for two probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' So, by optimizing 휃 to minimize 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푃푡(휃) || 푃(휃))], we maximize a lower bound of 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), which is our primary goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, we can simply introduce a variational distribution to approxi- mate 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) as long as we simultaneously mini- mize the aforementioned KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Such a variational distribution corresponds to a recognition model that tries to predict the next latent states in addition to recognizing the current and past ones, which is more useful than one that would directly approximate 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Notice that: 푝푍|푋1∶푡(푧| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푍|푋(푧푡| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푍|푍,푋(푧1∶푡−1|푧푡, 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푍|푍,푋(푧푡+1∶푇 |푧1∶푡, 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푍|푋(푧푡| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푡−1 ∏ 푘=1 푝푍|푍,푋(푧푘|푧푘+1, 푥1∶푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푇 ∏ 푘=푡+1 푝푍|푍(푧푘|푧푘−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), (7) omitting variable indices in distribution indices for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Based on this decomposition, let us introduce two variational distributions 푄1 푡 (휙) = 푞푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) and 푄2 푡 (휙) = 푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) taking their parameters in the parameter set 휙 such that: 푞푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) ≈ 푝푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) ≈ 푝푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We assume that both 푝푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) and 푝푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) have an approximate Gaussian form with an approximately diagonal covariance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푄1 푡 (휙) = \ue23a (휇푏(푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), 휎푏(푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼푑) 푄2 푡 (휙) = \ue23a (휇푠(푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), 휎푠(푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼푑), where 휇푏, 휎푏, 휇푠 and 휎푠 are deep neural networks taking their parameters in the parameter set 휙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Taking back Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (7), we get: 푝푍|푋1∶푡(푧| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ≈ 푞푍|푋(푧푡| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푡−1 ∏ 푘=1 푞푍|푍,푋(푧푘|푧푘+1, 푥1∶푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푇 ∏ 푘=푡+1 푝푍|푍(푧푘|푧푘−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푞푍|푋(푧1∶푡|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푝푍|푍(푧푡+1∶푇 | 푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푞푍|푋1∶푡(푧| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) = 푄푡(휃, 휙), which means that posing our two variational distributions 푄1 푡 (휙) and 푄2 푡 (휙) is equivalent to posing the variational distribution 푄푡(휃, 휙) ≈ 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Therefore, we want to optimize 휙 and 휃 to minimize 피 푡∼ \ue241[푡min, 푇 −1] [ 퐷퐾퐿 ( 푄푡(휃, 휙) |||| |||| 푃(휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )] while optimizing 휙 to minimize 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃푡(휃))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 9 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Actually, to achieve both these objectives, we only need to minimize \ue238LP-VAE(푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) = 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃))] (8) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' both 휙 and 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' See Appendix A for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' De- veloping the KL divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (8) to make our recurrent distributions appear, we finally obtain: 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|퐵1∶푡(푍1∶푡|푏1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 | 푍푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] − 피 푍∼푄푡(휃,휙) [ log 푝푍1∶푡(푍1∶푡 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 |푍푡 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + log 푝푋,푌 |푍(푥, 푦|푍 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] = 퐷퐾퐿 ( 푞푍1∶푡|퐵1∶푡(⋅|푏1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푍1∶푡(⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) − 피 푍∼푄푡(휃,휙) [log 푝푋,푌 |푍(푥, 푦|푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] (9) which leads to 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 피 푍∼푄푡(휃,휙) [ log 푞푍푖|퐵푖 (푍푡|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + 푡−1 ∑ 푘=1 log 푞푍푖|퐵푖,푍푖+1 (푍푘|푏푘, 푍푘+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍푖 (푍1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푡∑ 푘=2 log 푝푍푖+1|푍푖 (푍푘|푍푘−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푇∑ 푘=1 log 푝푋푖,푌푖|푍푖 (푥푘, 푦푘|푍푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] (10) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 8 illustrates the process of evaluating (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We can easily give an interpretation to this loss: we can identify two global objectives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (9) that are reminiscent of the original VAE [28] in terms of interpretation: the 퐷퐾퐿 term is an encoder loss for the recognition model of parameters 휙, while the second term is a decoder loss for the generative model of parameters 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It can be viewed as a precision loss (second term) optimized against a regularization (first term) to prevent from overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We can even go deeper in interpretation to highlight what differs from the original VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Contrary to the original VAE, our model generates a sequence of observations instead of an isolated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, we have a Markovian transition model that predicts a latent state from the previous one with its own set of parameters separated from the decoder ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Therefore, it seems natural to have a third loss term for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We can make it appear by splitting the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (9), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 퐷퐾퐿 ( 푞푍1∶푡|퐵1∶푡(⋅|푏1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푍1∶푡(⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) − 피 푍∼푄푡(휃,휙) [ log 푝(푋,푌 )1∶푡|푍1∶푡((푥, 푦)1∶푡|푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] − 피 푍∼푄푡(휃,휙) [ log 푝(푋,푌 )푡+1∶푇 |푍푡+1∶푇 ((푥, 푦)푡+1∶푇 |푍푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] The first term is an encoder loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The second term is a decoder loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The third term is a prediction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This pre- diction loss can also be viewed as a loss optimized against a regularization since the 퐷퐾퐿 term affects the inference of 푍푡 by the recognition model from which the next latent states are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' LP-VAE with actions The models we described up to this point represents the environment evolving around the observing agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, our agent also acts on this environment and influences the observations gathered to train our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, we need to modify it in order to integrate this subtlety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Let 퐴푡 be the action applied at time 푡 on perceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This action describes a mask on the information contained in 푌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This partial information is then transmitted to the observing agent, influencing 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It has no influence on the environment evolving around the agent, only on its perception of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that 푌푡 and 푍푡 are not affected by 퐴푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Moreover, we will now consider that the random variable 푋푡 is the ego-vehicle perception at time 푡, eventually augmented with information from 푌푡, in accordance with 퐴푡, and combined with the discounted memory of the previous partial obser- vations 푋1∶푡−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 9 provides the corresponding Bayesian network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We set the following constraints: 푍푖 ∼ \ue23a (0, 퐼푑) 푝푍푖+1|푍푖(⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푧(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 휎2 푧(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼푑) 푝푌푖|푍푖(⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푦(푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 훼푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼|푋푡|) 푝푋푖|푋푖−1,푌푡,푍푡,퐴푡(⋅|푥푡−1, 푦푡, 푧푡, 푎푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = \ue23a (휇푥(푥푡−1, 푦푡, 푧푡, 푎푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), 훼푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='퐼|푋푡|) where all parameters 휇⋅ and 휎⋅ are deep neural networks taking their parameters in 휃, and 훼⋅ ∈ [ 1 2휋 , +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our dataset 퐷 is composed of 푁 independent sequences of partial and complete observations with a randomly chosen bounding box 퐴푡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 퐷 = (푥1∶푇 , 푦1∶푇 , 푎2∶푇 )1∶푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fortu- nately, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (7) still holds in this new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Moreover, we know that the environment does not depend on the actions 퐴2∶푇 taken on its perception of it and that the actions only mask regions of 푌푡 while not altering the remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, since 푋푡 contains the information transmitted from 푌푡 in accordance with 퐴푡, the actions 퐴2∶푡 do not bring any infor- mation for the inference of the latent states 푍1∶푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Given the Bayesian network in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 9, the actions 퐴 without knowing Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 10 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown 푏1 푏2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푏푡−1 푏푡 푥1 푥2 푥푡−1 푥푡 푧1 푧푡−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푧2 푧푡 푧푇 푧푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푧푡+1 푍2 푍푡−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푍푡 푋푌1 푋푌2 푋푌푡−1 푋푌푡 푋푌푡+1 푋푌푇 −1 푋푌푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Figure 8: Illustration of the forward computations allowing for the evaluation of the LP-VAE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A diamond indicates a deterministically inferred variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A rectangle indicates the deterministic inference of distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A circle indicates the deterministic inference of distribution parameters and a sample from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The blue network is the belief network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The red network is the smoothing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The black network is the Markovian transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The brown network is the decoding network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푁 휃 푌1 푌2 푌푇 −1 푌푇 퐴2 퐴푇 −1 퐴푇 푋1 푋2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푋푇 −1 푋푇 푍1 푍푇 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푍2 푍푇 Figure 9: Bayesian network of our generative model of parameters in 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We have 푁 replications of this model, corresponding to the 푁 sequences of length 푇 in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The parameter set 휃 influences the inference of all variables in the model for the 푁 sequences we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 푋푡+1∶푇 do not bring any information for the inference of the latent states 푍푡+1∶푇 either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We have: 푝푍|푋1∶푡,퐴(⋅| 푥1∶푡, 푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) Thus, we keep the LP-VAE variational distributions 푄1 푡 (휙) = 푞푍푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), 푄2 푡 (휙) = 푞푍푡|푋1∶푡,푍푡+1(⋅|푥1∶푡, 푧푡+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙), 푄푡(휃, 휙) ≈ 푝푍|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, for our local predictability constraint (See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (5)), we consider 푝푍|푋,푌 ,퐴(⋅| 푥, 푦, 푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) instead of 푝푍|푋,푌 (⋅| 푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Notice that 푝푍|푋,푌 ,퐴(⋅| 푥, 푦, 푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푝푋,푌 ,푍|퐴(푥, 푦, ⋅ |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) 푝푋,푌 |퐴(푥, 푦 |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 푃(휃) 푝푋,푌 |퐴(푥, 푦 |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) We take as loss function \ue238LP-VAE(푥, 푦 |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) instead of \ue238LP-VAE(푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙), where \ue238LP-VAE(푥, 푦 |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃, 휙) = 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃))] (11) This loss maximizes a lower bound of 푝푋,푌 |퐴(푥, 푦 |푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Developing the KL divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (11) in accordance with our new model, we get: 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|퐵1∶푡 (푍1∶푡|푏1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + log 푝푍푡+1∶푇 |푍푡 (푍푡+1∶푇 |푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푍1∶푡 (푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푍푡+1∶푇 |푍푡 (푍푡+1∶푇 |푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푌 |푍 (푦 |푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 11 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown − log 푝푋|푌 ,푍,퐴 (푥 |푦, 푍, 푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|퐵1∶푡 (푍1∶푡|푏1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍1∶푡 (푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푌 |푍 (푦 |푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋|푌 ,푍,퐴 (푥 |푦, 푍, 푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] which leads to 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 피 푍∼푄푡(휃,휙) [ log 푞푍푖|퐵푖 (푍푡|푏푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍푖 (푍1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + 푡−1 ∑ 푘=1 log 푞푍푖|퐵푖,푍푖+1 (푍푘|푏푘, 푍푘+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − 푡∑ 푘=2 log 푝푍푖+1|푍푖 (푍푘|푍푘−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푇∑ 푘=1 log 푝푌푖|푍푖 (푦푘|푍푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − 푇∑ 푘=2 log 푝푋푖|푋푖−1,푌푖,푍푖,퐴푖 (푥푘|푥푘−1, 푦푘, 푍푘, 푎푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋1|푌1,푍1 (푥1|푦1, 푍1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ] (12) In practice however, we will neglect the term − log 푝푋1|푌1,푍1 (푥1|푦1, 푍1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' First, it avoids to optimize parameters that would only be used in the learning phase, while not corresponding to an important component (the complete observation 푦1 being already con- sidered and containing 푥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' But maybe more importantly, since 푋푡 keeps a memory of past observations in this for- mulation of the LP-VAE, 푥1 may also contain information on actions preceding 푎2∶푇 that should be given as well if 푥1 is actually not the start of an episode of interactions in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Not generating 푥1 allows us to start the inference of latent states at any point of the episode, independently from the previous actions and observations that produced 푥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that we can re-use different subsequences of the same training sequence in the learning phase, without having to make sure that 푥1 do not contain information related to past observations and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Implementation as neural networks 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Belief state computation The grids 퐺푡 introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1 are not directly taken as input of our LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Beforehand, we train a Con- volutional VAE (CVAE) to learn a compressed, essentialized representation of these observations in which spatial features have been extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This CVAE is itself separated into 4 independent parts in order to preserve the semantics of these features: a CVAE for the pedestrian channel, another for the car channel, another for static elements (road lines, road, other) and a last one for the ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The projection of 퐺푡 into the latent space of this Convolutional VAE is the 푋푡 taken by our LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, we feed 푋푡, 푋푡−1 and the ego-motion 푉푡 to a Multilayer perceptron (MLP) in order to extract features about the motion of road users around the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The output of this MLP serves as input to a Recurrent Neural Network (RNN) composed of Long Short- Term Memory (LSTM) cells to form and update a belief over the dynamics of other road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The concatenation of the hidden state of this RNN with 푋푡 and the driving controls 퐶푡 represents the belief state 퐵푡 at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 10 visually sums up this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Inference of Gaussian parameters In [20], they proposed to use what they called D maps1 to infer the Gaussian parameters of any of the distributions over the latent state 푧푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It is a part of a LSTM cell (new features multiplied by the input gate), as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 11, where the output is passed to two fully connected (FC) layers in parallel without activation function, one to determine 휇푧푡 and the other to determine log(휎푧푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Yet, in our sequential setting, this D map becomes a truly recurrent unit, chaining itself multiple times from 푡1 to 1 in the smoothing network and from 푡1 to 푡2 in the prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' As for any recurrent network, this poses the issue of vanishing gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fur- thermore, it lacks the semantics of a transition model: some components could disappear from the frame (forget gate) and some other could become visible or simply move from their initial state (input gate, followed by an addition to the initial components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These are exactly the transformations applied to the cell state of a LSTM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, using the cell state of a LSTM cell as latent state mean 휇푧푡 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 11, where ℎ = 푧푡+1 and input = 푏푡, solves both the vanishing gradient issue and the lack of model semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Giving ℎ as both hidden and cell states also has the effect of implementing peephole connections [29], giving the cell state some control over the input, forget and output gates (the three sigmoïd layers), which better captures sporadic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In addition, uncertainty should be encoded within the latent state to be self-sufficient for a transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This encourages the computation of the standard deviation 휎푧푡 from 휇푧푡 with some filtering gate (output gate), which is exactly what a LSTM cell does to output a quantity based on its cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Similarly, we use this LSTM cell in the prediction network for 푝푍푖+1|푍푖(⋅|푧푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), where ℎ = 푧푡 and input = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the belief network, we keep this D map as there is no propagation in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Decoding So far, we determined the networks outputting distribu- tion parameters describing the latent states 푍 used in the evaluation of \ue238LP-VAE, both for the generative model and the recognition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It remains to propose the decoding network that is part of the generative model and produces 푋 and 푌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Given the conditional distributions appearing in \ue238LP-VAE, we need a decoder inferring 푌푡 from 푍푡 and another one inferring 푋푡 from 푋푡−1, 푌푡, 퐴푡 and 푍푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1In [20], they used a 16-layer model where the information transits from layer to layer through the states of a LSTM, possibly in place of this D map, in their DeepMind Lab experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Note however that it is recurrent through layers, not time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is different from what is proposed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 12 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown CNN CNN CNN CNN [pedestrian, car, road line, road, other, Ω] 6-channels mass grid at 푡 [pedestrian] [car] [roal line, road, other] [Ω] 푥푡 푥푡−1 푥푡 푣푡 MLP ℎ푡−1 LSTM ℎ푡 푐푡 푏푡 ℎ푡 Belief state computation Figure 10: Illustration of the process of computing the observation 푋푡 and the belief state 퐵푡 from 퐺푡, 푋푡−1, 푉푡 and 퐶푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Four independent Convolutional VAEs are trained to learn a sufficient representation of pedestrian, car, {road lines, road, other} and ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These encodings form 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A Multilayer perceptron (MLP) tries to learn features about the motion of road users around the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The output of this MLP serves as input to a Recurrent Neural Network (RNN) composed of Long Short-Term Memory (LSTM) cells to form and update a belief over the dynamics of other road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The concatenation of the hidden state of this RNN with 푋푡 and the driving controls 퐶푡 represents the belief state 퐵푡 at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' LSTM cell D map ℎ input × + 휎 FC 휎 FC × tanh tanh × FC 휎 FC FC 휇푧푡 log(휎푧푡) Figure 11: Proposed replacement for D maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The FC rectan- gles indicate a single Fully Connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Circles indicate point-wise operations, where 휎 is the sigmoïd activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, since 푋푡 and 푌푡 are not given in the original space but in a learned compressed one, extracting features from 푌푡 according to the bounding box 퐴푡 is not directly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' One has to decode 푌푡, extract features according to 퐴푡, decode 푋푡 and then fuse it with the leaked features from 푌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the sake of efficiency, we will learn to directly extract these features that we denote by the random variable 푀푡 in the learned compressed space and to fuse them with 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, in parallel to \ue238LP-VAE, we minimize an extra loss term − 푇∑ 푘=2 log 푝푀푖|퐴푖,푌푖 (푚푘|푎푘, 푦푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) , 푧푡 푥푡−1 휎 × + tanh 휎 × tanh 휎 × 푦푡 푎푡 휎 × + tanh 휎 × 푚푡 휎 × + tanh 휎 × 푥푡 Decoder Figure 12: Illustration of our decoding architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The decoder block infers 푥푡 the partial observation, 푦푡 the spatially complete observation and 푚푡 the masked 푦푡 (as dictated by the bounding box 푎푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It takes as inputs a latent state 푧푡, a previous partial observation 푥푡−1 and a bounding box 푎푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A rectangle indicates a fully connected layer, while the symbol at its center indicates the activation function applied to its output (휎 for sigmoid, tanh for hyperbolic tangent and nothing for the identity function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each updating network is composed of a forget gate (first 휎) and a D map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' input features (tanh), an input gate (last 휎) and a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' where 푚푡 corresponds to 푦푡 masked in accordance with 푎푡 and compressed by the same CVAE as for 푦푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Note that our dataset becomes 퐷 = (푥1∶푇 , 푦1∶푇 , 푚2∶푇 , 푎2∶푇 )1∶푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 13 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown CNN CNN CNN CNN [pedestrian, car, road line, road, other, Ω] 6-channels mass grid at 푡 [pedestrian] [car] [roal line, road, other] [Ω] 푥푡 or 푦푡 or 푚푡 Grid decoder Figure 13: Illustration of the decoding of 푋푡 or 푌푡 or 푀푡 by the decoder of the CVAE that gave 푋푡 to get back into the observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The CNN blocks are Transposed CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We choose to infer 푌푡 from 푍푡 through a D map as intro- duced in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' All other inferences are done through an updating module that is inspired by the updating of a LSTM cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The masking of 푌푡 is orchestrated by 퐴푡, producing 푀푡 by filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, 푋푡−1 is updated in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The first update is assumed to change its reference frame and to determine which parts of 푌푡 are visible to the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This implicitly produces the 푋푡 corresponding to the null action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' the action that consists in doing nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We consider this transformation deterministic, given 푦푡 and 푧푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The second update transmits the excerpt 푀푡 from 푌푡 to this prior perception, producing the actual 푋푡 influenced by 퐴푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 12 depicts these networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 13 illustrates the decoding of 푋푡 by the decoder of the Convolutional VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Data acquisition & RL Environment To conduct our experiments, we chose to work with the open-source driving simulator CARLA [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our semantic grids 퐺푡 are computed online from a frontal 320 × 480 depth camera with FOV of 135◦ and its corresponding pixel- wise semantic classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These simulated sensors are attached to a simulated vehicle autonomously wandering in a city with other vehicles, bikes and pedestrians (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More precisely, 퐺푡 is obtained by counting the number of occurrences of each class in each possible configuration of 4 × 4 consecutive pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' All classes corresponding to static objects are merged into the class other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, in each cell of the resulting 80 × 120 × 5 grid, these numbers are divided by 16 and we add a channel representing ignorance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Ω) to store the quantity needed to make the sum on all channels equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also discount the resulting mass functions by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01 to simulate noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' all masses are multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='99 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01 is added to the mass on ignorance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Finally, thanks to the depth and information about the camera, we create a 3D point cloud of this frontal perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, to get the 2D grid 퐺푡, we ignore points higher than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 meters and we take the highest of the remaining ones (if more than one point at the same ground coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For this reason, it sometimes happens that the ground under a vehicle is perceived, but not its top, leading to road cells surrounded by car cells, as can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2 Left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' An important road elevation may also conflict with the threshold of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This view can be obtained by a LIDAR and a 3D semantic classifier [30] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our top-down semantic grids corresponding to complete observations 푦푡 in our model are obtained with a facing ground camera above the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, it contains itself some occlusions due to trees, poles, buildings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, it is rather a hint about the true 푦푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This grid can also be obtained by the fusion of multiple view points, from a fleet of autonomous vehicles or infrastructure sensors, which can be acquired in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A drone may be able to acquire this information as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In any case, this ground truth grid is in fact itself uncertain and so is computed as 퐺푡 with an ignorance channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We created a dataset composed of 1560 sequences of 50 timesteps (5 seconds) each, where each perception is 80×120×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' There are 30 runs in each of four cities available in CARLA, including small towns, big towns and fast lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each run is 35 seconds long and a sequence is recorded every 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 seconds, leading to 13 sequences per run, hence the size of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This dataset provides the grids corresponding to 푋푡 and 푌푡 in the action-independent model of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To provide the grids corresponding to 푋푡 as defined in the full model of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5, we created a second dataset from the first one by choosing random regions of 푌푡 to be given to 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also added a visual memory that keeps a buffer of grid cells, transforms their coordinates according to the given motion of the ego-vehicle, discounts their mass functions to account for information ageing and fuses them Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 14 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Binary classification per class Mass P C RL R O Ω score LP-VAE 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% STD-VAE 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='9% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% Table 1 Mass score and binary classification accurracy per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' P indicates the pedestrian channel, C the car channel, RL the road lines channel, R the road channel, O the other channel and Ω the complete out-of-sight channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It is clear that STD-VAE outperforms LP-VAE for simple grid completion, though the total mass score is not so different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' with the current perception grid, resulting in this 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In fact, the first dataset combined with our visual memory and our fusion procedure of Algorithm 1 for ̃퐺푡 and 퐺푀 푡 constitutes the environment in which our agent will learn a communication policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Models During training, we give between 8 and 10 timesteps of observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8 and 1 second) and it is asked to predict between 5 and 10 timesteps ahead, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 and 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We use the Mean Squared Error (MSE) loss function to compute the Gaussian negative log likelihoods of observing the grids corresponding to 푥푡 and 푚푡 given latent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, this is analog to taking 훼 = 1 2 and ignoring the constant term log (√ 2휋훼 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the negative log likelihoods on the grid corresponding to 푦푡, we binarize it by taking the class with maximum mass and use a cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To account for the fact that the instances of 푌푡 in our dataset are not perfect, we simply do a pointwise multiplication between this loss and the complement to 1 of its ignorance channel (last channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' That way, if 푦푡 does not have any information about a cell, no loss on 푦푡 is actually back-propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Furthermore, we weight this cross-entropy loss differently from one channel to another to account for class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We used the weight vector [100, 10, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, on average, there are far less cells containing pedestrians than cells containing the road or any other static class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Doing so, without weights, the network would consider pedestrian as noise and neglect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In the following, we compare STD-VAE and LP-VAE for complete grid inference and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Grid completion In this experiment, we use the decoder network de- scribed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 12 on the current latent state 푍푡 inferred from 퐵푡 to retrieve 푌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, we use the network described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 13 to transform 푌푡 into the complete mass grid 퐺푌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To compare STD-VAE and LP-VAE, we employed two metrics: binary classification accuracy per class and a mass score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our Mass score metric is computed as the mean of 퐺푌 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' ̂ 퐺푌 푡 over all cells in the grid, where 퐺푌 푡 is the true binary complete grid classification and ̂ 퐺푌 푡 is a mass grid inferred by some model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Since 퐺푌 푡 is binary, it acts as an indicator function for the correct class and the mass score represents the mean mass given to the right class by the model generating ̂ 퐺푌 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Results are showed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Prediction In this experiment, we compare prediction accuracy between LP-VAE and STD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For this, we study mass variations on the super-class {road, road line}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' the sum of the road and road line grid channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, this super- class represents the road layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Its absence in a cell indicates either road users or the other class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, its mass variations accounts for the dynamics of the whole scene, independently of classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In practice, for each model, we infer a prediction se- quence of 10 complete grids ̂푦1∶10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1 second in the future), based on 10 observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' the past second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' From it, we compute the corresponding sequence of 9 grid variations ̂푦′ 푡 = ̂푦푡+1 − ̂푦푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We execute the same process with the true complete grids, which produces grids 푦′ 1∶9 of values ranging in {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We test separately the accu- racy on positive and negative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the former, we do a pointwise multiplication between the true complete positive grids max(0, 푦′ 1∶9) and the inferred positive ones max(0, ̂푦′ 1∶9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the latter, we do a pointwise multiplication between the true complete negative grids max(0, −푦′ 1∶9) and the inferred negative ones max(0, − ̂푦′ 1∶9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We then sum all cells of each grid in the sequence, over 4992 sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 49 920 inferred grids and compare it to the separate sums of positive and negative true changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Results are displayed in the first two columns of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, note that this binary mask can be quite hard to match, as both the exact location of these changes and their amplitude must be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To alleviate this constraint, we repeat this test with blurring filters applied to each grid of 푦′ 1∶9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The resulting grids, noted ̃푦′ 1∶9, are then renormalized so that ∑ max(0, 푦′ 1∶9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' max(0, ̃푦′ 1∶9) = ∑ max(0, 푦′ 1∶9) and ∑ max(0, −푦′ 1∶9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' max(0, − ̃푦′ 1∶9) = ∑ max(0, −푦′ 1∶9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This allows for slight misplacements of cells in predicted grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We repeated this test twice with Gaussian filters, with ker- nels 5x5 and 11x11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' These experiments correspond to the last 4 columns of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our LP-VAE outperforms STD- VAE in every of these tests, no matter how hard the constraint on change location is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This means that the predicted changes of LP-VAE are not just better located, but also better shaped than the ones of STD-VAE, as expected by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 14 illustrates this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 15 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown True 푦′ No blur Gaussian blur 5x5 Gaussian blur 11x11 + + + LP-VAE ̂푦′ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='81% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='94% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='41% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='61% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='81% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='36% STD-VAE ̂푦′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='10% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='37% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='89% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='41% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='66% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='52% Table 2 Prediction accurracies between STD-VAE and LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' As expected, LP-VAE significantly outperforms STD-VAE on predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (a) (b) Figure 14: (a) Left column: partial grid 퐺푡 corresponding to 푋푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Right column: complete grid 퐺푌 푡 corresponding to 푌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Top row: true classification grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Bottom row: classification grids predicted by LP-VAE from 푋 alone, 4 time steps in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (b) Prediction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Black represents the absence of variation, white some mass change in the cells of the road and road line channels of the grid in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Left column corresponds to the true variations, blurred by a 11x11 Gaussian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The central column corresponds to the prediction dynamics of STD- VAE, multiplied by the ones of the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Same for the right column but for LP-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The first row represents positive changes, while the second row represents negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Policy learning Here, we finally compare different policies learned with PPO, with and without model to test the benefits of using belief states in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each policy is the best found among iterations of training with 3000 transitions amounting to 500 000 time steps in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We used a batch size of 60, with 10 epochs on each transition dataset, with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='0003 and an entropy coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also made the time horizon vary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' we made the hyperparameter 훾 vary from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7, in order to see if a medium/long term strategy performs better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The network learned with PPO has two parts: one for inferring the Value of a state, representing the mean of all potential future rewards, and one for inferring the best action from this same state, representing the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each of these networks is composed of two fully connected hidden layers of 128 and 64 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Different communication behaviors can be obtained by adjusting reward parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In particular, increasing 퐾 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 2 will make requests bigger, increasing 푤 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 1 will make requests more focused on completely unknown areas, increasing 휂 will make requests more focused on pedestrians and cars, less rewarding in general and so less frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We chose the following values: 휂 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3, 퐾 = 36 and 푤 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also added a penalty of -15 for no cooperation at all (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' choice of a bounding box with no pixel in it, which means no transmission cost either) to force the agent to play the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Moreover, approximating the top-down dimensions of cars and pedestrians, we took the following reward densities per squared meter: 푟푚 obj = [540∕(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7 ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6), 540∕(3 ∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8), 20, 20, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, we converted them into rewards per squared cell by multiplying them by our grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' More precisely, we set our cameras in CARLA so that the height corresponds to 40 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, our reward densities per squared cell are 푟obj = ( 40 80)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='푟푚 obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Our final rewards are obtained by normalizing 푟obj to [0, 1] by dividing it by its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For the spatial filter, we used the parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 훼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5, 훽퐹 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8, 훽퐿 = 1 and 휁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In order to evaluate and compare the performance of different policy learning schemes, we take as metrics the mean request size and the mean informational gain over all time steps of a test set with same size and characteristics as the training set described Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We applied these metrics to 3 class groups: pedestrians (P), cars (C) and {road lines, road} (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In these conditions, we compared 3 schemes: PPO on top of the LP-VAE belief state 퐵푡, PPO on top of the STD-VAE belief state 퐵푡 and PPO on top of 푋푡 alone (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' only the features extracted from the current mass grid 퐺푡 by a Convolutional VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Each of them has been trained with 훾 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' only immediate rewards matter), 훾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='35 and 훾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7, to see if we could benefit from medium/long term strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We also compare these policies with a simple random policy that has a 50% chance of making a request and chooses uniformly random size and Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 16 of 19 0 0 0 50 50 50 0 50 100 0 50 100 0 50 100 0 0 0 50 50 50 0 50 100 0 50 100 0 50 100Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown Information gain Request P C R size Random 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 22% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='9% 13% LP-VAE 퐵푡 22% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 6% 훾 = 0 STD-VAE 퐵푡 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='9% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4% 5% 푋푡 alone 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 6% LP-VAE 퐵푡 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% 6% 훾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='35 STD-VAE 퐵푡 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% 5% 푋푡 alone 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='4% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% 5% LP-VAE 퐵푡 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5% 5% 훾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='7 STD-VAE 퐵푡 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2% 4% 푋푡 alone 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='3% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='8% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='6% 4% Table 3 Learned communication policy performances relatively to a broadcasting policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' The information gain is a mean percentage representing the mass actually gained after request, over the total mass that can be gained, at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' position of bounding box when it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Table 3 presents our results, in percentage relatively to the maximal information gain and request size possible inherent to a broadcasting policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' All of our learned policies only ask for about 5% of the space around the ego-vehicle, while receiving about 25% of the relevant information the agent lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Requiring about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='5 times more information from the vehicular network for about the same relevant information gain or lower, the random policy is vastly less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It only outperforms the others for pedestrians, which is consistent with the highly random behavior of pedestrians in CARLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, PPO + 푋푡 alone and 훾 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' greedy policy) is the policy that performs best overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Surprisingly enough, taking into account future rewards actually harms performance in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A lower discounting factor in the memory module (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' observations that are kept longer in memory) would proba- bly make policies perform best with 훾 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Furthermore, note that LP-VAE always performs better than the other learned policies when 훾 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This is consistent with the fact that LP-VAE has better prediction capabilities and thus provides useful information in its belief state for predicting future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Conclusions In this paper, we tried to elaborate an efficient peer-to- peer communication policy for collaborative perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For this, we made agents learn what could be hidden in their blind spots through a generative sequence model that we proposed, named Locally Predictable VAE (LP-VAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We compared its performance with another generative sequence model for RL applications called TD-VAE that we slightly adapted to our problem by making it both jumpy and se- quential, referring to it as STD-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' We demonstrated that LP-VAE produces better predictions than STD-VAE, which translated into better performance for policies learned on top of its belief state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' However, we discovered in the end that our best communication policy was a greedy one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' one that does not need prediction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Combined with the fact that we augmented each observation with the discounted memories of past observations, it followed that only a state-less Convolutional VAE was needed for this greedy policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Overall, our best learned policies only require about 5% of the space around the ego-vehicle, while gaining about 25% of the relevant information the agent lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, we proved that learning to value the unknown is much more efficient than employing a broadcasting policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' It is also more efficient than blindly asking for random areas around the ego-vehicle since it requires about 13% of the total information, while gaining less than 25% of the relevant information the agent lacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' In addition, we defined interpretable hyperparameters shaping the reward function corresponding to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This makes it possible to obtain various communication policies, with different trade- offs between request size and information gain, as well as different class valuations, spatial priorities and valuation of ignorance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' more or less emphasis on total ignorance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' For future works, it would be interesting to compare LP-VAE and STD-VAE in RL tasks where future rewards are more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Also, we would like to test our communication policies in a truly multi-agent context, where the agent would need to take into account the availability of nearby communicating vehicles, and with real sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' LP-VAE loss A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Minimization of 퐷퐾퐿 (푄푡(휃, 휙) || 푃푡(휃)) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Indeed, we have, for some instant 푡: 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) + log 푝푍푡+1∶푇 |푍푡(⋅| ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푍1∶푡(푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푍푡+1∶푇 |푍푡(푍푡+1∶푇 |푍푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋,푌 |푍(푥, 푦|푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍1∶푡(푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋|푍(푥|푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푌 |푋,푍(푦|푥, 푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 17 of 19 Decentralized cooperative perception for autonomous vehicles: Learning to value the unknown = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푍1∶푡(푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋1∶푡|푍1∶푡(푥1∶푡|푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푌 |푋,푍(푦|푥, 푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] = 피 푍∼푄푡(휃,휙) [ log 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) − log 푝푋1∶푡,푍1∶푡(푋1∶푡, 푍1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) − log 푝푌 |푋,푍(푦|푥, 푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃)] Suppose that both 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) and 푝푌 |푋,푍(푦|푥, 푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) range in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' This can be easily verified if they can be written as a factorization of probability density functions that each ranges in [0, 1], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Gaus- sian distributions with diagonal covariance matrices where each term of the diagonal is in [ 1 2휋 , +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Then, both − log 푝푋푡+1∶푇 |푍푡+1∶푇 (푥푡+1∶푇 |푍푡+1∶푇 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) and − log 푝푌 |푋,푍(푦|푥, 푍;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) are nonnegative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) ≥ 퐷퐾퐿 ( 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Thus, by minimizing 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)), we minimize an upper bound of 퐷퐾퐿 ( 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Furthermore, since we have 퐷퐾퐿 ( 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) = 퐷퐾퐿 ( 푞푍1∶푡|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푍1∶푡|푋1∶푡(⋅| 푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) − log 푝푋1∶푡(푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) = 퐷퐾퐿 (푄푡(휃, 휙) || 푃푡(휃)) − log 푝푋1∶푡(푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃), we know that by optimizing 휙 to minimize 퐷퐾퐿 ( 푞푍1∶푡|푋1∶푡(⋅|푥1∶푡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙) || 푝푋1∶푡,푍1∶푡(푥1∶푡, ⋅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) ) , we mini- mize 퐷퐾퐿 (푄푡(휃, 휙) || 푃푡(휃)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' To sum up, minimizing 퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃)) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휙 minimizes an upper bound of 퐷퐾퐿 (푄푡(휃, 휙) || 푃푡(휃)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Maximization of 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Replacing 푃푡(휃) by 푄푡(휃, 휙) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' (6), we get: 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃))] = − log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) + 피 푡∼ \ue241[푡min, 푇 −1] [ 퐷퐾퐿 ( 푄푡(휃, 휙) |||| |||| 푃 (휃) 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) )] ≥ − log 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃) Therefore, by optimizing 휙 to minimize 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃))], we minimize 피 푡∼ \ue241[푡min, 푇 −1] [ 퐷퐾퐿 ( 푄푡(휃, 휙) |||| |||| 푃(휃) 푝푋,푌 (푥,푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content='휃) )] , and by opti- mizing 휃 to minimize 피 푡∼ \ue241[푡min, 푇 −1] [퐷퐾퐿 (푄푡(휃, 휙) || 푃(휃))], we maximize a lower bound of 푝푋,푌 (푥, 푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' 휃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' ■ References [1] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Ma, S.' metadata={'source': 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Validation, Visualization, Writing - Original Draft, Writing Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Franck Davoine: Supervision, Writing Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Véronique Cherfaoui: Supervision, Writing - Review & Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' Chaveroche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 19 of 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfTfx4/content/2301.01250v1.pdf'} diff --git a/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf 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Department of Electrical and Computer Engineering, Georgia Tech. +2. Department of Computer Science, Purdue University +hwang3028@gatech.edu, panli@gatech.edu, panli@purdue.edu +ABSTRACT +A general framework of unsupervised learning for combinatorial optimization +(CO) is to train a neural network (NN) whose output gives a problem solution +by directly optimizing the CO objective. Albeit with some advantages over tra- +ditional solvers, the current framework optimizes an averaged performance over +the distribution of historical problem instances, which misaligns with the actual +goal of CO that looks for a good solution to every future encountered instance. +With this observation, we propose a new objective of unsupervised learning for +CO where the goal of learning is to search for good initialization for future prob- +lem instances rather than give direct solutions. We propose a meta-learning-based +training pipeline for this new objective. Our method achieves go empirical perfor- +mance. We observe that even just the initial solution given by our model before +fine-tuning can significantly outperform the baselines under various evaluation +settings including evaluation across multiple datasets, and the case with big shifts +in the problem scale. The reason we conjecture is that meta-learning-based train- +ing lets the model loosely tied to each local optima for a training instance while +being more adaptive to the changes of optimization landscapes across instances. 1 +1 +INTRODUCTION +Combinatorial optimization (CO), aiming to find out the optimal solution from discrete search space, +has pivotal position in scientific and engineering fields (Papadimitriou & Steiglitz, 1998; Crama, +1997). Most CO problems are NP-complete or NP-hard. Conventional heuristics or approximation +requires insightful comprehension of the particular problem. Starting from the seminal work from +Hopfield & Tank (1985), researchers apply neural networks (NNs) (Smith, 1999; Vinyals et al., +2015) to solve CO problems. The motivation is that NNs may learning heuristics through solving +historical problems, which could be useful to solve similar problems in the future. +Many NN-based methods (Selsam et al., 2018; Joshi et al., 2019; Hudson et al., 2021; Gasse et al., +2019; Khalil et al., 2016) require optimal solutions to the CO problem as supervision in training. +However, optimal solutions are hard to get in practice and the obtained model often does not gener- +alize well (Yehuda et al., 2020). Methods based on reinforcement learning (RL) (Mazyavkina et al., +2021; Bello et al., 2016; Khalil et al., 2017; Yolcu & P´oczos, 2019; Chen & Tian, 2019; Yao et al., +2019; Kwon et al., 2020; 2021; Delarue et al., 2020; Nandwani et al., 2021) do not need labels while +they often suffer from notoriously unstable training. Recently, unsupervised learning methods have +attracted much attention (Toenshoff et al., 2021; Amizadeh et al., 2018; Yao et al., 2019; Karalias & +Loukas, 2020; Wang et al., 2022). A common strategy of these methods is to design an NN whose +output gives a solution to the CO problem and then train the NN via gradient descent by directly +optimizing the CO objectives over a set of training instances. This strategy is superior in its faster +training, good generalization, and strong capability of dealing with large-scale problems. +Despite the prominent progress, current unsupervised learning methods always optimize NNs to- +wards an averaged good performance over training instances. This means even if a testing instance +comes from the same distribution of the training instances, the solution to this single instance may +1Our code is available at: https://github.com/Graph-COM/Meta_CO +1 +arXiv:2301.03116v1 [cs.LG] 8 Jan 2023 + +Preprint +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^3 nodes +0.775 +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +0.950 +Approximation rate +Degree-based Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^4 nodes +0.775 +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +0.950 +Approximation rate +Degree-based Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^5 nodes +0.775 +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +0.950 +Approximation rate +Degree-based Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +Figure 1: Approximation Rates of different methods in the MIS problem. Meta-EGN and EGN (Kar- +alias & Loukas, 2020) are trained on RRGs with 1000 nodes and with node degree randomly sam- +pled from 3, 7, 10, 20. Meta-EGN and EGN are evaluated over larger RRGs with 103 ∼ 105 nodes. +More details about the setting are in Secs. 5.1 and 5.4. Meta-EGN outperforms DGA (Angelini & +Ricci-Tersenghi, 2019) by about 0.3% − 0.5% in approximation rates on average. +not have good quality, let alone the case when the testing instance is out-of-distribution (OOD). This +induces a concern when we apply NNs in practice because practical problems often expect to have +a good solution to every encountered instance. For example, allocating surveillance cameras is cru- +cial for each-time exhibition in every art gallery. Solvers when applied to this problem (O’rourke, +1987; Yabuta & Kitazawa, 2008) should output a good solution every time. Traditional CO solvers +are designed towards this goal. However, it is time-consuming and unable to learn heuristics from +historical instances. So, can we leverage the benefit of learning from history while with the goal of +achieving an instance-wise good solution instead of an averaged good solution? +This motivates us to study a new formulation of unsupervised learning for CO. We regard the ob- +jective of learning from history as to search a good initialization for each future instance rather than +give a direct solution. Since in practice, future instances are unavailable during the training stage, +we propose to view each training instance as a pseudo-new instance for the rest training instances. +Then, our learning objective is to learn a good initialization of this model, such that further opti- +mization of the initialization could achieve good solutions on each of these pseudo-new instances. +We observe meta learning is suitable to implement this idea, and propose to adopt MAML (Finn +et al., 2017) in our training pipeline as a proof of concept. Note that the step of optimization on each +pseudo-new instance shares a similar spirit with fine-tuning a model over each down-streaming task +as traditional meta learning does. However, each task in our case corresponds to optimization over +each training instance. +We name our method Meta-EGN by extending the previous framework EGN (Karalias & Loukas, +2020) via meta learning. Our key observation is that with this new objective, even the initial solution +given by Meta-EGN (before fine-tuning on a test instance) is substantially better than the solution +given by EGN and other methods that optimize the averaged performance over training instances. +Our conjectured reason is that the new objective, by taking into account fine-tuning the model over +new instances, trains the model to avoid being trapped into a local minima induced by each training +instance while being more adaptive to the changes of optimization landscapes across instances. +We demonstrate the benefits of Meta-EGN via experiments within three benchmark CO problems +(max clique, vertex cover and max independent set) on multiple synthetic graphs and three real- +world graph datasets, with the number of nodes ranging from 100 to 5000. Meta-EGN significantly +outperforms state-of-the-art learning-based baselines (Karalias & Loukas, 2020; Toenshoff et al., +2021), greedy algorithms, and the commercial CO solver Gurobi9.5 (Gurobi Optimization, 2022) in +most cases. Meta-EGN also shows super OOD generalization performance when the training and +test datasets are different or have graphs of entirely different sizes. +Moreover, recently, Angelini & Ricci-Tersenghi (2022) have shown that the learning-based method +in (Schuetz et al., 2022) could not achieve comparable results with the degree-based greedy algo- +rithm (DGA) (Angelini & Ricci-Tersenghi, 2019) in the max independent set (MIS) problem on +large-scaled random-regular graphs (RRGs), which raises attentions from machine learning com- +munity. We observe the issues come from two aspects: (1) graph neural networks (GNNs) used +to encode the regular graph suffer from the node ambiguity issue due to their limited expressive +power (Xu et al., 2019); (2) the model in (Schuetz et al., 2022) did not learn from history but was +directly optimized over each testing case, which tends to be trapped into a local optimum. By ad- +dressing these two issues, Meta-EGN can consistently outperform DGA while maintaining the same +time complexity to generate solutions. Fig. 1 show the results. +2 + +Preprint +2 +RELATED WORK +In the following, we review two groups of works: unsupervised learning for CO and meta learning. +Previous works on unsupervised learning for CO have studied max-cut (Yao et al., 2019) and TSP +problems (Hudson et al., 2021), while these works depend on carefully selected problem-specific +objectives. Some works have investigated satisfaction problems (Amizadeh et al., 2018; Toenshoff +et al., 2019). Applying these approaches to general CO problems requires problem reductions. +The works most relevant to us are (Karalias & Loukas, 2020), (Wang et al., 2022) and (Schuetz +et al., 2022). Karalias & Loukas (2020) propose an unsupervised learning framework EGN for +general CO problems based on the Erd˝os’s probabilistic method, which bonds the quality of the final +solutions with probability. Wang et al. (2022) generalize EGN and prove that if the CO objective +can be relaxed into an entry-wise concave form, a solution of good quality can be deterministically +achieved. This further inspires the design of proxy objectives for the CO problems that may not have +closed-form objectives, such as those in circuit design. Schuetz et al. (2022) have recently extended +EGN to large-scale max independent set problems on random-regular graphs. +Meta learning is proposed to learn hyper-parameters or initialization from historical tasks and +achieve fast adaption to new tasks. +Finn et al. (2017) propose model-agnostic meta learning +(MAML), which aimed to obtain good parameter initialization and accommodated to few-shot learn- +ing tasks with limited steps of fine-tuning. Nichol et al. (2018) accelerate MAML by adopting first- +order approximation on the gradient estimation. Rajeswaran et al. (2019) introduce implicit-MAML +that adopts an objective with fine-tuning till the stationary points on new tasks. Implicit-MAML does +not fit our case because we try to avoid long-time fine-tuning. Hsu et al. (2018) study unsupervised +learning under the meta learning framework and focused exclusively on vision tasks. To the best of +our knowledge, our work is the first one to apply meta learning to unsupervised learning for CO. +3 +PRELIMINARIES: NOTATIONS AND PROBLEM FORMULATION +Combinatorial Optimization on Graphs. +We follow the settings considered in (Karalias & +Loukas, 2020; Wang et al., 2022; Schuetz et al., 2022) and study CO problems on graphs whose +solutions can be represented as a subset of nodes of the input graph instance, although our method +could be applied to a broader range of problems. Suppose G is the universe of graph instances. Let +G(V, E) ∈ G denote a graph instance where V = {1, 2, ..., n} is the node set and E is the edge +set. Let X = (Xi)1≤i≤n ∈ {0, 1}n denote the discrete optimization variables defined on V , where +Xi = 1 denotes that node i is selected in the output node subset. A CO problem on G consists of a +cost function f(·; G) : {0, 1}n → R≥0 and a feasible set Ω ⊆ {0, 1}n that stands for the finite set +of all feasible X’s, and asks to solve +min +X∈{0,1}n f(X; G) +s.t. +X ∈ Ω +(1) +Unsupervised Learning for CO. The Erd¨os-Goes-Neural (EGN) framework of unsupervised learn- +ing for CO proposed in (Karalias & Loukas, 2020) is reviewed as follows. Here, we use the notation +system in a follow-up work (Wang et al., 2022) as it is more clear. Learning for CO problem is to +learn an algorithm Aθ(·) : G → {0, 1}n typically parameterized by an NN where θ denotes the +parameters of the NN such that given a graph instance G, X = Aθ(G) gives a solution of Eq. 1. In +practice, directly optimizing the parameters θ is hard in general. +Therefore, we may consider a relaxed cost function fr(·; G) : [0, 1]n → R≥0 where fr(X; G) = +f(X; G) on any discrete points X ∈ {0, 1}n and a relaxed constraint gr(·; G) : [0, 1]n → R≥0 +where {X ∈ {0, 1}n : gr(X; G) = 0} and {X ∈ {0, 1}n : gr(X; G) ≥ 1} defines the feasible +set Ω and the infeasible set Ωc respectively. Also, suppose the NN in Aθ can give soft solutions +¯X ∈ [0, 1]n. Then, we may train θ by minimizing a label-independent loss function: +min +θ +l(θ; G) ≜ fr( ¯X; G) + βgr( ¯X; G), +¯X = Aθ(G), for some β > 0. +(2) +The significant observation made by (Wang et al., 2022), which generalizes the argument in (Karalias +& Loukas, 2020), is a type of performance guarantee on the condition that fr and gr are entry-wise +concave, which is satisfied in all the cases studied in this work: If the loss that achieves l(θ; G) < β +for some β > maxX∈{0,1}n f(X; G), then the discrete solution X obtained by rounding the soft +solution ¯X = Aθ(G) according to Def. 1 is feasible X ∈ Ω and of good quality f(X; G) ≤ l(θ; G). +3 + +Preprint +Definition 1 (Rounding). For a soft solution ¯X ∈ [0, 1]n and an arbitrary order of the en- +tries (w.l.o.g 1,2,...,n), fix all the other entries unchanged and round ¯Xi into 0 or 1 as Xi = +arg minj=0,1 fr(X1, ..., Xi−1, j, ¯Xi+1, ..., ¯Xn) + βgr(X1, ..., Xi−1, j, ¯Xi+1, ..., ¯Xn), replace ¯Xi +with Xi and repeat this operation until all the entries are discrete. +4 +META LEARNING FOR ERD ¨OS GOES NEURAL (META-EGN) +The above performance guarantee lays the theoretical foundation for EGN. However, the following +practical issue motivates us to incorporate meta learning into EGN. +4.1 +MOTIVATION: WHAT NEEDED IS LEARNING FOR INSTANCE-WISE GOOD SOLUTIONS +It is often time consuming to perform online optimization of l(θ; G) for each encountered instance +G. This also mismatchs the goal of learning, i.e., learning heuristics from history/data. Therefore, a +pipeline commonly adopted is as follows. Suppose there is a set of training instances Gi, 1 ≤ i ≤ m, +IID sampled from a distribution PG. We optimize θ by following +min +θ +m +� +i=1 +l(θ; Gi), +(3) +which is similar to empirical risk minimization (ERM) in standard supervised learning. When a test +instance G appears, we apply the learned Aθ to get a soft solution and round it to the final solution. +This pipeline cannot guarantee the quality for this instance G. Even if the training instances Gi, +1 ≤ i ≤ m are in a large quantity (so in-distribution generalization is not a problem), and even if the +test instance G also follows PG, we may not guarantee a low l(θ; G) for one particular G because +ERM only guarantees a low averaged performance EG∼PG[l(θ; G)]. This issue may also voilate +the condition to have performance guarantee as reviewed in Sec. 3, as it is instance-wise. Here, +we highlight that in practice even the minimal averaged loss minθ EG∼PG[l(θ; G)] is often strictly +greater than averaged instance-wise minimal loss EG∼PG[minθ l(θ; G)], because practical NNs are +not expressive enough to remember the optimal solution to every instance. +Unfortunately, many practical CO problems actually expect instance-wise good solutions. This +is because every instance in practice is crucial. A terrible solution for one instance may raise a +security issue (e.g., the surveillance-camera allocation problem) or cause huge economic losses +(e.g., the routing problem in a transportation system). With this observation, our work is to address +the problem by studying unsupervised learning for instance-wise good solutions to CO problems. +4.2 +TRAINING TOWARDS INSTANCE-WISE OPTIMALITY VIA META LEARNING +Our idea to address the problem is to regard the goal of learning from history as to search good +initialization for future instances rather than give direct solutions. Such good initialization can +be quickly fine-tuned by further optimizing the model for each instance, which ultimately gives +instance-wise good solutions. However, in practice, we do not have access to any future/test in- +stances. So, can we just use historical/training instances to implement the above idea? Our strategy +is to view each training instance Gi as a pseudo-test instance to test and optimize the quality of +initialization given by the model. Specifically, this strategy gives us an objective +min +θ +m +� +i=1 +˜li(θ), +where ˜li(θ) = min +θi l(θi; Gi) with θi = θ as initialization. +(4) +Eq. 4 has some abuse of notations. The minimum ˜li(θ) depending on the initialization θ is because +of the non-convex nature of minθi l(θi; Gi), where the initialization θi = θ matters significantly. +We further simplify Eq. 4 with some practical consideration. In fact, we may not allow further +optimizing θ with so many gradient-descent steps for each instance, especially during the online +test stage. As a proof of concept, we consider the case with only one-step gradient descent, which +already gives good empirical results. Specifically, our training objective follows +Our Objective: min +θ +m +� +i=1 +li(θ), +where li(θ) = l(θi; Gi) with θi = θ − α∇θl(θ; Gi). +(5) +4 + +Preprint +Algorithm 1 Train Meta-EGN and Test Meta-EGN with/without Fine-tuning +Require: Training instances Ξ = {G1, G2, ..., Gm}; Hyperparameters: α, γ. +1: Randomly initialize θ(0) +2: for each randomly sampled mini-batch Bj ⊂ Ξ, j = 0, 1, ..., K − 1 do +▷ Training starts +3: +For each Gi ∈ Bj, compute the adapted parameter: θ(j) +i += θ(j) − α∇θ(j)l(θ(j); Gi) +4: +Update: θ(j+1) ← θ(j) − γ∇θ(j) � +Gi∈Bj l(θ(j) +i ; Gi) +5: end for +6: return θ ← θ(K) +▷ Training ends +7: For a given testing instance G′: +▷ Testing starts +8: if fine-tuning is allowed then +9: +Fine-tune the parameters: θG′ ← θ − α∇θl(θ; G′) +10: +Use Def. 1 to round the relaxed solution given by AθG′ (G′) +▷ With fine-tuning +11: else +12: +Use Def. 1 to round the relaxed solution given by Aθ(G′) +▷ Without fine-tuning +13: end if +▷ Testing ends +0 +1000 +2000 +3000 +4000 +Epochs +300 +250 +200 +150 +100 +50 +Loss value +Meta-EGN-Pre +Meta-EGN-Post +EGN +(a) Training Loss +0 +1000 +2000 +3000 +4000 +Epochs +300 +250 +200 +150 +100 +50 +Loss value +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +(b) Validation Loss +0 +1000 +2000 +3000 +4000 +Epochs +0.760 +0.780 +0.800 +0.820 +0.840 +0.860 +0.880 +0.900 +Approximation Rate +EGN +EGN Fine-tune +Meta-EGN +Meta-EGN Fine-tune +(c) Validation Approximation Rate +Figure 2: Training/validating dynamics of Meta-EGN and EGN Karalias & Loukas (2020) for the +MIS problem. Detailed experiment settings follow Sec. 5.1. +Here, θ is to give a good initialization Aθ(Gi) over each instance Gi while θi is with one-step +fine-tune to achieve a Gi-specified good solution Aθi(Gi). +Optimization in Eq. 5 can be implemented via the meta learning pipeline MAML (Finn et al., 2017). +We name the obtained model Meta-EGN and summarize its training and testing in Alg. 1. In step 3, +Meta-EGN performs the one-step gradient descent on each training instance. Note that we consider +two testing cases with or without fine-tuning, because the latter saves much inference time. A simple +extension of the Theorem 1 in (Wang et al., 2022) gives a performance guarantee for Meta-EGN in +Theorem 1 as follows. Here, for a test instance G, we even allow l(θ; G) to violate the original +condition l(θ; G) < β in (Wang et al., 2022) to some extend. After one-step fine-tuning in step 9, +the performance guarantee is still achievable. The detailed proof is in Appendix. A. +Theorem 1 (Performance Guarantee). Suppose the relaxations fr and gr are entry-wise concave +as required in (Wang et al., 2022). Let θ denote the learned parameter after training. Given a test +instance G, suppose locally l(·; G) is L-smooth at θ, i.e., ∥∇θ′l(θ′; G) − ∇θl(θ; G)∥ ≤ L∥θ′ − θ∥ +for all θ′ that satisfies ∥θ′ − θ∥ ≤ ϵ. Then, if l(θ; G) < β + △ (even if l(θ; G) ≥ β), for any +α ∈ (0, 2/L) Meta-GNN with one-step finetuning outputs a feasible solution X of good quality +f(X; G) ≤ l(θ; G) − △. +Here, △ = ∥∇θl(θ; G)∥ϵ + +1 +2Lα2−4αϵ2 if ϵ < α∥∇θl(θ; G)∥ or +△ = (α − Lα2 +2 )∥∇θl(θ; G)∥2 o.w.. +To better understand Meta-EGN, we show its training/testing dynamics in Fig. 2. As we expected, +the training loss of EGN is somewhere in-between the losses of Meta-EGN before and after the +fine-tune step. Training EGN is stabler and converges faster than training Meta-EGN. However, +what is unexpected is that in validation, Meta-EGN has a much lower loss and achieves much better +performance than EGN even before fine-tuning. +This implies that Meta-EGN holds better generalization than EGN. We conjecture the reasons are +as follows. First, the optimization landscape for CO problems is extremely non-convex (Mezard & +Montanari, 2009) due to the intersected feasible-infeasible regions and the high penalty coefficient +β. EGN that has low losses for training instances may give a high loss even when the optimization +5 + +Preprint +Table 1: Comparison between different unsupervised frameworks. G denotes the test instance and +Gi, 1 ≤ i ≤ m are training instances. The standard EGN pipeline does not adopt any fine-tuning. +EGN +(Karalias & Loukas, 2020) +P-I GNN +(Schuetz et al., 2022) +Meta-EGN +(Ours) +Classical Solver +Gurobi Optimization (2022) +Obj. to optimize the NN +�m +i=1 l(θ; Gi) +l(θ; G) +�m +i=1 l(θ − ∇θl(θ; Gi); Gi) +f(X; G) s.t. X ∈ Ω +Training or not +Yes +No +Yes +No +Fine-tune timing +No +Long +Short/No +Long +Generalization +Good +- +Better +- +Table 2: The discrete objectives (Eq. 1) and their relaxations +(Eq. 2) for the three problems to be studied. +MC +Discrete Obj. +maxX +� +1≤i≤n Xi +s.t. +(i, j) ∈ E if Xi, Xj = 1 +Relaxation +lMC(θ; G) ≜ −(β + 1) � +(i,j)∈E ¯Xi ¯Xj + β +2 +� +i̸=j ¯Xi ¯Xj +MVC +Discrete Obj. +minX +� +1≤i≤n Xi +s.t. +Xi + Xj ≥ 1 if (i, j) ∈ E +Relaxation +lMVC(θ; G) ≜ � +1≤i≤n ¯Xi + β � +(i,j)∈E(1 − ¯Xi)(1 − ¯Xj) +MIS +Discrete Obj. +maxX +� +1≤i≤n Xi +s.t. +XiXj = 0 if (i, j) ∈ E +Relaxation +lMIS(θ; G) ≜ − � +1≤i≤n ¯Xi + β � +(i,j)∈E ¯ +Xi ¯ +Xj +Figure 3: Performance v.s. hyper- +parameter ρ of the RB model +0.05 +0.15 +0.25 +0.35 +0.45 +0.55 +0.65 +0.75 +0.85 +0.95 +The hyper-parameter p in RB model +1.006 +1.008 +1.010 +1.012 +1.014 +1.016 +1.018 +1.020 +1.022 +Approximation Rate +EGN on RB500 +Meta-EGN on RB500 +Gurobi9.5 on RB500 +landscape is just slightly shifted (from training to a test instance). However, the parameters of Meta- +EGN are loosely tied to a local minimum for each training instance. Instead, those parameters, as +being aware of follow-up instance-wise fine-tuning steps, are likely to fall into some location close +to a local minimum for each instance while being not trapped in anyone of them, which makes the +model robust to landscape shifts across instances. Second, a CO problem could vary a lot across +graph instances even for those generated from the same distribution, especially when the instances +are large. So, it is reasonable to view the problem over each instance as a separate but relevant task. +Meta learning has shown good generalization when data distributions shift across tasks, which has +empirical evidence in CV and NLP applications (Jeong & Kim, 2020; Conklin et al., 2021). +As a summary, we provide a comparison between different unsupervised frameworks to solve CO +problems in Table 1. Note that PI-GNN (Schuetz et al., 2022) is directly fine-tuned on each test +instance without training so the fine-tuning time is long. Also, although PI-GNN also pursues +instance-wise good solutions, its performance could be bad because it does not learn from train- +ing instances. The instance-wise solutions could be just bad local minima. +5 +EXPERIMENTS +We study three CO problems, namely max clique (MC) to find the largest set of nodes where each +pair of nodes are connected, minimum vertex covering (MVC) to find the smallest set of nodes that +every edge is connected to at least one nodes in the set, and max independent set (MIS) to find the +largest set where any two vertices in the set are not adjacent. Their objectives (Eq. 1) and relaxations +(Eq. 2) are listed in Table 2. For the detailed derivation, see Appendix C. +5.1 +SETTINGS +Datasets: We conduct experiments on the MC, MVC problems over three real datasets Twit- +ter (Leskovec & Krevl, 2014), COLLAB and IMDB (Yanardag & Vishwanathan, 2015) and two +synthetic datasets with 200 and 500 nodes generated by the RB model (Xu, 2007). We name them +RB200 and RB500, respectively. We make RB200 and RB500 extremely hard by setting a small +hyper-parameter ρ = 0.25 of the RB model (Xu, 2007). The difficulty-ρ relationship on the MVC +problem with 500 vertices is shown in Fig. 3, where the models are pre-trained on the RB graphs with +uniformly sampled ρ ∈ [0.3, 1.0] and tested on different RB graphs generated with single ρ’s. We +keep all the other hyper-parameters the same. As ρ increases, Meta-EGN and Gurobi9.5 all tend to +achieve better performance. Meta-EGN could outperform Gurobi9.5 in hard instances ρ ∈ (0, 0.55] +while remains a gap on the easy ones. To verify performances for data-scale generalization, we also +generate large-graph datasets RB1000, RB2000 and RB5000 with ρ = 0.25. As for the MIS prob- +lem, random-regular graphs (RRGs) are often used as benchmarks because they are challenging. +Our experiments also use RRGs by following the settings of (Schuetz et al., 2022) with the node +6 + +Preprint +Table 3: ApR (time: second/graph) on the MC problem. ApR is the larger the better. ‘report’ denotes +the reported performance in Karalias & Loukas (2020), ‘re-impl’ denotes re-implementation, ‘f-t’ +stands for fine-tune. Pareto-optimal results are in bold. +Twitter +COLLAB +IMDB +RB 200 +RB 500 +EGN (report) +0.924 ± 0.133 (0.17) +0.982 ± 0.063 (0.10) +1.000 (0.08) +- +- +EGN (re-impl) +0.926 ± 0.113(0.17) +0.982 ± 0.069 (0.10) +1.000 (0.08) +0.820 ± 0.188 (0.26) +0.829 ± 0.192 (0.29) +EGN (re-impl) f-t +0.949 ± 0.102(0.49) +0.986 ± 0.060(0.27) +1.000 (0.20) +0.846 ± 0.180 (0.81) +0.864 ± 0.181 (0.89) +RUN-CSP +0.909 ± 0.145 (0.21) +0.912 ± 0.188 (0.14) +0.823 ± 0.191 (0.11) +0.858 ± 0.731 (2.05) +0.748 ± 0.689 (2.16) +Meta-EGN +0.976 ± 0.048(0.17) +0.988 ± 0.059 (0.10) +1.000 (0.08) +0.834 ± 0.178 (0.26) +0.834 ± 0.198 (0.29) +Meta-EGN f-t +0.990 ± 0.028(0.49) +0.993 ± 0.038 (0.27) +1.000 (0.20) +0.874 ± 0.169 (0.81) +0.878 ± 0.181 (0.89) +Toenshoff-Greedy +0.917 ± 0.126 (0.08) +0.969 ± 0.087 (0.06) +0.987 ± 0.050 (1e-3) +0.786 ± 0.195 (2.25) +0.793 ± 0.202 (2.38) +Gurobi9.5 (≤0.20s) +0.737 ± 0.267 (0.17) +0.871 ± 0.242 (0.04) +1.000 (1e-3) +- +- +Gurobi9.5 (≤1.00s) +1.000 (0.37) +0.979 ± 0.117 (0.06) +1.000 (1e-3) +- +- +Gurobi9.5 (≤2.50s) +1.000 (0.37) +0.997 ± 0.036 (0.06) +1.000 (1e-3) +0.667 ± 0.188 (2.10) +0.663 ± 0.188 (2.41) +Gurobi9.5 (≤4.00s) +1.000 (0.37) +1.000 (0.06) +1.000 (1e-3) +0.755 ± 0.225 (3.96) +0.742 ± 0.213 (3.88) +Table 4: ApR (time: second/graph) on the MVC problem. ApR is the smaller the better.. ‘f-t’ stands +for one-step fine-tune. Pareto-optimal results are in bold. +Twitter +COLLAB +IMDB +RB 200 +RB 500 +EGN +1.033 ± 0.023(0.29) +1.013 ± 0.022 (0.15) +1.000 (0.08) +1.031 ± 0.004 (0.26) +1.021 ± 0.002 (0.48) +EGN f-t +1.028 ± 0.021(0.80) +1.008 ± 0.015 (0.38) +1.000 (0.32) +1.030 ± 0.005 (0.80) +1.021 ± 0.002 (1.59) +RUN-CSP +1.180 ± 0.435 (0.16) +1.208 ± 0.198 (0.19) +1.188 ± 0.178 (0.08) +1.124 ± 0.001 (0.28) +1.062 ± 0.005 (1.65) +Meta-EGN +1.019 ± 0.017(0.29) +1.003 ± 0.010 (0.15) +1.000 (0.08) +1.028 ± 0.005 (0.26) +1.016 ± 0.002 (0.48) +Meta-EGN f-t +1.017 ± 0.017(0.80) +1.002 ± 0.010 (0.38) +1.000 (0.32) +1.027 ± 0.006 (0.80) +1.016 ± 0.002 (1.59) +Greedy +1.014 ± 0.014 (1.95) +1.209 ± 0.198 (1.79) +1.180 ± 0.077 (0.02) +1.124 ± 0.002 (5.02) +1.062 ± 0.005 (15.59) +Gurobi9.5 (≤0.25s) +1.028 ± 0.054 (0.09) +1.002 ± 0.010 (0.10) +1.000 (0.01) +- +- +Gurobi9.5 (≤0.50s) +1+1e-3 ± 0.001 (0.13) +1.000 (0.10) +1.000 (0.01) +- +- +Gurobi9.5 (≤1.00s) +1.000 (0.13) +1.000 (0.10) +1.000 (0.01) +1.011 ± 0.003 (0.63) +1.019 ± 0.003 (0.69) +Gurobi9.5 (≤2.00s) +1.000 (0.13) +1.000 (0.10) +1.000 (0.01) +1.008 ± 0.002 (1.16) +1.019 ± 0.003 (1.68) +number ranging from 102 to 105 and the node degree selected from the set D = {3, 7, 10, 20}. Here, +node degree equaling 20 is the hardest setting (Angelini & Ricci-Tersenghi, 2022). A summary of +these datasets are in Table. 10 in Appendix. +Data Splitting & The Evaluation Metric: For the real datasets, training/validation/test instances +are randomly divided with the ratio of 8:1:1; For RB200 and RB500, 2000/100/100 graphs are +generated for training/validation/test instances; For RB1000, RB2000, RB5000, we generate 100 +test instances. As to RRG datasets, the training set contains 3000 RRGs, of which each has 1000 +nodes and the node degree is uniformly sampled from D. We generate 30/20 graphs for each node +degree configuration in D for validation/test. Our evaluation metric uses approximation rate (ApR). +All results are summarized based on 5 times independent experiments with different random seeds. +Baselines: Our baselines include unsupervised learning methods, heuristics and traditional CO +solvers. For the MC and MVC problems, we take our direct baseline EGN (Karalias & Loukas, +2020), and also take RUN-CSP (Toenshoff et al., 2021) as another baseline. We do not consider +other learning-based methods because they generally perform worse than EGN (Karalias & Loukas, +2020). As to the heuristics, we use the greedy algorithms as the heuristic baselines. For traditional +CO solvers, we compare against the best commercial CO problem solver Gurobi9.5 (Gurobi Opti- +mization, 2022) via converting the problems into integer programming form. We track the time t +that the models use from the start of inferring to the end of rounding to output feasible solutions. We +set this time t as the time budget of Gurobi9.5 for purely solving the integer programming, and list +the actual time usage of Gurobi9.5 which includes pre-processing plus t. As to the MIS problem, +we take PI-GNN (Schuetz et al., 2022) and EGN Karalias & Loukas (2020) as the learning-based +baselines. We take the random greedy algorithm (RGA) and degree-based greedy algorithm (DGA) +as introduced in Angelini & Ricci-Tersenghi (2019) as the heuristic baselines. When we consider +fine-tuning EGN and Meta-EGN over a test instance, we use 1-step gradient descent as fine-tuning. +Implementation: For the MC and MVC problems, we use 4-layer GIN (Xu et al., 2019) as the +backbone network for both meta-EGN and EGN. We use 1e-3 as both the outer learning rate (γ) +of Meta-EGN and the learning rate of EGN. Here, the backbone and the learning rate are same as +those in (Karalias & Loukas, 2020). For the MIS problem, we use 6-layer GIN. The outer learning +rate (γ) of Meta-EGN and the learning rate of EGN are set as 1e-4. The inner learning rate (α) of +Meta-EGN is always set as 5e-5. We run all experiments by using a Xeon(R) Gold 6248 CPU with +7 + +Preprint +Table 5: Scale generalization performance on the MC and MVC problems. ApR is the larger the +better for MC while the smaller the better for MVC. All the models are trained on RB500 training +data. ‘Fast/Medium/Accurate’ denotes GNNs (without fine-tuning) using 1/4/8 random single node +seed(s) per testing instance. ‘Fine-tuning’ use 1-step Fine-tuning the best trial among the 8 node +seed(s). ‘Gap’ represents the averaged gap defined as c × (# of nodes in the optimal solution - # +of nodes by the given method) where c = 1 for MC and c = −1 for MVC. ‘Rank’ is the averaged +ranks of solutions among the three methods. Optimal solutions are generated via Gurobi9.5 with the +time limit 3000 seconds. Approximation rate for MC larger than 1, highlighted by ∗, indicates the +model outperforms Gurobi9.5 solver with 3000s time budget. Pareto-optimal results are in bold. +Dataset +Method +Fast (1) +Medium (4) +Accurate (8) +Fine-tune +ApR(s/g) +Gap +Rank +ApR(s/g) +Gap +Rank +ApR(s/g) +Gap +Rank +ApR(s/g) +Gap +Rank +MC +RB1000 +EGN +0.6462±0.282(0.05) +11.48 +2.406 +0.8433±0.229(0.17) +6.47 +2.237 +0.9099±0.205(0.33) +4.86 +2.025 +0.9631±0.186(0.98) +4.13 +1.693 +Meta-EGN +0.7692±0.276(0.05) +8.57 +1.943 +0.9388±0.196(0.17) +4.61 +1.543 +0.9408±0.205(0.33) +4.97 +1.581 +0.9745±0.195(0.98) +4.01 +1.625 +Gurobi9.5 +0.8851±0.197(6.11) +5.08 +1.650 +0.8851±0.197(6.18) +5.08 +2.218 +0.8851±0.197(6.48) +5.08 +2.393 +0.8851±0.197(7.01) +5.08 +2.681 +RB2000 +EGN +0.6793±0.290(0.10) +12.38 +2.408 +0.8968±0.184(0.29) +5.23 +2.136 +0.9454±0.160(0.58) +3.81 +2.208 +0.9714±0.154(2.03) +3.61 +1.983 +Meta-EGN +0.8077±0.114(0.10) +8.13 +1.991 +0.9783±0.157(0.29) +3.48 +1.591 +0.9958±0.146(0.58) +1.28 +1.466 +1.0112±0.134(2.03)∗ +0.55 +1.483 +Gurobi9.5 +0.9510±0.145(24.14) +3.01 +1.600 +0.9510±0.145(24.56) +3.01 +2.091 +0.9510±0.145(25.01) +3.01 +2.325 +0.9510±0.145(25.66) +3.01 +2.533 +RB5000 +EGN +0.9603±0.159(0.33) +2.42 +2.130 +1.0203±0.139(1.02)∗ +-1.26 +2.060 +1.0272±0.140(2.50)∗ +-1.68 +1.980 +1.0475±0.188(9.66)∗ +-2.86 +1.970 +Meta-EGN +1.0288±0.138(0.33)∗ +-1.62 +1.820 +1.0684±0.233(1.02)∗ +-4.02 +1.820 +1.0727±0.234(2.50)∗ +-4.42 +1.790 +1.0778±0.233(9.66)∗ +-4.72 +1.710 +Gurobi9.5 +1.0000(201.55) +0.00 +2.050 +1.0000(202.36) +0.00 +2.120 +1.0000(205.64) +0.00 +2.230 +1.0000(214.35) +0.00 +2.320 +Gurobi9.5 +1.0000(3000) +0.00 +- +1.0000(3000) +0.00 +- +1.0000(3000) +0.00 +- +1.0000(3000) +0.00 +- +MVC +RB1000 +EGN +1.0161±0.0048(0.20) +16.46 +2.250 +1.0135±0.0013(0.72) +13.73 +1.920 +1.0138±0.0013(1.37) +13.29 +1.860 +1.0138±0.0013(3.05) +13.28 +1.960 +Meta-EGN +1.0145±0.0016(0.20) +14.81 +1.935 +0.0131±0.0012(0.72) +13.40 +1.700 +1.0125±0.0012(1.37) +12.75 +1.545 +1.0124±0.0012(3.05) +12.69 +1.455 +Gurobi9.5 +1.0143±0.0018(1.92) +14.58 +1.835 +1.0143±0.0018(2.58) +14.58 +2.380 +1.0143±0.0018(3.08) +14.58 +2.595 +1.0143±0.0018(4.96) +14.58 +2.585 +RB2000 +EGN +1.0114±0.0026(0.34) +22.02 +2.350 +1.0096±0.0008(1.32) +18.57 +1.765 +1.0094±0.0007(2.69) +18.17 +1.765 +1.0093±0.0007(6.27) +17.98 +1.890 +Meta-EGN +0.0103±0.0015(0.34) +19.94 +1.740 +1.0095±0.0008(1.32) +18.41 +1.635 +1.0092±0.0007(2.69) +17.82 +1.510 +1.0090±0.0006(6.27) +17.38 +1.360 +Gurobi9.5 +1.0104±0.0010(5.63) +20.18 +2.910 +1.0104±0.0010(6.65) +20.18 +2.600 +1.0104±0.0010(8.04) +20.18 +2.725 +1.0104±0.0010(13.24) +20.18 +2.750 +RB5000 +EGN +1.0071±0.0014(1.01) +34.19 +2.170 +1.0064±0.0004(3.99) +30.83 +1.985 +1.0062±0.0004(7.95) +29.87 +1.865 +1.0062±0.0004(18.41) +29.68 +1.960 +Meta-EGN +1.0067±0.0005(1.01) +32.51 +2.045 +1.0062±0.0005(3.99) +29.96 +1.600 +1.0061±0.0004(7.95) +29.44 +1.555 +1.0060±0.0003(18.41) +29.15 +1.470 +Gurobi9.5 +1.0066±0.0006(24.60) +31.88 +1.785 +1.0066±0.0006(28.72) +31.88 +2.415 +1.0066±0.0006(32.16) +31.88 +2.580 +1.0066±0.0006(42.62) +31.88 +2.570 +Table 6: Generalization performance from Twitter to RB2000 on the MC and MVC problems. +Pareto-optimal results are in bold. +Method +MC (Approximation Rate ↑ (time)) +MVC (Approximation Rate ↓ (time)) +Fast (1) +Medium (4) +Accurate (8) +Fine-tune +Fast (1) +Medium (4) +Accurate (8) +Fine-tune +EGN +0.594±0.210(0.07) +0.788±0.201(0.16) +0.819±0.195(0.29) +0.831±0.192(0.89) +1.055±0.005(0.11) +1.053±0.004(0.37) +1.052±0.004(0.48) +1.050±0.004(1.59) +Meta-EGN +0.690±0.201(0.07) +0.793±0.197(0.16) +0.833±0.193(0.29) +0.876±0.182(0.89) +1.036±0.005(0.11) +1.030±0.003(0.37) +1.029±0.002(0.48) +1.021±0.003(1.59) +Gurobi9.5 +0.663±0.188(2.92) +0.663±0.188(2.92) +0.669±0.191(3.08) +0.742±0.213(3.88) +1.019±0.003(1.12) +1.019±0.003(1.30) +1.019±0.003(1.35) +1.017±0.002(2.40) +26 threads and a Quadro RTX 6000 GPU. All codes run on the PyTorch platform (Paszke et al., +2019). For more details, see Appendix. C. +Overcoming the limited expressive power of GNNs: GNNs are known with limited expressive +power (Xu et al., 2019; Morris et al., 2019). Specifically over RRGs, the GIN backbone will asso- +ciate each node with the same representation, unless node representations are initialized not equal. +To keep fair comparison, for the MC and MVC problem, we follow Karalias & Loukas (2020) and +adopt the initialization based on a single random node seed (one selected node is initialized as 1, +others as 0). We use 8 single random node seeds for EGN and Meta-EGN in the experiments of +Sec. 5.2 and report the best among the 8 trials. We try different numbers of random node seeds in +the experiments of Sec. 5.3. For the large-scale MIS problem studied in Sec. 5.4, we find such single +node initialization is too local to generate valid global solutions. So, we adopt initialization based +on the solutions of greedy algorithms DGA (for Figs. 1,2) and RGA (for Fig. 4). Then, EGN and +Meta-EGN can be viewed as to learn heuristics to improve the greedy solutions. Note that learning +heuristics to tune these solutions is non-trivial (Andrade et al., 2012; Rahman & Virag, 2017). +5.2 +META-EGN BOOSTS THE PERFORMANCE WITHOUT DISTRIBUTION SHIFTS +We first compare the performances of different methods when the datasets used for training and test +are from the same distribution. Table 3 and Table 4 show the results for the MC problem and the +MVC problem respectively. In both problems and across the five datasets, Meta-EGN siginificantly +outperforms EGN and RUN-CSP, both before and after the fine-tuning step. In comparison with +the traditional CO solvers, Meta-EGN narrows the gap from Gurobi9.5 on those real small graphs. +For RB graphs, Meta-EGN outperforms Gurobi9.5 for the MC problem. For the MVC problem, +Meta-EGN outperforms Gurobi9.5 on RB500. +We notice that both EGN and Meta-EGN perform generally well on the MC problem while not as +competitive on the MVC problem. This results from the initialization of GNN inputs. The MC +problem outputs clusters that are more local while MVC asks for global assignments, which makes +such single-seed-based initialization less fit for the MVC problem. +8 + +Preprint +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^3 nodes +0.70 +0.75 +0.80 +0.85 +0.90 +Approximation rate +Random Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^4 nodes +0.70 +0.75 +0.80 +0.85 +0.90 +Approximation rate +Random Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^5 nodes +0.70 +0.75 +0.80 +0.85 +0.90 +Approximation rate +Random Greedy +Meta-EGN +Meta-EGN Fine-tune +EGN +EGN Fine-tune +PI-GNN +Figure 4: ApRs in the MIS problem on RRGs. Meta-EGN and EGN are both trained with the output +of Random Greedy Algorithm (RGA) as initialization. +5.3 +META-EGN BOOSTS THE PERFORMANCE WITH DISTRIBUTION SHIFTS +Problem Scale Shift: Here, we use large-scale RB graphs of 1000-5000 nodes to test EGN and +Meta-EGN that are trained based on RB500. +Table 5 shows the results. +Both methods show +good generalization while Meta-EGN is always better. As the scale increases, Meta-EGN outper- +forms Gurobi9.5. For example, it takes Meta-EGN with 4 random initializations only 1.02s to beat +Gurobi9.5 that runs for 3000 seconds on RB5000 dataset in the MC problem. Moreover, Meta-EGN +can even outperform Gurobi9.5 on the MVC problem when the problem scale be comes large. +Real-Synthetic Distribution Shift: Here, we train EGN and Meta-EGN on Twitter and test them on +RB500. Table 6 shows the results. Compare Table 6 with Tables 3,4. We observe better generaliza- +tion performance of Meta-EGN compared to EGN. For example, for the MC problem, Meta-EGN +has almost the same performance whether there is a dataset shift or not (0.833 v.s. 0.834 before +fine-tuning, 0.876 v.s. 0.878 after fine-tuning) while EGN has a bigger gap in performance when +there is a shift (0.819 v.s. 0.829 before fine-tuning, 0.831 v.s. 0.864 after fine-tuning). For the MVC +problem, although the performance drop of Meta-EGN is larger, such a drop is still much smaller +than that of EGN. +5.4 +MAX INDEPENDENT SET: A RESPONSE TO (ANGELINI & RICCI-TERSENGHI, 2022) +10^2 +10^3 +10^4 +10^5 +Number of Nodes (degree=20) +10^-2 +10^-1 +1 +10^1 +10^2 +Time (s/graph) +DGA +Meta-EGN Fine-tune DGA +Meta-EGN DGA +Figure 5: Time cost v.s. Graph Scales. +For the MIS problem on large-scale RRGs, +Angelini & +Ricci-Tersenghi (2022) have recently posted a concern on +learning-based methods by arguing that PI-GNN in Schuetz +et al. (2022) could not achieve comparable results with +the heuristic algorithm DGA (Angelini & Ricci-Tersenghi, +2019). We see the reason comes from an improper usage of +learning-based methods in Schuetz et al. (2022) as stated in +Sec. 1: 1) PI-GNN is trained directly on each single test- +ing instance without learning from the training dataset that +contains varies graphs, which is likely to be trapped into the +local optima; 2) GNN generally suffers from a node ambi- +guity issue on RRGs. To resolve the problem, we utilize the outputs of DGA and RGA as the +initialization of GNN inputs (EGN, Meta-EGN) and expect to learn heuristics from historical data +to further tune the solutions given by the greedy algorithms. We train GNN models on RRGs with +1000 nodes with node degrees randomly sampled from 3, 7, 10, 20, and test on larger RRGs (up-to +105 nodes). Experiments show that Meta-EGN can further improve DGA (in Fig. 1) and RGA (in +Fig. 4), while EGN fails to better tune DGA. Note that here EGN and Meta-EGN adopt the exactly +same backbones. We attribute the improvement to meta-learning-based training as adopted by Meta- +EGN. See Table 7 in Appendix for more details of the numerical improvement by Meta-EGN. We +also see in these cases, one-step fine-tuning does not contribute much to the performance of EGN or +Meta-EGN, indicating the model before fine-tuning has been very close to a local minimum. +We also check the extra time cost by running Meta-EGN to improve DGA solutions in Fig. 5 and +Fig. 6 in Appendix. The extra time cost is just 1% (without fine-tuning) - 30% (with fine-tuning) of +the time cost of DGA. In theory, the extra time cost without fine-tuning should be O(|E|) for GNN +inference plus O(|V |) for rounding, which is in the same order of DGA, while the GNN parallel +inference substantially reduces the time. +9 + +Preprint +6 +CONCLUSION +This work proposes an unsupervised learning framework Meta-EGN with the goal of optimizing +NNs towards instance-wise good solutions to CO problems. Meta-EGN leverages MAML to achieve +the goal. Meta-EGN views each training instance as a separate task and learns a good initialization +for all these tasks. Meta-EGN significantly improves the performance of its baseline and has shown +good generalization when the data used for training and test has different scales or distributions. In +addition, Meta-EGN can learn to improve the greedy heuristics while paying almost no extra time +cost in the problem of maximum independent set on large-scale random regular graphs. +7 +ACKNOWLEDGEMENT +We would like to express our deepest appreciation to Dr. Tianyi Chen for the insightful discussion +on the meta-learning framework from a theoretical aspect and Dr. Ruqi Zhang for the constructive +advice on the fine-tuning strategies. We would also like to extend our deepest gratitude to Dr. +Hanjun Dai and Dr. Jialin Liu for sharing their invaluable insights into the general ideas of learning +for combinatorial optimization. Also many thanks to our fundings, H. Wang and P. Li are partially +supported by 2021 JPMorgan Faculty Award and the NSF award OAC-2117997. +REFERENCES +Saeed Amizadeh, Sergiy Matusevych, and Markus Weimer. Learning to solve circuit-sat: An un- +supervised differentiable approach. 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In Proceedings of the 21th ACM +SIGKDD international conference on knowledge discovery and data mining, pp. 1365–1374, +2015. +Weichi Yao, Afonso S Bandeira, and Soledad Villar. Experimental performance of graph neural +networks on random instances of max-cut. In Wavelets and Sparsity XVIII, volume 11138, pp. +242–251. SPIE, 2019. +Gal Yehuda, Moshe Gabel, and Assaf Schuster. It’s not what machines can learn, it’s what we cannot +teach. In International conference on machine learning, pp. 10831–10841. PMLR, 2020. +Emre Yolcu and Barnab´as P´oczos. Learning local search heuristics for boolean satisfiability. Ad- +vances in Neural Information Processing Systems, 32, 2019. +12 + +Preprint +A +PROOF OF THEOREM 1 +We first prove Theorem 1, then we specify the value of α to obtain Theorem 2 as a specific case of +Theorem 1. The proof of Theorem 1 is divided into two parts. +In part 1, we prove that if l(θ; G) < β + △ (even if l(θ; G) ≥ β), for any α ∈ (0, 2/L) Meta-GNN +with one-step finetuning outputs a feasible solution X of good quality f(X; G) ≤ l(θ; G) − △. +Here, △ = ∥∇θl(θ; G)∥ϵ + +1 +2Lα2−4αϵ2 if ϵ < α∥∇θl(θ; G)∥ or △ = (α − Lα2 +2 )∥∇θl(θ; G)∥2 o.w.. +In part 2, we prove that once Meta-EGN achieves the loss value l(θ′; G) after the one-step finetuning, +the rounding process would output a feasible X whose objective satisfies f(X; G) ≤ l(θ′; G). +Part 1:We could get +l(θ′; G) +(a) +≤ l(θ; G) + ∇θl(θ; G)(θ′ − θ) + 1 +2L∥θ′ − θ∥2 +2 +(b) += l(θ; G) + 1 +2L∥θ′ − θ∥2 +2 − α∥∇θl(θ; G)∥2 +2 += l(θ; G) + (Lα2 +2 +− α)∥∇θl(θ; G)∥2 +2, +(6) +where (a) is due to the local L-smoothness of l(·; G), (b) is due to the definition of one-step finetuning +θ′ = θ − α∇θl(θ; G). +If ϵ < α∥∇θl(θ; G)∥: +Let △ = ∥∇θl(θ; G)∥ϵ + +1 +2Lα2−4αϵ2, we have: +min +ϵ +−△ = min +ϵ +− +1 +2Lα2 − 4αϵ2 − ∥∇θl(θ; G)∥ϵ = (Lα2 +2 +− α)∥∇θl(θ; G)∥2, +(7) +thus +l(θ′; G) ≤ l(θ; G) − △. +(8) +If ϵ ≥ α∥∇θl(θ; G)∥: +Let △ = (α − Lα2 +2 )∥∇θl(θ; G)∥2, we would directly have: +l(θ′; G) ≤ l(θ; G) − △. +(9) +By this, we finish the first part of the proof for Theorem 1. +Part 2: The proof in this part follows the rounding analysis in Wang et al. (2022). Consider the +rounding procedure from continuous space ¯X = Aθ(G), ¯X ∈ [0, 1]n into the discrete feasible +solution X ∈ {0, 1}n. Let ¯Xi, Xi, i = {0, 1, ..., n} denote their entries. W.l.o.g, suppose the +rounding order is from 1 to n and we have finished the rounding before the t-th node, we now +analyze the rounding of t-th node: +fr([X1, ..., Xt−1, ¯Xt, ¯Xt+1, ..., ¯Xn]; G) + βgr([X1, ..., Xt−1, ¯Xt, ¯Xt+1, ..., ¯Xn]; G) +(d) +≥ ¯Xt(fr([X1, ..., Xt−1, 1, ¯Xt+1, ... ¯Xn]; G) + βgr([X1, ..., Xt−1, 1, ¯Xt+1, ..., ¯Xn]; G)) ++ (1 − ¯Xt)(fr([X1, ..., Xt−1, 0, ¯Xt+1, ..., ¯Xn]; G) + βgr([X1, ..., Xt−1, 0, ¯Xt+1, ..., ¯Xn]; G)) +≥ ¯Xt( min +jt={0,1} fr([X1, ..., Xt−1, jt, ¯Xt+1, ..., ¯Xn]; G) + βgr([X1, ..., Xt−1, jt, ¯Xt+1, ..., ¯Xn]; G)) ++ (1 − ¯Xt)( min +jt={0,1} fr([X1, ..., Xt−1, jt, ¯Xt+1, ..., ¯Xn]; G) ++ βgr([X1, ..., Xt−1, jt, ¯Xt+1, ..., ¯Xn]; G)) +(e) +=fr([X1, ..., Xt−1, Xt, ¯Xt, ..., ¯Xn]; G) + βgr([X1, ..., Xt−1, Xt, ¯Xt, ..., ¯Xn]; G) +(10) +where (d) is due to lr(θ; G)’s entry-wise concavity w.r.t ¯X and Jensen’s inequality, (e) is due +to Xt = arg minj=0,1 fr(X1, ..., Xt−1, t, ¯Xt+1, ..., ¯Xn) + βgr(X1, ..., Xt−1, t, ¯Xt+1, ..., ¯Xn) (the +13 + +Preprint +definition of our rounding process). The loss value is monotonically non-increasing through the +whole rounding process according to the equation above, thus we could get: +l(θ′) ≥ f(X; G) + βg(X; G) +(11) +By this, we finish the proof of the second part. +A.1 +A SPECIFIC CASE +Note that in the first part of the proof above, if we specify the value of α as 1 +L in equation (6), we +could have: +l(θ′; G) ≤ l(θ; G) − ∥∇θl(θ; G)∥2 +2 +2L +(12) +If ϵ < 1 +L∥∇θl(θ; G)∥: +Let △ = ∥∇θl(θ; G)∥ϵ − L +2 ϵ2, we have: +min +ϵ +−△ = min +ϵ +L +2 ϵ2 − ∥∇θl(θ; G)∥ϵ = −∥∇θl(θ; G)∥2 +2L +, +(13) +thus +l(θ′; G) ≤ l(θ; G) − △. +(14) +If ϵ ≥ 1 +L∥∇θl(θ; G)∥: +Let △ = +1 +2L∥∇θl(θ; G)∥2, we would directly have: +l(θ′; G) ≤ l(θ; G) − △. +(15) +By this, we obtain Theorem 2, a specific case of Theorem 1 as follows: +Theorem 2 (A Specific case of Theorem 1). Suppose the relaxations fr and gr are entry-wise con- +cave as required in (Wang et al., 2022). Let θ denote the learned parameter after training. Given a +test instance G, suppose locally l(·; G) is L-smooth at θ, i.e., ∥∇θ′l(θ′; G)−∇θl(θ; G)∥ ≤ L∥θ′−θ∥ +for all θ′ that satisfies ∥θ′ − θ∥ ≤ ϵ. Then, if l(θ; G) < β + △ (even if l(θ; G) ≥ β), there ex- +ists α such that Meta-GNN with one-step finetuning outputs a feasible solution X of good quality +f(X; G) ≤ l(θ; G) − △. +Here, △ = ∥∇θl(θ; G)∥ϵ − L +2 ϵ2 if ϵ < +1 +L∥∇θl(θ; G)∥ or △ = +1 +2L∥∇θl(θ; G)∥2 o.w.. +B +SUPPLEMENTARY EXPERIMENT RESULTS +B.1 +SUPPLEMENTARY TIME COST V.S. GRAPH SCALE IN THE MIS +we show the degree 3, 7, 10 in the following Fig. 6. They show the same time-cost vs scale relation +as that in Fig. 5. The extra time cost of GNN is O(|E|) for inference plus O(|V |) for rounding, +which is in the same order of DGA. +10^2 +10^3 +10^4 +10^5 +Number of Nodes (degree=3) +10^-2 +10^-1 +1 +10^1 +10^2 +Time (s/graph) +DGA +Meta-EGN Fine-tune DGA +Meta-EGN DGA +10^2 +10^3 +10^4 +10^5 +Number of Nodes (degree=7) +10^-2 +10^-1 +1 +10^1 +10^2 +Time (s/graph) +DGA +Meta-EGN Fine-tune DGA +Meta-EGN DGA +10^2 +10^3 +10^4 +10^5 +Number of Nodes (degree=10) +10^-2 +10^-1 +1 +10^1 +10^2 +Time (s/graph) +DGA +Meta-EGN Fine-tune DGA +Meta-EGN DGA +Figure 6: Time cost v.s. Graph Scales on degree 3, 7, 10 +14 + +Preprint +Table 7: Improvement of Meta-EGN over DGA and RGA in the MIS on RRGs, ‘Imp in ApR’ +denotes the average improvement in approximation rate and ‘Imp in #Node’ denotes the average +number of nodes that Meta-EGN could find more than the heuristics. +Scale/Degree +3 +7 +10 +20 +Imp in ApR +Imp in #Node +Imp in ApR +Imp in #Node +Imp in ApR +Imp in #Node +Imp in ApR +Imp in #Node +Meta-EGN improves DGA by +103 +0.0043 +1.950 +0.0060 +2.014 +0.0044 +1.254 +0.0084 +1.657 +104 +0.0050 +22.768 +0.0062 +20.811 +0.0067 +19.109 +0.0079 +15.588 +105 +0.0032 +145.718 +0.0045 +151.051 +0.0051 +145.69 +0.0050 +98.660 +Meta-EGN improves RGA by +103 +0.0944 +42.986 +0.1208 +40.549 +0.1292 +36.849 +0.1239 +24.447 +104 +0.0932 +424.404 +0.1125 +377.628 +0.1151 +328.276 +0.1173 +231.456 +105 +0.0871 +3966.272 +0.1045 +3507.751 +0.1083 +3088.824 +0.0969 +1912.030 +B.2 +HOW MUCH DOES META-EGN MODIFY DGA AND RGA HEURISTICS IN THE MIS +We display the average approximation rate improvement and the average node number increase by +Meta-EGN over DGA and RGA in Table. 7. +B.3 +TRAINING THE MODELS ON SUBSETS OF THE TRAINING DATA +We display the average approximation rates of the models that are only trained on subsets of the +original training data in the max clique problem on Twitter. The training dataset is randomly sampled +from the original training dataset and the testing dataset remains the same as that in Table. 3. Both +the methods have worse performance as the number of training instances reduces, while Meta-EGN +only has a 0.7% performance decrease from the full-size training dataset with 750 samples to the +training subset with only 64 instances. In contrast, EGN decreases its performance by 1.7%. +Table 8: The approximation rate of the max clique problem on Twitter. Models are only trained on +subsets of the dataset, ‘training subset’ denotes the number of instances in the training data. +training subset +64 +128 +256 +512 +Full (750) +EGN +0.909±0.122 +0.911±0.118 +0.914±0.118 +0.922±0.115 +0.926±0.113 +Meta-EGN +0.970±0.058 +0.973±0.055 +0.975±0.055 +0.975±0.051 +0.976±0.048 +B.4 +TRAINING THE MODELS ON RRGS WITH SINGLE DEGREES IN THE MIS +We train the EGN and Meta-EGN models on RRGs with only 3 or 20 degrees and test them on +RRGs with the rest degrees from {3, 7, 10, 20}. Models take the output of DGA as the initialization +graph node feature. We show the performance of both the models without fine-tuning in Fig. 7. +When only trained on RRGs with degree 3 (See the left two figures in Fig. 7), both the models could +not generalize well, as neither of them could outperform the initialization input of DGA. Note that +Meta-EGN still achieves better performance than EGN in this case. As to the models only trained +on RRGs with degree 20 (See the right two figures in Fig. 7), we observe that both the model have +relatively good generalization ability across different degrees, yet Meta-EGN could still marginally +outperform EGN in this case. We attribute this phenomenon to the fact that solving the MIS on +RRGs with degree 20 is much more complicated than those with degree 3 and thus may contain +adequate heuristics for solving RRGs with lower degrees. +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^3 nodes +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Approximation rate +Degree-based Greedy +EGN +Meta-EGN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^4 nodes +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +Approximation rate +Degree-based Greedy +EGN +Meta-EGN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^3 nodes +0.88 +0.90 +0.92 +0.94 +Approximation rate +Degree-based Greedy +EGN +Meta-EGN +d=20 +d=10 +d=7 +d=3 +Degree of the RRGs with 10^4 nodes +0.88 +0.89 +0.90 +0.91 +0.92 +0.93 +0.94 +0.95 +Approximation rate +Degree-based Greedy +EGN +Meta-EGN +Figure 7: The left two figures show the ApRs on RRGs with 103 and 104 nodes of the models trained +only on RRGs with degree 3. The right two figures show the ApRs on RRGs with 103 and 104 nodes +of the models trained only on RRGs with degree 20. +15 + +Preprint +B.5 +COMPARISON ON THE TRAINING TIME OF THE MODELS +We display the wall clock training time for the two methods to convergence in Table. 9 (from start to +the the best epoch on validation set). We observe that Meta-EGN generally takes two to three times +to converge compared with EGN, but their training time cost basically remain on the same order of +magnitude. +Table 9: The wall clock training time to convergence of EGN and Meta-EGN in different problems. +Dataset/Time +(min:second) +MC +MVC +MIS +Twitter +RB200 +RB500 +Twitter +RB200 +RB500 +RRGs +EGN +46:50 +104:37 +282:57 +100:58 +83:27 +128:39 +733:02 +Meta-EGN +101:55 +210:04 +609:47 +276:38 +168:25 +282:15 +1088:55 +C +SUPPLEMENTARY IMPLEMENTATION DETAILS +C.1 +EXPERIMENT DETAILS +All the codes run on the PyTorch platform 1.9.0 (Paszke et al., 2019) and PyTorch Geometric frame- +work 1.7.2 (Fey & Lenssen, 2019). The details of each datasets is shown in Table. 10, all of the real +datasets are publicly available, we follow the code in (Toenshoff et al., 2021) to generate the RB +model. +Table 10: The number of instances in each dataset. ‘20/scale/degree’ means that we generate 20 +testing instances for each different scale-degree pair. We generate RB1000, RB2000 and RB5000 +only for testing. +Dataset +Twitter +COLLAB +IMDB +RB200 +RB500 +RB1000 +RB2000 +RB5000 +RRGs +Training +750 +3600 +800 +2000 +2000 +- +- +- +3000 +Validation +100 +450 +100 +100 +100 +- +- +- +30 +Testing +100 +450 +100 +100 +100 +100 +100 +100 +20/scale degree +To balance the training time per epoch of EGN and Meta-EGN, we define the epoch as follows: For +each epoch of EGN training, the whole dataset is split into mini-batches. EGN performs standard +mini-batch training along these batches and optimizes over each mini-batch. As to Meta-EGN, for +each training epoch Meta-EGN only randomly samples a single batch and do the meta learning +algorithm on the batch. The batch sizes of the methods are controlled the same. +C.2 +DETAILED DERIVATION OF THE LOSS FUNCTION RELAXATION +In this part, we display the detailed loss function relaxation of the three problems in our study (the +MC, the MVC and the MIS). The basic idea of training loss design and relaxation follow (Karalias +& Loukas, 2020; Wang et al., 2022). In the following derivation, we use i, j to represent the nodes +in graphs, we use Xi, Xj ∈ {0, 1} to denote the discrete assignment of the binary optimization vari- +ables, and we use ¯Xi, ¯Xj ∈ [0, 1] to denote the relaxed soft assignment of the binary optimization +variables. +The maximum clique (MC): A clique is a set of nodes S ∈ V such that any two distinct nodes in +the set are adjacent. The MC aims to find out the clique with the largest number of nodes. We could +formulated the optimization objective as follows: +max +X +� +1≤i≤n +Xi +s.t. +(i, j) ∈ E if Xi, Xj = 1, +(16) +Xi, Xj denotes whether to take the node into the clique set (Xi = 1) or not (Xi = 0). By setting a +proper penalty coefficient β, we could formulate the loss function relaxation as follows (the detailed +16 + +Preprint +derivation follows the corresponding case study in Karalias & Loukas (2020)). +lMC(θ; G) ≜ −(β + 1) +� +(i,j)∈E +¯Xi ¯Xj + β +2 +� +i̸=j +¯Xi ¯Xj. +(17) +The minimum vertex covering (MVC): A vertex cover is a set of nodes S ∈ V that any edge in +the graph is connected to at least a node from the set. The MVC aims to find out the cover set with +the smallest number of nodes. The optimization objective could be summarized as follows: +min +X +� +1≤i≤n +Xi +s.t. +Xi + Xj ≥ 1 if (i, j) ∈ E, +(18) +where Xi, Xi denotes whether to take the node into the cover set (Xi = 1) or not (Xi = 0). We +design the constraint function g to represent the total number of edges that have not been covered +given a set of variable assignment X, and thus we write g as: +gMVC(X; G) ≜ +� +(i,j)∈E +(1 − Xi)(1 − Xj). +(19) +Then we relax the constraint g and add it into the training objective by multiplying a proper penalty +coefficient β, following the relaxation principle in Wang et al. (2022): +lMVC(θ; G) ≜ +� +1≤i≤n +¯Xi + β +� +(i,j)∈E +(1 − ¯Xi)(1 − ¯Xj). +(20) +By this, we aim to minimize the value of the loss function above in order to minimize the node +number of the cover set as well as consider the covering property in the constraint. +The maximum independent set (MIS): An independent set is a set of nodes where any two distinct +nodes in the set are not adjacent to each other. The MIS aims to find out the independent set with +the largest number of nodes. We could formulate the objective of the MIS as follows: +max +X +� +1≤i≤n +Xi +s.t. +XiXj = 0 if (i, j) ∈ E, +(21) +where Xi, Xj denotes whether to take the node into the independent set (Xi = 1) or not (Xi = 0). +We formulate the constraint g as the total number of edges whose two connected nodes at the end- +points are both assigned into the independent set. Therefore we could write the constraint as follows: +gMIS ≜ +� +(i,j)∈E +XiXj. +(22) +We then relax the constraint g into continuous space and add it into the c function with a proper +penalty coefficient β, following the relaxation principle in (Wang et al., 2022), and thus we could +write the training loss function as: +lMIS(θ; G) ≜ − +� +1≤i≤n +¯Xi + β +� +(i,j)∈E +¯ +Xi ¯ +Xj. +(23) +By this, we aim to minimize the value of the loss function above in order to maximize the node +number of the independent set as well as consider the independent property in the constraint. +C.3 +SEPARATED ALGORITHM TABLES +We separate the algorithm table of Meta-EGN into training and testing parts to make it clearer. The +algorithm table is shown in Alg. 2 for training and Alg. 3. +C.4 +IMPLEMENTATION OF THE HEURISTICS +We run all of the greedy algorithms with PyThon 3.8 in this paper. A potential method to boost the +time cost of these greedy algorithm is to use c++. +17 + +Preprint +Algorithm 2 Train Meta-EGN +Require: Training instances Ξ = {G1, G2, ..., Gm}; Hyperparameters: α, γ. +1: Randomly initialize θ(0) +2: for each randomly sampled mini-batch Bj ⊂ Ξ, j = 0, 1, ..., K − 1 do +▷ Training starts +3: +For each Gi ∈ Bj, compute the adapted parameter: θ(j) +i += θ(j) − α∇θ(j)l(θ(j); Gi) +4: +Update: θ(j+1) ← θ(j) − γ∇θ(j) � +Gi∈Bj l(θ(j) +i ; Gi) +5: end for +6: return θ ← θ(K) +▷ Training ends +Algorithm 3 Test Meta-EGN with/without Fine-tuning +Require: Testing instance G′; Hyperparameter: α; Pre-trained parameter initialization θ. +1: For a given testing instance G′: +▷ Testing starts +2: if fine-tuning is allowed then +3: +Fine-tune the parameters: θG′ ← θ − α∇θl(θ; G′) +4: +Use Def. 1 to round the relaxed solution given by AθG′ (G′) +▷ With fine-tuning +5: else +6: +Use Def. 1 to round the relaxed solution given by Aθ(G′) +▷ Without fine-tuning +7: end if +8: return the rounded solution +▷ Testing ends +Random Greedy Algorithm for MIS (RGA): RGA takes a time to reach a solution that is linear +in the problem size n. It starts from an empty independent set S. At each step 1 ≤ t ≤ n, a node i +is chosen at random from the graph Gt and added to the independent set. Then all the neighbors of +i are removed from Gt to formulated a new graph Gt+1. The process iterates until Gt∗ is empty at +step t∗, the solution is S. +Degree-based Greedy Alforithm for MIS (DGA): DGA modifies RGA by sorting the degrees of +the nodes before each iteration starts, and always put the node with the smallest degree into the +independent set. +Teonshoff Greedy for MC: Toenshoff et al. (2021) convert the testing instances into its complement +graph, and then run DGA to solve the MIS problem. It takes the solution to the MIS problem on the +complement graph as the solution for MC on the original graph. +Greedy for MVC: Greedy for MVC starts from an empty covering set S. At each step 1 ≤ t ≤ n, it +first sorts the degrees of the nodes in the graph Gt and always add the node i with the largest degree +into the covering set S. Then all the edges that connect with i are removed from Gt to formulate a +new graph Gt+1. The process stops until Gt∗ is empty at step t∗, the solution is S. +D +DISCUSSION ON LIMITATIONS +• As mentioned in the end of Sec. 5.1, both EGN and Meta-EGN perform generally well in MC, +which outputs the cliques that are more local in comparison with the vertex covering in MVCs that +require more global assignments. The random initialization seed with one node randomly set as 1 +and the others as 0 would potentially limit the performance of EGN and Meta-EGN in more global +CO tasks. +• We use Meta-EGN and EGN to modify the solution of DGA and RGA in the MIS problem. In +addition, there are also many other Monte Carlo (MC) algorithms (i.e. simulated annealing and +parallel tempering) that could produce better results than DGA or RGA in RRGs (Angelini & Ricci- +Tersenghi, 2022). An intuitive idea is to test whether we could learn Meta-EGN to further fine-tune +these more advanced MC algorithms in the MIS problem on RRGs. +We leave the research on modifying Meta-EGN to better deal with the CO problems that require +global assignments and using Meta-EGN to improve other advanced MC algorithms as a future +study. +18 + diff --git a/EdE1T4oBgHgl3EQfWgTb/content/tmp_files/load_file.txt b/EdE1T4oBgHgl3EQfWgTb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc155da35233681289a0ee79dc7c15e3af1a6ef2 --- /dev/null +++ b/EdE1T4oBgHgl3EQfWgTb/content/tmp_files/load_file.txt @@ -0,0 +1,1885 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf,len=1884 +page_content='Preprint UNSUPERVISED LEARNING FOR COMBINATORIAL OP- TIMIZATION NEEDS META LEARNING Haoyu Wang1, Pan Li1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Department of Electrical and Computer Engineering, Georgia Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Department of Computer Science, Purdue University hwang3028@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='edu, panli@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='edu, panli@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='edu ABSTRACT A general framework of unsupervised learning for combinatorial optimization (CO) is to train a neural network (NN) whose output gives a problem solution by directly optimizing the CO objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Albeit with some advantages over tra- ditional solvers, the current framework optimizes an averaged performance over the distribution of historical problem instances, which misaligns with the actual goal of CO that looks for a good solution to every future encountered instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' With this observation, we propose a new objective of unsupervised learning for CO where the goal of learning is to search for good initialization for future prob- lem instances rather than give direct solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We propose a meta-learning-based training pipeline for this new objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our method achieves go empirical perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We observe that even just the initial solution given by our model before fine-tuning can significantly outperform the baselines under various evaluation settings including evaluation across multiple datasets, and the case with big shifts in the problem scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The reason we conjecture is that meta-learning-based train- ing lets the model loosely tied to each local optima for a training instance while being more adaptive to the changes of optimization landscapes across instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 1 INTRODUCTION Combinatorial optimization (CO), aiming to find out the optimal solution from discrete search space, has pivotal position in scientific and engineering fields (Papadimitriou & Steiglitz, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Crama, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Most CO problems are NP-complete or NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Conventional heuristics or approximation requires insightful comprehension of the particular problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Starting from the seminal work from Hopfield & Tank (1985), researchers apply neural networks (NNs) (Smith, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2015) to solve CO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The motivation is that NNs may learning heuristics through solving historical problems, which could be useful to solve similar problems in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Many NN-based methods (Selsam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gasse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2016) require optimal solutions to the CO problem as supervision in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, optimal solutions are hard to get in practice and the obtained model often does not gener- alize well (Yehuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Methods based on reinforcement learning (RL) (Mazyavkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Bello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Khalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Yolcu & P´oczos, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Chen & Tian, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Delarue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Nandwani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021) do not need labels while they often suffer from notoriously unstable training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Recently, unsupervised learning methods have attracted much attention (Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Amizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Karalias & Loukas, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A common strategy of these methods is to design an NN whose output gives a solution to the CO problem and then train the NN via gradient descent by directly optimizing the CO objectives over a set of training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This strategy is superior in its faster training, good generalization, and strong capability of dealing with large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Despite the prominent progress, current unsupervised learning methods always optimize NNs to- wards an averaged good performance over training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This means even if a testing instance comes from the same distribution of the training instances, the solution to this single instance may 1Our code is available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='com/Graph-COM/Meta_CO 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='03116v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='LG] 8 Jan 2023 Preprint d=20 d=10 d=7 d=3 Degree of the RRGs with 10^3 nodes 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='950 Approximation rate Degree-based Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^4 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='950 Approximation rate Degree-based Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^5 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='950 Approximation rate Degree-based Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN Figure 1: Approximation Rates of different methods in the MIS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN and EGN (Kar- alias & Loukas, 2020) are trained on RRGs with 1000 nodes and with node degree randomly sam- pled from 3, 7, 10, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN and EGN are evaluated over larger RRGs with 103 ∼ 105 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' More details about the setting are in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN outperforms DGA (Angelini & Ricci-Tersenghi, 2019) by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3% − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5% in approximation rates on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' not have good quality, let alone the case when the testing instance is out-of-distribution (OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This induces a concern when we apply NNs in practice because practical problems often expect to have a good solution to every encountered instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For example, allocating surveillance cameras is cru- cial for each-time exhibition in every art gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Solvers when applied to this problem (O’rourke, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Yabuta & Kitazawa, 2008) should output a good solution every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Traditional CO solvers are designed towards this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, it is time-consuming and unable to learn heuristics from historical instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' So, can we leverage the benefit of learning from history while with the goal of achieving an instance-wise good solution instead of an averaged good solution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This motivates us to study a new formulation of unsupervised learning for CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We regard the ob- jective of learning from history as to search a good initialization for each future instance rather than give a direct solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Since in practice, future instances are unavailable during the training stage, we propose to view each training instance as a pseudo-new instance for the rest training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then, our learning objective is to learn a good initialization of this model, such that further opti- mization of the initialization could achieve good solutions on each of these pseudo-new instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We observe meta learning is suitable to implement this idea, and propose to adopt MAML (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2017) in our training pipeline as a proof of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that the step of optimization on each pseudo-new instance shares a similar spirit with fine-tuning a model over each down-streaming task as traditional meta learning does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, each task in our case corresponds to optimization over each training instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We name our method Meta-EGN by extending the previous framework EGN (Karalias & Loukas, 2020) via meta learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our key observation is that with this new objective, even the initial solution given by Meta-EGN (before fine-tuning on a test instance) is substantially better than the solution given by EGN and other methods that optimize the averaged performance over training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our conjectured reason is that the new objective, by taking into account fine-tuning the model over new instances, trains the model to avoid being trapped into a local minima induced by each training instance while being more adaptive to the changes of optimization landscapes across instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We demonstrate the benefits of Meta-EGN via experiments within three benchmark CO problems (max clique, vertex cover and max independent set) on multiple synthetic graphs and three real- world graph datasets, with the number of nodes ranging from 100 to 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN significantly outperforms state-of-the-art learning-based baselines (Karalias & Loukas, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021), greedy algorithms, and the commercial CO solver Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (Gurobi Optimization, 2022) in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN also shows super OOD generalization performance when the training and test datasets are different or have graphs of entirely different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Moreover, recently, Angelini & Ricci-Tersenghi (2022) have shown that the learning-based method in (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) could not achieve comparable results with the degree-based greedy algo- rithm (DGA) (Angelini & Ricci-Tersenghi, 2019) in the max independent set (MIS) problem on large-scaled random-regular graphs (RRGs), which raises attentions from machine learning com- munity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We observe the issues come from two aspects: (1) graph neural networks (GNNs) used to encode the regular graph suffer from the node ambiguity issue due to their limited expressive power (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2) the model in (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) did not learn from history but was directly optimized over each testing case, which tends to be trapped into a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' By ad- dressing these two issues, Meta-EGN can consistently outperform DGA while maintaining the same time complexity to generate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 show the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2 Preprint 2 RELATED WORK In the following, we review two groups of works: unsupervised learning for CO and meta learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Previous works on unsupervised learning for CO have studied max-cut (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019) and TSP problems (Hudson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021), while these works depend on carefully selected problem-specific objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Some works have investigated satisfaction problems (Amizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Applying these approaches to general CO problems requires problem reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The works most relevant to us are (Karalias & Loukas, 2020), (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) and (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Karalias & Loukas (2020) propose an unsupervised learning framework EGN for general CO problems based on the Erd˝os’s probabilistic method, which bonds the quality of the final solutions with probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022) generalize EGN and prove that if the CO objective can be relaxed into an entry-wise concave form, a solution of good quality can be deterministically achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This further inspires the design of proxy objectives for the CO problems that may not have closed-form objectives, such as those in circuit design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022) have recently extended EGN to large-scale max independent set problems on random-regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta learning is proposed to learn hyper-parameters or initialization from historical tasks and achieve fast adaption to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2017) propose model-agnostic meta learning (MAML), which aimed to obtain good parameter initialization and accommodated to few-shot learn- ing tasks with limited steps of fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2018) accelerate MAML by adopting first- order approximation on the gradient estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Rajeswaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2019) introduce implicit-MAML that adopts an objective with fine-tuning till the stationary points on new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Implicit-MAML does not fit our case because we try to avoid long-time fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2018) study unsupervised learning under the meta learning framework and focused exclusively on vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' To the best of our knowledge, our work is the first one to apply meta learning to unsupervised learning for CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3 PRELIMINARIES: NOTATIONS AND PROBLEM FORMULATION Combinatorial Optimization on Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We follow the settings considered in (Karalias & Loukas, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) and study CO problems on graphs whose solutions can be represented as a subset of nodes of the input graph instance, although our method could be applied to a broader range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Suppose G is the universe of graph instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Let G(V, E) ∈ G denote a graph instance where V = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', n} is the node set and E is the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Let X = (Xi)1≤i≤n ∈ {0, 1}n denote the discrete optimization variables defined on V , where Xi = 1 denotes that node i is selected in the output node subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A CO problem on G consists of a cost function f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) : {0, 1}n → R≥0 and a feasible set Ω ⊆ {0, 1}n that stands for the finite set of all feasible X’s, and asks to solve min X∈{0,1}n f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' X ∈ Ω (1) Unsupervised Learning for CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The Erd¨os-Goes-Neural (EGN) framework of unsupervised learn- ing for CO proposed in (Karalias & Loukas, 2020) is reviewed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, we use the notation system in a follow-up work (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) as it is more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Learning for CO problem is to learn an algorithm Aθ(·) : G → {0, 1}n typically parameterized by an NN where θ denotes the parameters of the NN such that given a graph instance G, X = Aθ(G) gives a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In practice, directly optimizing the parameters θ is hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Therefore, we may consider a relaxed cost function fr(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) : [0, 1]n → R≥0 where fr(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) = f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) on any discrete points X ∈ {0, 1}n and a relaxed constraint gr(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) : [0, 1]n → R≥0 where {X ∈ {0, 1}n : gr(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) = 0} and {X ∈ {0, 1}n : gr(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≥ 1} defines the feasible set Ω and the infeasible set Ωc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Also, suppose the NN in Aθ can give soft solutions ¯X ∈ [0, 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then, we may train θ by minimizing a label-independent loss function: min θ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ fr( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr( ¯X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G), ¯X = Aθ(G), for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2) The significant observation made by (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022), which generalizes the argument in (Karalias & Loukas, 2020), is a type of performance guarantee on the condition that fr and gr are entry-wise concave, which is satisfied in all the cases studied in this work: If the loss that achieves l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) < β for some β > maxX∈{0,1}n f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G), then the discrete solution X obtained by rounding the soft solution ¯X = Aθ(G) according to Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 is feasible X ∈ Ω and of good quality f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3 Preprint Definition 1 (Rounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For a soft solution ¯X ∈ [0, 1]n and an arbitrary order of the en- tries (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='g 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=',n), fix all the other entries unchanged and round ¯Xi into 0 or 1 as Xi = arg minj=0,1 fr(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xi−1, j, ¯Xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn) + βgr(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xi−1, j, ¯Xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn), replace ¯Xi with Xi and repeat this operation until all the entries are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4 META LEARNING FOR ERD ¨OS GOES NEURAL (META-EGN) The above performance guarantee lays the theoretical foundation for EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, the following practical issue motivates us to incorporate meta learning into EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 MOTIVATION: WHAT NEEDED IS LEARNING FOR INSTANCE-WISE GOOD SOLUTIONS It is often time consuming to perform online optimization of l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) for each encountered instance G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This also mismatchs the goal of learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', learning heuristics from history/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Therefore, a pipeline commonly adopted is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Suppose there is a set of training instances Gi, 1 ≤ i ≤ m, IID sampled from a distribution PG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We optimize θ by following min θ m � i=1 l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi), (3) which is similar to empirical risk minimization (ERM) in standard supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' When a test instance G appears, we apply the learned Aθ to get a soft solution and round it to the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This pipeline cannot guarantee the quality for this instance G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Even if the training instances Gi, 1 ≤ i ≤ m are in a large quantity (so in-distribution generalization is not a problem), and even if the test instance G also follows PG, we may not guarantee a low l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) for one particular G because ERM only guarantees a low averaged performance EG∼PG[l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This issue may also voilate the condition to have performance guarantee as reviewed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3, as it is instance-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, we highlight that in practice even the minimal averaged loss minθ EG∼PG[l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)] is often strictly greater than averaged instance-wise minimal loss EG∼PG[minθ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)], because practical NNs are not expressive enough to remember the optimal solution to every instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Unfortunately, many practical CO problems actually expect instance-wise good solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This is because every instance in practice is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A terrible solution for one instance may raise a security issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', the surveillance-camera allocation problem) or cause huge economic losses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', the routing problem in a transportation system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' With this observation, our work is to address the problem by studying unsupervised learning for instance-wise good solutions to CO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 TRAINING TOWARDS INSTANCE-WISE OPTIMALITY VIA META LEARNING Our idea to address the problem is to regard the goal of learning from history as to search good initialization for future instances rather than give direct solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Such good initialization can be quickly fine-tuned by further optimizing the model for each instance, which ultimately gives instance-wise good solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, in practice, we do not have access to any future/test in- stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' So, can we just use historical/training instances to implement the above idea?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our strategy is to view each training instance Gi as a pseudo-test instance to test and optimize the quality of initialization given by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Specifically, this strategy gives us an objective min θ m � i=1 ˜li(θ), where ˜li(θ) = min θi l(θi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) with θi = θ as initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4 has some abuse of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The minimum ˜li(θ) depending on the initialization θ is because of the non-convex nature of minθi l(θi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi), where the initialization θi = θ matters significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We further simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4 with some practical consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In fact, we may not allow further optimizing θ with so many gradient-descent steps for each instance, especially during the online test stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As a proof of concept, we consider the case with only one-step gradient descent, which already gives good empirical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Specifically, our training objective follows Our Objective: min θ m � i=1 li(θ), where li(θ) = l(θi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) with θi = θ − α∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (5) 4 Preprint Algorithm 1 Train Meta-EGN and Test Meta-EGN with/without Fine-tuning Require: Training instances Ξ = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Gm};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hyperparameters: α, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1: Randomly initialize θ(0) 2: for each randomly sampled mini-batch Bj ⊂ Ξ, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', K − 1 do ▷ Training starts 3: For each Gi ∈ Bj, compute the adapted parameter: θ(j) i = θ(j) − α∇θ(j)l(θ(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) 4: Update: θ(j+1) ← θ(j) − γ∇θ(j) � Gi∈Bj l(θ(j) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) 5: end for 6: return θ ← θ(K) ▷ Training ends 7: For a given testing instance G′: ▷ Testing starts 8: if fine-tuning is allowed then 9: Fine-tune the parameters: θG′ ← θ − α∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G′) 10: Use Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 to round the relaxed solution given by AθG′ (G′) ▷ With fine-tuning 11: else 12: Use Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 to round the relaxed solution given by Aθ(G′) ▷ Without fine-tuning 13: end if ▷ Testing ends 0 1000 2000 3000 4000 Epochs 300 250 200 150 100 50 Loss value Meta-EGN-Pre Meta-EGN-Post EGN (a) Training Loss 0 1000 2000 3000 4000 Epochs 300 250 200 150 100 50 Loss value Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune (b) Validation Loss 0 1000 2000 3000 4000 Epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='900 Approximation Rate EGN EGN Fine-tune Meta-EGN Meta-EGN Fine-tune (c) Validation Approximation Rate Figure 2: Training/validating dynamics of Meta-EGN and EGN Karalias & Loukas (2020) for the MIS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Detailed experiment settings follow Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, θ is to give a good initialization Aθ(Gi) over each instance Gi while θi is with one-step fine-tune to achieve a Gi-specified good solution Aθi(Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Optimization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5 can be implemented via the meta learning pipeline MAML (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We name the obtained model Meta-EGN and summarize its training and testing in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In step 3, Meta-EGN performs the one-step gradient descent on each training instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that we consider two testing cases with or without fine-tuning, because the latter saves much inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A simple extension of the Theorem 1 in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) gives a performance guarantee for Meta-EGN in Theorem 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, for a test instance G, we even allow l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) to violate the original condition l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) < β in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) to some extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' After one-step fine-tuning in step 9, the performance guarantee is still achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The detailed proof is in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Theorem 1 (Performance Guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Suppose the relaxations fr and gr are entry-wise concave as required in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Let θ denote the learned parameter after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Given a test instance G, suppose locally l(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) is L-smooth at θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ∥∇θ′l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − ∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ ≤ L∥θ′ − θ∥ for all θ′ that satisfies ∥θ′ − θ∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then, if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) < β + △ (even if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≥ β), for any α ∈ (0, 2/L) Meta-GNN with one-step finetuning outputs a feasible solution X of good quality f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, △ = ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ + 1 2Lα2−4αϵ2 if ϵ < α∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ or △ = (α − Lα2 2 )∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='. To better understand Meta-EGN, we show its training/testing dynamics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As we expected, the training loss of EGN is somewhere in-between the losses of Meta-EGN before and after the fine-tune step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Training EGN is stabler and converges faster than training Meta-EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, what is unexpected is that in validation, Meta-EGN has a much lower loss and achieves much better performance than EGN even before fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This implies that Meta-EGN holds better generalization than EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We conjecture the reasons are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' First, the optimization landscape for CO problems is extremely non-convex (Mezard & Montanari, 2009) due to the intersected feasible-infeasible regions and the high penalty coefficient β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' EGN that has low losses for training instances may give a high loss even when the optimization 5 Preprint Table 1: Comparison between different unsupervised frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G denotes the test instance and Gi, 1 ≤ i ≤ m are training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The standard EGN pipeline does not adopt any fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' EGN (Karalias & Loukas, 2020) P-I GNN (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) Meta-EGN (Ours) Classical Solver Gurobi Optimization (2022) Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' to optimize the NN �m i=1 l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) �m i=1 l(θ − ∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' X ∈ Ω Training or not Yes No Yes No Fine-tune timing No Long Short/No Long Generalization Good Better Table 2: The discrete objectives (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1) and their relaxations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2) for the three problems to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' MC Discrete Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' maxX � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (i, j) ∈ E if Xi, Xj = 1 Relaxation lMC(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ −(β + 1) � (i,j)∈E ¯Xi ¯Xj + β 2 � i̸=j ¯Xi ¯Xj MVC Discrete Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' minX � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Xi + Xj ≥ 1 if (i, j) ∈ E Relaxation lMVC(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ � 1≤i≤n ¯Xi + β � (i,j)∈E(1 − ¯Xi)(1 − ¯Xj) MIS Discrete Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' maxX � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' XiXj = 0 if (i, j) ∈ E Relaxation lMIS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ − � 1≤i≤n ¯Xi + β � (i,j)∈E ¯ Xi ¯ Xj Figure 3: Performance v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' hyper- parameter ρ of the RB model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='95 The hyper-parameter p in RB model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='006 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='008 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='022 Approximation Rate EGN on RB500 Meta-EGN on RB500 Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 on RB500 landscape is just slightly shifted (from training to a test instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' However, the parameters of Meta- EGN are loosely tied to a local minimum for each training instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Instead, those parameters, as being aware of follow-up instance-wise fine-tuning steps, are likely to fall into some location close to a local minimum for each instance while being not trapped in anyone of them, which makes the model robust to landscape shifts across instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Second, a CO problem could vary a lot across graph instances even for those generated from the same distribution, especially when the instances are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' So, it is reasonable to view the problem over each instance as a separate but relevant task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta learning has shown good generalization when data distributions shift across tasks, which has empirical evidence in CV and NLP applications (Jeong & Kim, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Conklin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As a summary, we provide a comparison between different unsupervised frameworks to solve CO problems in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that PI-GNN (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) is directly fine-tuned on each test instance without training so the fine-tuning time is long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Also, although PI-GNN also pursues instance-wise good solutions, its performance could be bad because it does not learn from train- ing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The instance-wise solutions could be just bad local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5 EXPERIMENTS We study three CO problems, namely max clique (MC) to find the largest set of nodes where each pair of nodes are connected, minimum vertex covering (MVC) to find the smallest set of nodes that every edge is connected to at least one nodes in the set, and max independent set (MIS) to find the largest set where any two vertices in the set are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Their objectives (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1) and relaxations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2) are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the detailed derivation, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 SETTINGS Datasets: We conduct experiments on the MC, MVC problems over three real datasets Twit- ter (Leskovec & Krevl, 2014), COLLAB and IMDB (Yanardag & Vishwanathan, 2015) and two synthetic datasets with 200 and 500 nodes generated by the RB model (Xu, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We name them RB200 and RB500, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We make RB200 and RB500 extremely hard by setting a small hyper-parameter ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='25 of the RB model (Xu, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The difficulty-ρ relationship on the MVC problem with 500 vertices is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3, where the models are pre-trained on the RB graphs with uniformly sampled ρ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='0] and tested on different RB graphs generated with single ρ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We keep all the other hyper-parameters the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As ρ increases, Meta-EGN and Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 all tend to achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN could outperform Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 in hard instances ρ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='55] while remains a gap on the easy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' To verify performances for data-scale generalization, we also generate large-graph datasets RB1000, RB2000 and RB5000 with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As for the MIS prob- lem, random-regular graphs (RRGs) are often used as benchmarks because they are challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our experiments also use RRGs by following the settings of (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) with the node 6 Preprint Table 3: ApR (time: second/graph) on the MC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ApR is the larger the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘report’ denotes the reported performance in Karalias & Loukas (2020), ‘re-impl’ denotes re-implementation, ‘f-t’ stands for fine-tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Pareto-optimal results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Twitter COLLAB IMDB RB 200 RB 500 EGN (report) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='924 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='133 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='063 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) EGN (re-impl) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='926 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='113(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='069 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='820 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='188 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='829 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='192 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='29) EGN (re-impl) f-t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='949 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='102(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='49) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='986 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='060(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='846 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='180 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='81) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='864 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='181 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='89) RUN-CSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='909 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='145 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='21) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='912 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='188 (0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='748 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='689 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='16) Meta-EGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='976 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='048(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='988 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='059 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='178 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='198 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='29) Meta-EGN f-t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='990 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='028(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='49) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='993 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='038 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='874 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='169 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='81) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='878 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='181 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='89) Toenshoff-Greedy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='917 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='126 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='969 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='087 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='06) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='987 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='050 (1e-3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='195 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='793 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='202 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='38) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='20s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='737 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='267 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='871 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='242 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='04) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (1e-3) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='00s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='37) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='979 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='117 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='06) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (1e-3) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='50s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='37) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='997 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='036 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='06) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (1e-3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='667 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='188 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='663 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='188 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='41) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='00s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='37) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='06) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (1e-3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='755 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='225 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='96) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='742 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='213 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='88) Table 4: ApR (time: second/graph) on the MVC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ApR is the smaller the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='. ‘f-t’ stands for one-step fine-tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Pareto-optimal results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Twitter COLLAB IMDB RB 200 RB 500 EGN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='033 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='023(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='29) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='022 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='15) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='26) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='48) EGN f-t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='021(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='38) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='32) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='030 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='59) RUN-CSP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='435 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='16) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='208 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='198 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='19) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='188 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='178 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='124 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='28) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='062 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='005 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='65) Meta-EGN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='017(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='29) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='003 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='15) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='08) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='26) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='48) Meta-EGN f-t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='017(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='38) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='32) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='016 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='59) Greedy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='014 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='95) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='209 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='198 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='79) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='077 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='124 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='062 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='005 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='59) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='25s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='054 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='09) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='01) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='50s) 1+1e-3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='01) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='00s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='63) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='69) Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (≤2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='00s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='000 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='16) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='68) number ranging from 102 to 105 and the node degree selected from the set D = {3, 7, 10, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, node degree equaling 20 is the hardest setting (Angelini & Ricci-Tersenghi, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A summary of these datasets are in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 10 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Data Splitting & The Evaluation Metric: For the real datasets, training/validation/test instances are randomly divided with the ratio of 8:1:1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For RB200 and RB500, 2000/100/100 graphs are generated for training/validation/test instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For RB1000, RB2000, RB5000, we generate 100 test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As to RRG datasets, the training set contains 3000 RRGs, of which each has 1000 nodes and the node degree is uniformly sampled from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We generate 30/20 graphs for each node degree configuration in D for validation/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Our evaluation metric uses approximation rate (ApR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' All results are summarized based on 5 times independent experiments with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Baselines: Our baselines include unsupervised learning methods, heuristics and traditional CO solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the MC and MVC problems, we take our direct baseline EGN (Karalias & Loukas, 2020), and also take RUN-CSP (Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021) as another baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We do not consider other learning-based methods because they generally perform worse than EGN (Karalias & Loukas, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As to the heuristics, we use the greedy algorithms as the heuristic baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For traditional CO solvers, we compare against the best commercial CO problem solver Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 (Gurobi Opti- mization, 2022) via converting the problems into integer programming form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We track the time t that the models use from the start of inferring to the end of rounding to output feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We set this time t as the time budget of Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 for purely solving the integer programming, and list the actual time usage of Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 which includes pre-processing plus t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As to the MIS problem, we take PI-GNN (Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022) and EGN Karalias & Loukas (2020) as the learning-based baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We take the random greedy algorithm (RGA) and degree-based greedy algorithm (DGA) as introduced in Angelini & Ricci-Tersenghi (2019) as the heuristic baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' When we consider fine-tuning EGN and Meta-EGN over a test instance, we use 1-step gradient descent as fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Implementation: For the MC and MVC problems, we use 4-layer GIN (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019) as the backbone network for both meta-EGN and EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We use 1e-3 as both the outer learning rate (γ) of Meta-EGN and the learning rate of EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, the backbone and the learning rate are same as those in (Karalias & Loukas, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the MIS problem, we use 6-layer GIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The outer learning rate (γ) of Meta-EGN and the learning rate of EGN are set as 1e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The inner learning rate (α) of Meta-EGN is always set as 5e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We run all experiments by using a Xeon(R) Gold 6248 CPU with 7 Preprint Table 5: Scale generalization performance on the MC and MVC problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ApR is the larger the better for MC while the smaller the better for MVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' All the models are trained on RB500 training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘Fast/Medium/Accurate’ denotes GNNs (without fine-tuning) using 1/4/8 random single node seed(s) per testing instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘Fine-tuning’ use 1-step Fine-tuning the best trial among the 8 node seed(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘Gap’ represents the averaged gap defined as c × (# of nodes in the optimal solution - # of nodes by the given method) where c = 1 for MC and c = −1 for MVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘Rank’ is the averaged ranks of solutions among the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Optimal solutions are generated via Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 with the time limit 3000 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Approximation rate for MC larger than 1, highlighted by ∗, indicates the model outperforms Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 solver with 3000s time budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Pareto-optimal results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Dataset Method Fast (1) Medium (4) Accurate (8) Fine-tune ApR(s/g) Gap Rank ApR(s/g) Gap Rank ApR(s/g) Gap Rank ApR(s/g) Gap Rank MC RB1000 EGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='6462±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='282(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='05) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='48 2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='017±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='002(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='40) 26 threads and a Quadro RTX 6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' All codes run on the PyTorch platform (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For more details, see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Overcoming the limited expressive power of GNNs: GNNs are known with limited expressive power (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Specifically over RRGs, the GIN backbone will asso- ciate each node with the same representation, unless node representations are initialized not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' To keep fair comparison, for the MC and MVC problem, we follow Karalias & Loukas (2020) and adopt the initialization based on a single random node seed (one selected node is initialized as 1, others as 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We use 8 single random node seeds for EGN and Meta-EGN in the experiments of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 and report the best among the 8 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We try different numbers of random node seeds in the experiments of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the large-scale MIS problem studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='4, we find such single node initialization is too local to generate valid global solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' So, we adopt initialization based on the solutions of greedy algorithms DGA (for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1,2) and RGA (for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then, EGN and Meta-EGN can be viewed as to learn heuristics to improve the greedy solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that learning heuristics to tune these solutions is non-trivial (Andrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Rahman & Virag, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 META-EGN BOOSTS THE PERFORMANCE WITHOUT DISTRIBUTION SHIFTS We first compare the performances of different methods when the datasets used for training and test are from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 3 and Table 4 show the results for the MC problem and the MVC problem respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In both problems and across the five datasets, Meta-EGN siginificantly outperforms EGN and RUN-CSP, both before and after the fine-tuning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In comparison with the traditional CO solvers, Meta-EGN narrows the gap from Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 on those real small graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For RB graphs, Meta-EGN outperforms Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 for the MC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the MVC problem, Meta-EGN outperforms Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 on RB500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We notice that both EGN and Meta-EGN perform generally well on the MC problem while not as competitive on the MVC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' This results from the initialization of GNN inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The MC problem outputs clusters that are more local while MVC asks for global assignments, which makes such single-seed-based initialization less fit for the MVC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 8 Preprint d=20 d=10 d=7 d=3 Degree of the RRGs with 10^3 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 Approximation rate Random Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^4 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 Approximation rate Random Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^5 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 Approximation rate Random Greedy Meta-EGN Meta-EGN Fine-tune EGN EGN Fine-tune PI-GNN Figure 4: ApRs in the MIS problem on RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN and EGN are both trained with the output of Random Greedy Algorithm (RGA) as initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3 META-EGN BOOSTS THE PERFORMANCE WITH DISTRIBUTION SHIFTS Problem Scale Shift: Here, we use large-scale RB graphs of 1000-5000 nodes to test EGN and Meta-EGN that are trained based on RB500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 5 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Both methods show good generalization while Meta-EGN is always better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As the scale increases, Meta-EGN outper- forms Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For example, it takes Meta-EGN with 4 random initializations only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='02s to beat Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 that runs for 3000 seconds on RB5000 dataset in the MC problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Moreover, Meta-EGN can even outperform Gurobi9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 on the MVC problem when the problem scale be comes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Real-Synthetic Distribution Shift: Here, we train EGN and Meta-EGN on Twitter and test them on RB500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 6 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Compare Table 6 with Tables 3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We observe better generaliza- tion performance of Meta-EGN compared to EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For example, for the MC problem, Meta-EGN has almost the same performance whether there is a dataset shift or not (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='833 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='834 before fine-tuning, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='876 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='878 after fine-tuning) while EGN has a bigger gap in performance when there is a shift (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='819 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='829 before fine-tuning, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='831 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='864 after fine-tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the MVC problem, although the performance drop of Meta-EGN is larger, such a drop is still much smaller than that of EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='4 MAX INDEPENDENT SET: A RESPONSE TO (ANGELINI & RICCI-TERSENGHI, 2022) 10^2 10^3 10^4 10^5 Number of Nodes (degree=20) 10^-2 10^-1 1 10^1 10^2 Time (s/graph) DGA Meta-EGN Fine-tune DGA Meta-EGN DGA Figure 5: Time cost v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Graph Scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' For the MIS problem on large-scale RRGs, Angelini & Ricci-Tersenghi (2022) have recently posted a concern on learning-based methods by arguing that PI-GNN in Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022) could not achieve comparable results with the heuristic algorithm DGA (Angelini & Ricci-Tersenghi, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We see the reason comes from an improper usage of learning-based methods in Schuetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022) as stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1: 1) PI-GNN is trained directly on each single test- ing instance without learning from the training dataset that contains varies graphs, which is likely to be trapped into the local optima;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2) GNN generally suffers from a node ambi- guity issue on RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' To resolve the problem, we utilize the outputs of DGA and RGA as the initialization of GNN inputs (EGN, Meta-EGN) and expect to learn heuristics from historical data to further tune the solutions given by the greedy algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We train GNN models on RRGs with 1000 nodes with node degrees randomly sampled from 3, 7, 10, 20, and test on larger RRGs (up-to 105 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Experiments show that Meta-EGN can further improve DGA (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1) and RGA (in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 4), while EGN fails to better tune DGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that here EGN and Meta-EGN adopt the exactly same backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We attribute the improvement to meta-learning-based training as adopted by Meta- EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' See Table 7 in Appendix for more details of the numerical improvement by Meta-EGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We also see in these cases, one-step fine-tuning does not contribute much to the performance of EGN or Meta-EGN, indicating the model before fine-tuning has been very close to a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We also check the extra time cost by running Meta-EGN to improve DGA solutions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 6 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The extra time cost is just 1% (without fine-tuning) - 30% (with fine-tuning) of the time cost of DGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In theory, the extra time cost without fine-tuning should be O(|E|) for GNN inference plus O(|V |) for rounding, which is in the same order of DGA, while the GNN parallel inference substantially reduces the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 9 Preprint 6 CONCLUSION This work proposes an unsupervised learning framework Meta-EGN with the goal of optimizing NNs towards instance-wise good solutions to CO problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN leverages MAML to achieve the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN views each training instance as a separate task and learns a good initialization for all these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Meta-EGN significantly improves the performance of its baseline and has shown good generalization when the data used for training and test has different scales or distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In addition, Meta-EGN can learn to improve the greedy heuristics while paying almost no extra time cost in the problem of maximum independent set on large-scale random regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 7 ACKNOWLEDGEMENT We would like to express our deepest appreciation to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Tianyi Chen for the insightful discussion on the meta-learning framework from a theoretical aspect and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Ruqi Zhang for the constructive advice on the fine-tuning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We would also like to extend our deepest gratitude to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hanjun Dai and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Jialin Liu for sharing their invaluable insights into the general ideas of learning for combinatorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Also many thanks to our fundings, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Wang and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Li are partially supported by 2021 JPMorgan Faculty Award and the NSF award OAC-2117997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' REFERENCES Saeed Amizadeh, Sergiy Matusevych, and Markus Weimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Learning to 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' cn/kexu/benchmarks/graph-benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' htm, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' How powerful are graph neural networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In International Conference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Kenichi Yabuta and Hitoshi Kitazawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Optimum camera placement considering camera specification for security monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In 2008 IEEE International Symposium on Circuits and Systems (ISCAS), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2114–2117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' IEEE, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Pinar Yanardag and SVN Vishwanathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Deep graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1365–1374, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Weichi Yao, Afonso S Bandeira, and Soledad Villar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Experimental performance of graph neural networks on random instances of max-cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In Wavelets and Sparsity XVIII, volume 11138, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 242–251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' SPIE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gal Yehuda, Moshe Gabel, and Assaf Schuster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' It’s not what machines can learn, it’s what we cannot teach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In International conference on machine learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 10831–10841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Emre Yolcu and Barnab´as P´oczos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Learning local search heuristics for boolean satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Ad- vances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 12 Preprint A PROOF OF THEOREM 1 We first prove Theorem 1, then we specify the value of α to obtain Theorem 2 as a specific case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The proof of Theorem 1 is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In part 1, we prove that if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) < β + △ (even if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≥ β), for any α ∈ (0, 2/L) Meta-GNN with one-step finetuning outputs a feasible solution X of good quality f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, △ = ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ + 1 2Lα2−4αϵ2 if ϵ < α∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ or △ = (α − Lα2 2 )∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='. In part 2, we prove that once Meta-EGN achieves the loss value l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) after the one-step finetuning, the rounding process would output a feasible X whose objective satisfies f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Part 1:We could get l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) (a) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + ∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)(θ′ − θ) + 1 2L∥θ′ − θ∥2 2 (b) = l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + 1 2L∥θ′ − θ∥2 2 − α∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 2 = l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + (Lα2 2 − α)∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 2, (6) where (a) is due to the local L-smoothness of l(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G), (b) is due to the definition of one-step finetuning θ′ = θ − α∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' If ϵ < α∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥: Let △ = ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ + 1 2Lα2−4αϵ2, we have: min ϵ −△ = min ϵ − 1 2Lα2 − 4αϵ2 − ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ = (Lα2 2 − α)∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2, (7) thus l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (8) If ϵ ≥ α∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥: Let △ = (α − Lα2 2 )∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2, we would directly have: l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (9) By this, we finish the first part of the proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Part 2: The proof in this part follows the rounding analysis in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Consider the rounding procedure from continuous space ¯X = Aθ(G), ¯X ∈ [0, 1]n into the discrete feasible solution X ∈ {0, 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Let ¯Xi, Xi, i = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', n} denote their entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='g, suppose the rounding order is from 1 to n and we have finished the rounding before the t-th node, we now analyze the rounding of t-th node: fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, ¯Xt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, ¯Xt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) (d) ≥ ¯Xt(fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, 1, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, 1, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)) + (1 − ¯Xt)(fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, 0, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, 0, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)) ≥ ¯Xt( min jt={0,1} fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, jt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, jt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)) + (1 − ¯Xt)( min jt={0,1} fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, jt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, jt, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)) (e) =fr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, Xt, ¯Xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βgr([X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, Xt, ¯Xt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) (10) where (d) is due to lr(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)’s entry-wise concavity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t ¯X and Jensen’s inequality, (e) is due to Xt = arg minj=0,1 fr(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, t, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn) + βgr(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Xt−1, t, ¯Xt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ¯Xn) (the 13 Preprint definition of our rounding process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The loss value is monotonically non-increasing through the whole rounding process according to the equation above, thus we could get: l(θ′) ≥ f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) + βg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) (11) By this, we finish the proof of the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 A SPECIFIC CASE Note that in the first part of the proof above, if we specify the value of α as 1 L in equation (6), we could have: l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 2 2L (12) If ϵ < 1 L∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥: Let △ = ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ − L 2 ϵ2, we have: min ϵ −△ = min ϵ L 2 ϵ2 − ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ = −∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 2L , (13) thus l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (14) If ϵ ≥ 1 L∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥: Let △ = 1 2L∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2, we would directly have: l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (15) By this, we obtain Theorem 2, a specific case of Theorem 1 as follows: Theorem 2 (A Specific case of Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Suppose the relaxations fr and gr are entry-wise con- cave as required in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Let θ denote the learned parameter after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Given a test instance G, suppose locally l(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) is L-smooth at θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', ∥∇θ′l(θ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)−∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ ≤ L∥θ′−θ∥ for all θ′ that satisfies ∥θ′ − θ∥ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then, if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) < β + △ (even if l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≥ β), there ex- ists α such that Meta-GNN with one-step finetuning outputs a feasible solution X of good quality f(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≤ l(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) − △.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Here, △ = ∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ϵ − L 2 ϵ2 if ϵ < 1 L∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥ or △ = 1 2L∥∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G)∥2 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='. B SUPPLEMENTARY EXPERIMENT RESULTS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 SUPPLEMENTARY TIME COST V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' GRAPH SCALE IN THE MIS we show the degree 3, 7, 10 in the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' They show the same time-cost vs scale relation as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The extra time cost of GNN is O(|E|) for inference plus O(|V |) for rounding, which is in the same order of DGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 10^2 10^3 10^4 10^5 Number of Nodes (degree=3) 10^-2 10^-1 1 10^1 10^2 Time (s/graph) DGA Meta-EGN Fine-tune DGA Meta-EGN DGA 10^2 10^3 10^4 10^5 Number of Nodes (degree=7) 10^-2 10^-1 1 10^1 10^2 Time (s/graph) DGA Meta-EGN Fine-tune DGA Meta-EGN DGA 10^2 10^3 10^4 10^5 Number of Nodes (degree=10) 10^-2 10^-1 1 10^1 10^2 Time (s/graph) DGA Meta-EGN Fine-tune DGA Meta-EGN DGA Figure 6: Time cost v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Graph Scales on degree 3, 7, 10 14 Preprint Table 7: Improvement of Meta-EGN over DGA and RGA in the MIS on RRGs, ‘Imp in ApR’ denotes the average improvement in approximation rate and ‘Imp in #Node’ denotes the average number of nodes that Meta-EGN could find more than the heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Scale/Degree 3 7 10 20 Imp in ApR Imp in #Node Imp in ApR Imp in #Node Imp in ApR Imp in #Node Imp in ApR Imp in #Node Meta-EGN improves DGA by 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='0043 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='950 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1083 3088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='0969 1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='030 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 HOW MUCH DOES META-EGN MODIFY DGA AND RGA HEURISTICS IN THE MIS We display the average approximation rate improvement and the average node number increase by Meta-EGN over DGA and RGA in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3 TRAINING THE MODELS ON SUBSETS OF THE TRAINING DATA We display the average approximation rates of the models that are only trained on subsets of the original training data in the max clique problem on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The training dataset is randomly sampled from the original training dataset and the testing dataset remains the same as that in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Both the methods have worse performance as the number of training instances reduces, while Meta-EGN only has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='7% performance decrease from the full-size training dataset with 750 samples to the training subset with only 64 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In contrast, EGN decreases its performance by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 8: The approximation rate of the max clique problem on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Models are only trained on subsets of the dataset, ‘training subset’ denotes the number of instances in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' training subset 64 128 256 512 Full (750) EGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='909±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='911±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='914±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='922±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='926±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='113 Meta-EGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='970±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='973±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='975±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='975±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='976±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='048 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='4 TRAINING THE MODELS ON RRGS WITH SINGLE DEGREES IN THE MIS We train the EGN and Meta-EGN models on RRGs with only 3 or 20 degrees and test them on RRGs with the rest degrees from {3, 7, 10, 20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Models take the output of DGA as the initialization graph node feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We show the performance of both the models without fine-tuning in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' When only trained on RRGs with degree 3 (See the left two figures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 7), both the models could not generalize well, as neither of them could outperform the initialization input of DGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Note that Meta-EGN still achieves better performance than EGN in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As to the models only trained on RRGs with degree 20 (See the right two figures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 7), we observe that both the model have relatively good generalization ability across different degrees, yet Meta-EGN could still marginally outperform EGN in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We attribute this phenomenon to the fact that solving the MIS on RRGs with degree 20 is much more complicated than those with degree 3 and thus may contain adequate heuristics for solving RRGs with lower degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' d=20 d=10 d=7 d=3 Degree of the RRGs with 10^3 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='95 Approximation rate Degree-based Greedy EGN Meta-EGN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^4 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='95 Approximation rate Degree-based Greedy EGN Meta-EGN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^3 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='94 Approximation rate Degree-based Greedy EGN Meta-EGN d=20 d=10 d=7 d=3 Degree of the RRGs with 10^4 nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='95 Approximation rate Degree-based Greedy EGN Meta-EGN Figure 7: The left two figures show the ApRs on RRGs with 103 and 104 nodes of the models trained only on RRGs with degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The right two figures show the ApRs on RRGs with 103 and 104 nodes of the models trained only on RRGs with degree 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 15 Preprint B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='5 COMPARISON ON THE TRAINING TIME OF THE MODELS We display the wall clock training time for the two methods to convergence in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 9 (from start to the the best epoch on validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We observe that Meta-EGN generally takes two to three times to converge compared with EGN, but their training time cost basically remain on the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 9: The wall clock training time to convergence of EGN and Meta-EGN in different problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Dataset/Time (min:second) MC MVC MIS Twitter RB200 RB500 Twitter RB200 RB500 RRGs EGN 46:50 104:37 282:57 100:58 83:27 128:39 733:02 Meta-EGN 101:55 210:04 609:47 276:38 168:25 282:15 1088:55 C SUPPLEMENTARY IMPLEMENTATION DETAILS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1 EXPERIMENT DETAILS All the codes run on the PyTorch platform 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='0 (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2019) and PyTorch Geometric frame- work 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 (Fey & Lenssen, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The details of each datasets is shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 10, all of the real datasets are publicly available, we follow the code in (Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2021) to generate the RB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Table 10: The number of instances in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' ‘20/scale/degree’ means that we generate 20 testing instances for each different scale-degree pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We generate RB1000, RB2000 and RB5000 only for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Dataset Twitter COLLAB IMDB RB200 RB500 RB1000 RB2000 RB5000 RRGs Training 750 3600 800 2000 2000 3000 Validation 100 450 100 100 100 30 Testing 100 450 100 100 100 100 100 100 20/scale degree To balance the training time per epoch of EGN and Meta-EGN, we define the epoch as follows: For each epoch of EGN training, the whole dataset is split into mini-batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' EGN performs standard mini-batch training along these batches and optimizes over each mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' As to Meta-EGN, for each training epoch Meta-EGN only randomly samples a single batch and do the meta learning algorithm on the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The batch sizes of the methods are controlled the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='2 DETAILED DERIVATION OF THE LOSS FUNCTION RELAXATION In this part, we display the detailed loss function relaxation of the three problems in our study (the MC, the MVC and the MIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The basic idea of training loss design and relaxation follow (Karalias & Loukas, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In the following derivation, we use i, j to represent the nodes in graphs, we use Xi, Xj ∈ {0, 1} to denote the discrete assignment of the binary optimization vari- ables, and we use ¯Xi, ¯Xj ∈ [0, 1] to denote the relaxed soft assignment of the binary optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The maximum clique (MC): A clique is a set of nodes S ∈ V such that any two distinct nodes in the set are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The MC aims to find out the clique with the largest number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We could formulated the optimization objective as follows: max X � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (i, j) ∈ E if Xi, Xj = 1, (16) Xi, Xj denotes whether to take the node into the clique set (Xi = 1) or not (Xi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' By setting a proper penalty coefficient β, we could formulate the loss function relaxation as follows (the detailed 16 Preprint derivation follows the corresponding case study in Karalias & Loukas (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' lMC(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ −(β + 1) � (i,j)∈E ¯Xi ¯Xj + β 2 � i̸=j ¯Xi ¯Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (17) The minimum vertex covering (MVC): A vertex cover is a set of nodes S ∈ V that any edge in the graph is connected to at least a node from the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The MVC aims to find out the cover set with the smallest number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The optimization objective could be summarized as follows: min X � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Xi + Xj ≥ 1 if (i, j) ∈ E, (18) where Xi, Xi denotes whether to take the node into the cover set (Xi = 1) or not (Xi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We design the constraint function g to represent the total number of edges that have not been covered given a set of variable assignment X, and thus we write g as: gMVC(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ � (i,j)∈E (1 − Xi)(1 − Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (19) Then we relax the constraint g and add it into the training objective by multiplying a proper penalty coefficient β, following the relaxation principle in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2022): lMVC(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ � 1≤i≤n ¯Xi + β � (i,j)∈E (1 − ¯Xi)(1 − ¯Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (20) By this, we aim to minimize the value of the loss function above in order to minimize the node number of the cover set as well as consider the covering property in the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The maximum independent set (MIS): An independent set is a set of nodes where any two distinct nodes in the set are not adjacent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The MIS aims to find out the independent set with the largest number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We could formulate the objective of the MIS as follows: max X � 1≤i≤n Xi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' XiXj = 0 if (i, j) ∈ E, (21) where Xi, Xj denotes whether to take the node into the independent set (Xi = 1) or not (Xi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We formulate the constraint g as the total number of edges whose two connected nodes at the end- points are both assigned into the independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Therefore we could write the constraint as follows: gMIS ≜ � (i,j)∈E XiXj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (22) We then relax the constraint g into continuous space and add it into the c function with a proper penalty coefficient β, following the relaxation principle in (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', 2022), and thus we could write the training loss function as: lMIS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G) ≜ − � 1≤i≤n ¯Xi + β � (i,j)∈E ¯ Xi ¯ Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (23) By this, we aim to minimize the value of the loss function above in order to maximize the node number of the independent set as well as consider the independent property in the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='3 SEPARATED ALGORITHM TABLES We separate the algorithm table of Meta-EGN into training and testing parts to make it clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The algorithm table is shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 2 for training and Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='4 IMPLEMENTATION OF THE HEURISTICS We run all of the greedy algorithms with PyThon 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='8 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' A potential method to boost the time cost of these greedy algorithm is to use c++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 17 Preprint Algorithm 2 Train Meta-EGN Require: Training instances Ξ = {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', Gm};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hyperparameters: α, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1: Randomly initialize θ(0) 2: for each randomly sampled mini-batch Bj ⊂ Ξ, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=', K − 1 do ▷ Training starts 3: For each Gi ∈ Bj, compute the adapted parameter: θ(j) i = θ(j) − α∇θ(j)l(θ(j);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) 4: Update: θ(j+1) ← θ(j) − γ∇θ(j) � Gi∈Bj l(θ(j) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Gi) 5: end for 6: return θ ← θ(K) ▷ Training ends Algorithm 3 Test Meta-EGN with/without Fine-tuning Require: Testing instance G′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Hyperparameter: α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Pre-trained parameter initialization θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1: For a given testing instance G′: ▷ Testing starts 2: if fine-tuning is allowed then 3: Fine-tune the parameters: θG′ ← θ − α∇θl(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' G′) 4: Use Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 to round the relaxed solution given by AθG′ (G′) ▷ With fine-tuning 5: else 6: Use Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 1 to round the relaxed solution given by Aθ(G′) ▷ Without fine-tuning 7: end if 8: return the rounded solution ▷ Testing ends Random Greedy Algorithm for MIS (RGA): RGA takes a time to reach a solution that is linear in the problem size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' It starts from an empty independent set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' At each step 1 ≤ t ≤ n, a node i is chosen at random from the graph Gt and added to the independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then all the neighbors of i are removed from Gt to formulated a new graph Gt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The process iterates until Gt∗ is empty at step t∗, the solution is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Degree-based Greedy Alforithm for MIS (DGA): DGA modifies RGA by sorting the degrees of the nodes before each iteration starts, and always put the node with the smallest degree into the independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Teonshoff Greedy for MC: Toenshoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' (2021) convert the testing instances into its complement graph, and then run DGA to solve the MIS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' It takes the solution to the MIS problem on the complement graph as the solution for MC on the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Greedy for MVC: Greedy for MVC starts from an empty covering set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' At each step 1 ≤ t ≤ n, it first sorts the degrees of the nodes in the graph Gt and always add the node i with the largest degree into the covering set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' Then all the edges that connect with i are removed from Gt to formulate a new graph Gt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The process stops until Gt∗ is empty at step t∗, the solution is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' D DISCUSSION ON LIMITATIONS As mentioned in the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='1, both EGN and Meta-EGN perform generally well in MC, which outputs the cliques that are more local in comparison with the vertex covering in MVCs that require more global assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' The random initialization seed with one node randomly set as 1 and the others as 0 would potentially limit the performance of EGN and Meta-EGN in more global CO tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We use Meta-EGN and EGN to modify the solution of DGA and RGA in the MIS problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' In addition, there are also many other Monte Carlo (MC) algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' simulated annealing and parallel tempering) that could produce better results than DGA or RGA in RRGs (Angelini & Ricci- Tersenghi, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' An intuitive idea is to test whether we could learn Meta-EGN to further fine-tune these more advanced MC algorithms in the MIS problem on RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' We leave the research on modifying Meta-EGN to better deal with the CO problems that require global assignments and using Meta-EGN to improve other advanced MC algorithms as a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfWgTb/content/2301.03116v1.pdf'} diff --git a/FdFKT4oBgHgl3EQfay5f/content/tmp_files/2301.11809v1.pdf.txt b/FdFKT4oBgHgl3EQfay5f/content/tmp_files/2301.11809v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..10eb6315dbdc33dc0cacac86b2fbcfb583462ee1 --- /dev/null +++ b/FdFKT4oBgHgl3EQfay5f/content/tmp_files/2301.11809v1.pdf.txt @@ -0,0 +1,2444 @@ +1 + +Quantization of Fractional Singular Lagrangian systems with +Second-Order Derivatives Using Path Integral Method + +Eyad Hasan Hasan + +Tafila Technical University, Faculty of Science, Applied Physics Department, P. O. Box: 179, +Tafila 66110, Jordan + +Abstract + The fractional quantization of singular systems with second order Lagrangian is +examined. The fractional singular Lagrangian is presented. The equations of motion are written +as total differential equations within fractional calculus. Also, the set of Hamilton–Jacobi partial +differential equations is constructed in fractional form. The path integral formulation and path +integral quantization for these systems are constructed within fractional derivatives. We +examined a mathematical singular Lagrangian with two primary first-class constraints. +. + +Keywords: Path Integral Quantization, Fractional Calculus, Fractional Hamilton-Jacobi Function, +Singular Lagrangians, +PACS numbers: 11. 10. Ef, 45. 20. –j, 45. 10. Hj, 04.20. Fy +CONTENTS +1. Introduction 2 +2. Second-order fractional singular Lagrangian and quantization using fractional path integral +method 3 +3. Examples 7 +4. Conclusions 10 +References + +2 + + + +Email: iyad973@yahoo.com, dr_eyad2004@ttu.edu.jo +1. Introduction +The efforts to quantize singular Lagranians systems have been studied with increasing interest +and treated first by Dirac [1, 2], for quantizing the gravitational field, then his formalism has +been developed using a new formalism for investigating singular systems- the canonical- was +developed by Guler [3]. In this formalism, the equations of motion are written in fractional form +as total differential equations and the set of fractional Hamilton–Jacobi partial differential +equations is constructed. Then, this formalism has been used for quantizing singular Lagrangian +systems using the WKB approximation and path integral approach [4-8]. +Fractional calculus with singular systems had treated with more interests and importance [9- 18]. +Recently, the Euler-Lagrange equations for second-order Lagrangian systems are analyzed +within fractional derivatives and the fractional Hamilton-Jacobi formalism for these systems are +discussed [15, 16]. More recently, authors have constructed a formalism using the canonical +method for quantizing singular systems using the WKB approximation and path integral +approach for first-order derivatives [17, 18]. In this paper, we would like to extend our work to +Lagrangians with second-order derivatives. + Now, we will present the most important definitions of fractional derivatives [9]. +(i) The left Riemann–Liouville fractional derivative + + + + + + + +d +f +t +dt +d +n +t +f +D +t +a +n +n +t +a +) +( +) +( +) +( +1 +) +( +1 + + + + + + + + + + + + + +. (1) +(ii) The right Riemann–Liouville fractional derivative + + + + + + + +d +f +t +dt +d +n +t +f +D +b +t +n +n +b +t +) +( +) +( +) +( +1 +) +( +1 + + + + + + + + + + + + + +. (2) +where +N +n +, + + +1 +n + ˂ n and  is the Euler gamma function. Here  is an integer and these +derivatives can be defined as follows: + +3 + +) +( +) +( +t +f +dt +d +t +f +Dt +a + + + + + + + + + +, +) +( +) +( +t +f +dt +d +t +f +Db +t + + + + + + + + + +, (3) +Definition Given a function f : + + +) +,0 +[ +. Then for all t >0 , +)1,0 +( + + +, let + + + + + +) +( +) +( +lim +) +)( +( +1 +0 +t +f +t +t +f +t +f +D + + + + + +. (4) + +D is called the conformal fractional derivative of f of order of  [17]. +In this work, we aim to construct the formalism for quantizing singular Lagrangians systems +with second-order derivatives within framework of fractional derivatives. This paper is organized +as follows: In section 2, we will investigate the fractional singular Lagrangian and fractional path +integral approach. In section3, one illustrative example is examined. The work closes with some +concluding remarks in section 4. +2. Second-order fractional singular Lagrangian and quantization using fractional path +integral method + In this section, we will present a formalism for second-order singular Lagrangian and +canonical path integral approach within of fractional derivatives. We will start with a Lagrangian +depending on the fractional derivatives is given by + +) +, +, +, +( +2 +1 +t +q +D +q +D +q +D +L +L +i +i +i + + + + + . (5) +Where +iq +D + are the conformal fractional derivatives of the coordinates +iq [17]. +The Lagrangian and Hamiltonian formalism for second-order derivatives have been studied by +Ostrogradski [ 19] and the derivatives have been treated as coordinates. Thus, we can treat the +derivatives +iq +D +1 + + + and +iq +D + as coordinates. Therefore, the Poisson brackets can be defined +as + + +i +i +i +i +i +i +i +i +q +D +B +A +B +q +D +A +q +D +B +p +A +p +B +q +D +A +B +A + + + + + + + + + + + + + + + + + + + + + + + + + + + + +1 +1 +, +. + (6) +Here, the functions A and B are described in term of the canonical variables +iq +D +1 + + +, +iq +D +, +ip and +i + . Thus, the generalized momenta +ip and +i + are conjugated to the generalized + +4 + +coordinates +iq +D +1 + + + and +iq +D + respectively. Thus, one can write the fundamental Poisson +brackets as: + + +ij +j +i p +q +D + + + + +, +1 + and  + +ij +j +iq +D + + + + +, +, + +  + + +  + +i +i +j +i +j +i +j +i +p +q +D +q +D +q +D +q +D +q +D +q +D + + + + + + + +, +, +0 +, +, +1 +1 +1 + + + + + + + +, +where +N +j +i +,..., +1 +, + + + (7) +Now, the fractional of the Hessian matrix is defined as +j +i +ij +q +D +q +D +L +W + + +2 +2 +2 + + + + + + (8) +The fractional Lagrangian is called regular if it’s rank is N otherwise the Lagrangian is singular +, +R +N  + R < N . Thus, we can define the generalized momenta +i + corresponding to the +generalized coordinates +iq +D + as: +a +a +q +D +L + + +2 + + + +, +R +N +a + + +,..., +2.1 + (9) + + + + +q +D +L +2 + + + + . +N +R +N +,..., +1 + + + + + . (10) +Dirac showed in his formalism for investigating singular Lagrangian systems that the number of +degrees of freedom can be reduced from N to N-R due to the constraints [1,2]. + +Since the rank of the Hessian matrix is N-R, we can solve Eq. (9) to obtain N-R +accelerations +a +q +D  +2 + in terms of +a +i +i +q +D +q +D + + + +, +, +1 + + and + +q +D2 +as follows: +) +, +, +, +( +2 +1 +2 + + + + + + +q +D +q +D +q +D +w +q +D +a +i +i +a +a + + +. (11) +Substituting (10) in (9), we can obtain +. + (12) +) +, +, +, +( +1 +a +a +i +i +p +q +D +q +D +H + + + + + + + + + + + +5 + + We can obtain a similar expression for the momenta + +p and can be defined as: + (13) + + + + + + +Also, we can define the generalized momenta +ip corresponding to the generalized coordinates +iq +D +1 + + + as: + + + + + + + + + + + + +a +a +a +q +D +L +dt +d +q +D +L +p + + +2 +; (14) + + + + + + + + + + + + + + + + + +q +D +L +dt +d +q +D +L +p +2 +. (15) +We can define the fractional Hamiltonian + +H as + + + + + + + + + + + + +H +q +D +H +q +D +W +q +D +p +W +q +D +q +D +q +D +L +H +p +a +a +a +a +a +v +v +i +2 +2 +1 +) +, +, +, +( + + + + + + + + +. (16) + +R +,......, +1 + + +; +N +R +a +,..., +1 + + +. +Equations (12) and (13) become +; + (17) + + + + (18) +Also, equations (17) and (18) represent primary constraints [1, 2]. A natural of singular +Lagrangian indicates that the generalized momenta + +p and + + are not independent of +a +p and +a + . +Thus, we can write the set of Hamilton-Jacobi partial differential equations as +0 +, +, +, +, +( +1 +1 +1 + + + + + + + + + + + + + + + + + +v +a +v +a +i +i +q +D +S +q +D +S +q +D +S +q +D +S +q +D +q +D +H + + + + + + + +, (19) +,0 + + + +N +R +N +,..., +1 + + + +) +, +, +, +( +1 +a +a +i +i +p +p +q +D +q +D +H +p + + + + + + + + +0 +) +, +, +, +( +1 + + + + + +p +i +i +i +i +p +H +p +p +q +D +q +D +H + + + + + + +0 +) +, +, +, +( +1 + + + + + + + + + + + + + + +H +p +q +D +q +D +H +i +i +i +i + +6 + +Here, the fractional Hamilton's principle function is written as +) +, +, +, +, +( +1 +1 +t +q +D +q +D +q +D +q +D +S +S +a +a + + + + + + + + + + , we define +, +1 +a +a +q +D +S +p + + + + + + + + + +q +D +S +p +1 + + + + + , +a +a +q +D +S + + + + + +, + + + + +q +D +S + + + +and +. +t +S +p + + + + + +Thus, the action function and the equations of motion in fractional form can be written as total +differential equations as follows: +, +1 +1 + + + + + + + + +q +dD +p +H +q +dD +p +H +dt +p +H +q +dD +a +a +p +a +a + + + + + + + + + + + + + + + + (20) +, +1 + + + + + + + + + + + +q +dD +H +q +dD +H +dt +H +q +dD +a +a +p +a +a + + + + + + + + + + + + + + + (21) +, +1 +1 +1 +1 + + + + + + + + + + +q +dD +q +D +H +q +dD +q +D +H +dt +q +D +H +dp +i +i +i +i +p + + + + + + + + + + + + + + + + + + + (22) +, +1 + + + + + + + + + + + +q +dD +q +D +H +q +dD +q +D +H +dt +q +D +H +d +i +i +i +i +p + + + + + + + + + + + + + + + + (23) +. +) +( +) +( +) +( +1 + + + + + + + + + + + + + + + + + + + +q +dD +H +p +H +p +H +q +dD +H +p +H +p +H +dt +H +p +H +p +H +dS +a +a +a +a +a +p +a +a +p +a +p +a +a +a +a + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + (24) +If the total derivative of equation (19) is zero [3] +0 + + +H +d +; (25) +0 + + p +H +d + +; (26) +.0 + + + +H +d + (27) +This indicates that equations (20-24) are integrable, and the rank of Hessian matrix is +R +N  + . +Thus, the degrees of freedom are reduced from N to +R +N  +, the constraints reduce the canonical +phase space coordinates from +} +, +, +, +{ +1 +i +i +i +i +q +D +p +q +D + + + +to +} +, +, +, +{ +1 +a +a +a +a +q +D +p +q +D + + + +. Therefore, we can +represent the path integral approach for singular systems in the fractional form as + + + +7 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +q +dD +H +p +H +p +H +q +dD +H +p +H +p +H +dt +H +p +H +p +H +i +d +dp +q +dD +q +dD +t +q +D +q +D +q +D +q +D +K +a +a +a +a +a +p +a +a +p +a +p +a +a +a +a +R +N +a +a +a +a +a +a +a +1 +1 +1 +1 +1 +exp +) +, +, +, +, +( + + + + +(28) + + +N +i +,..., +2.1 + +, +R +N +a + + +,..., +2.1 +, +N +R +N +,..., +1 + + + + +. +4. Example +Second-Order Fractional Singular Lagrangian +Let us consider the following mathematical singular Lagrangian with two primary first-class +constraints. + + + +. +) +( +) +( +2 +1 +2 +2 +1 +3 +1 +3 +3 +2 +3 +2 +2 +2 +2 +2 +q +D +q +D +q +D +q +D +q +D +q +D +q +D +q +D +L + + + + + + + + + + + + + + + + (29) +The corresponding generalized momenta, Eqs. (9, 10) and (14, 15) are + +1 +3 +1 +q +D +p + + + +; (30) + +2 +3 +2 +1 +2 +q +D +q +D +p + + + + + +; (31) + +p +H +q +D +p +3 +3 +1 +3 + + + + + +; (32) + +1 +2 +1 +q +D  +  +; (33) + +2 +2 +2 +q +D  + + +; (34) + + + + +3 +3 +3 +H +q +D + + + +. (35) +Here, Equations (32) and (35) can be written as +0 +3 +1 +3 +3 + + + + + q +D +p +H +p + +; (36) + +8 + +0 +3 +3 +3 + + + + +q +D +H + + + +. (37) +and represent as primary constraints [1,2]. +The Hamiltonian +0 +H is calculated as + +). +( +2 +1 +) +( +2 +2 +2 +1 +2 +2 +1 +2 +1 +1 + + + + + + + + + + + +q +D +q +D +p +q +D +p +H  + (38) +The corresponding set of HJPDEs, Eqs. (19), reads +). +( +2 +1 +) +( +2 +2 +2 +1 +2 +2 +1 +2 +1 +1 + + + + + + + + + + + + + + + +q +D +q +D +p +q +D +p +p +H +p +H + + + + + (39) + +0 +3 +1 +3 +3 + + + + + q +D +p +H +p + +; (40) + +0 +3 +3 +3 + + + + +q +D +H + + + +. (41) +Here, the Poisson brackets  + +0 +, +3 + + + + +H +H +p +,  + +0 +, +3 + + + + +H +H  + and  + +0 +, +3 +3 + + + + +H +H +p +. There are no +secondary constraints, this means that these are first-class constraints [1, 2]. +The equations of motion Eqs. (20- 24) can be calculated as +, +1 +1 +1 +dt +q +D +q +dD + + + + + (42) +, +2 +2 +1 +dt +q +D +q +dD + + + + + (43) +, +1 +1 +dt +q +dD + + + + (44) +, +2 +2 +dt +q +dD + + + + (45) +,0 +1  +dp + (46) +, +2 +2 +dt +q +D +dp + + + + + (47) +, +3 +1 +3 +q +dD +dp + + + + + + (48) +, +1 +1 +dt +p +d + +  + (49) +, +) +( +2 +1 +2 +2 +dt +q +D +p +d + + + + + + + (50) + +9 + +, +3 +3 +q +dD +d + + + + + + (51) +3 +3 +3 +1 +3 +2 +2 +1 +2 +2 +2 +1 +) +2 +1 +. +2 +1 +( +q +dD +q +dD +p +dt +q +D +q +D +dS + + + + + + + + + + + + + + + (52) +3 +3 +3 +1 +3 +1 +2 +2 +1 +2 +2 +2 +1 +) +2 +1 +. +2 +1 +( +q +dD +q +D +q +dD +q +D +dt +q +D +q +D +dS + + + + + + + + + + + + + + + + +. (53) +3 +3 +3 +1 +3 +1 +2 +2 +1 +2 +2 +2 +1 +) +2 +1 +. +2 +1 +( +q +dD +q +D +q +dD +q +D +dt +q +D +q +D +S + + + + + + + + + + + + + + + + + +. (54) +Finally, By obtaining the fractional action function S , we can represent the path integral +approach in fractional form as + + + + + + + + + + + + + + + + + + + + + + + +3 +3 +3 +1 +3 +1 +2 +2 +2 +1 +2 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +1 +1 +3 +2 +1 +3 +1 +2 +1 +1 +1 +) +2 +2 +( +exp +) +, +, +, +, +, +, +( +q +dD +q +D +q +dD +q +D +dt +q +D +q +D +i +d +d +dp +dp +q +dD +q +dD +q +dD +q +dD +t +q +D +q +D +q +D +q +D +q +D +q +D +K + + + + + + + + + + + + + + + + + + + + +. (55) +Author Contributions +E H wrote the main manuscript text. +Funding +Funding information is not applicable/No funding was received. +Availability of data and materials + Data sharing not applicable to this paper. +Declarations +Conflict of interest +There is no conflict of interests with regards to the publication of this paper. + +Conclusion +In this work, we constructed a formalism for quantizing singular Lagrangian systems using path +integral approach within fractional calculus. We wrote the equations of motion and action +function in fractional form as total differential equations, besides the path integral approach is +constructed within fractional derivatives. Then, we discussed a mathematical example to +demonstrate our formalism. + + + + +10 + +References +[1] Dirac, P. A. M. 1950. Generalized Hamiltonian Dynamics. Canadian Journal of +Mathematical Physics, 2, 129-148. +[2] Dirac, P. A. M. 1964. Lectures on Quantum Mechanics, Belfer Graduate School of +Science, Yeshiva University, New York. +[3] Guler, Y. 1992b. Canonical Formulation of Singular Systems. IL Nuovo Cimento B, 107 +(10), 1143-1149. +[4] Muslih, S. I. 2001. Path Integral Formulation of Constrained Systems with Singular Higher- +Order Lagrangians. Hadronic Journal, 24, 713-721. +[5] Pimentel, R.G. and Teixeira 1996. Hamilton-Jacobi Formulation for Singular Systems with +Second-Order Lagrangians. IL Nuovo Ciemento B, 111, 841-854. +[6] Rabei, E. M., Hasan, E. H., and Ghassib, H. B (2004). Hamilton-Jacobi Treatment of +Constrained Systems with Second-Order Lagrangians, International Journal of Theoretical +Physics, Vol. 43 N0. (4), 1073-1096. +[7] Rabei, E. M., Hasan, E. H., and Ghassib, H. B. Quantization of Second-Order Constrained +Lagrangian Systems Using the WKB Approximation, International Journal of Geometric +methods in Modern Physics 2005, Vol. 2. P 485-504. + [8] Hasan, E. H., Rabei, E. M., and Ghassib, H. B. Quantization of Higher-Order Constrained +Lagrangian Systems Using the WKB Approximation. International Journal of Theoretical +Physics 2004, Vol. 43 N0. 11 p 2285-2298. +[9] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory +and Applications, Gordon and Breach, 1993. +[10] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E, 53 +(1996), 1890-1899. +[11] E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih and D. Baleanu, The Hamilton +formalism with fractional derivatives, Journal of Mathematical Analysis and Applications, +327 (2007), no. 2, 891-897. +[12] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational +problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. + +11 + + [13] E.H. Hasan, Fractional Variational Problems of Euler-Lagrange Equations with Holonomic +Constrained Systems, Applied Physics Research, 8 (2016), no. 3, 60-65. + [14] E. H. Hasan, Fractional Quantization of Holonomic Constrained Systems Using Fractional +WKB Approximation, Advanced Studies in Theoretical Physics, Vol. 10, No. 5, P 223-234, +2016. +[15] E. H. Hasan and J. H. Asad, Remarks on Fractional Hamilton-Jacobi Formalism with +second-order Discrete Lagrangian Systems, Journal of Advanced Physics, Vol. 6, No. 3, P430- +433, 2017 + [16] E. H. Hasan, On Fractional Solution of Euler-Lagrange Equations with Second-Order +Linear Lagrangians ". Journal of Advanced Physics, Vol. 7, No. 1, P110-113, 2018 +[17] E.M. Rabei and M. Alhorani, Quantization of fractional singular Lagrangian systems using +WKB approximation, international Journal of Modern Physics A, Vol. 33 (2018), no. 36, +1850222-1-1850222-9. +[18] E. H. Hasan, Path Integral Quantization of Singular Lagrangians using Fractional +Derivatives, International Journal of Theoretical Physics 2020, Vol. 59 pages 1157–1164 +[19] M. Ostrogradski: Mem. Ac. St. Petersbourg, 1 (1850) 385 + + diff --git a/FdFKT4oBgHgl3EQfay5f/content/tmp_files/load_file.txt b/FdFKT4oBgHgl3EQfay5f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..925af41002270e73d7913f39b73ba30896c29fac --- /dev/null +++ b/FdFKT4oBgHgl3EQfay5f/content/tmp_files/load_file.txt @@ -0,0 +1,458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf,len=457 +page_content='1 Quantization of Fractional Singular Lagrangian systems with Second-Order Derivatives Using Path Integral Method Eyad Hasan Hasan Tafila Technical University, Faculty of Science, Applied Physics Department, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Box: 179, Tafila 66110, Jordan Abstract The fractional quantization of singular systems with second order Lagrangian is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The fractional singular Lagrangian is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The equations of motion are written as total differential equations within fractional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Also, the set of Hamilton–Jacobi partial differential equations is constructed in fractional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The path integral formulation and path integral quantization for these systems are constructed within fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' We examined a mathematical singular Lagrangian with two primary first-class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Keywords: Path Integral Quantization, Fractional Calculus, Fractional Hamilton-Jacobi Function, Singular Lagrangians, PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Ef, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' –j, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Hj, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Fy CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Second-order fractional singular Lagrangian and quantization using fractional path integral method 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Examples 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Conclusions 10 References 2 Email: iyad973@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='com, dr_eyad2004@ttu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='jo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Introduction The efforts to quantize singular Lagranians systems have been studied with increasing interest and treated first by Dirac [1, 2], for quantizing the gravitational field, then his formalism has been developed using a new formalism for investigating singular systems- the canonical- was developed by Guler [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' In this formalism, the equations of motion are written in fractional form as total differential equations and the set of fractional Hamilton–Jacobi partial differential equations is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Then, this formalism has been used for quantizing singular Lagrangian systems using the WKB approximation and path integral approach [4-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Fractional calculus with singular systems had treated with more interests and importance [9- 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Recently, the Euler-Lagrange equations for second-order Lagrangian systems are analyzed within fractional derivatives and the fractional Hamilton-Jacobi formalism for these systems are discussed [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' More recently, authors have constructed a formalism using the canonical method for quantizing singular systems using the WKB approximation and path integral approach for first-order derivatives [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' In this paper, we would like to extend our work to Lagrangians with second-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Now, we will present the most important definitions of fractional derivatives [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (i) The left Riemann–Liouville fractional derivative \uf074 \uf074 \uf074 \uf061 \uf061 \uf061 d f t dt d n t f D t a n n t a ) ( ) ( ) ( 1 ) ( 1 \uf0f2 \uf02d \uf02d \uf02d \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf047 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (1) (ii) The right Riemann–Liouville fractional derivative \uf074 \uf074 \uf074 \uf061 \uf061 \uf061 d f t dt d n t f D b t n n b t ) ( ) ( ) ( 1 ) ( 1 \uf0f2 \uf02d \uf02d \uf02d \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6\uf02d \uf02d \uf047 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (2) where N n\uf0ce , \uf061 \uf0a3 \uf02d1 n ˂ n and \uf047 is the Euler gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Here \uf061 is an integer and these derivatives can be defined as follows: 3 ) ( ) ( t f dt d t f Dt a \uf061 \uf061 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6 \uf03d , ) ( ) ( t f dt d t f Db t \uf061 \uf061 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e8 \uf0e6\uf02d \uf03d , (3) Definition Given a function f : \uf0c2 \uf0ae \uf0a5) ,0 [ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Then for all t >0 , )1,0 ( \uf0ce \uf061 , let \uf065 \uf065 \uf061 \uf065 \uf061 ) ( ) ( lim ) )( ( 1 0 t f t t f t f D \uf02d \uf02b \uf03d \uf02d \uf0ae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (4) \uf061 D is called the conformal fractional derivative of f of order of \uf061 [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' In this work, we aim to construct the formalism for quantizing singular Lagrangians systems with second-order derivatives within framework of fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' This paper is organized as follows: In section 2, we will investigate the fractional singular Lagrangian and fractional path integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' In section3, one illustrative example is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The work closes with some concluding remarks in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Second-order fractional singular Lagrangian and quantization using fractional path integral method In this section, we will present a formalism for second-order singular Lagrangian and canonical path integral approach within of fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' We will start with a Lagrangian depending on the fractional derivatives is given by ) , , , ( 2 1 t q D q D q D L L i i i \uf061 \uf061 \uf061\uf02d \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (5) Where iq D\uf061 are the conformal fractional derivatives of the coordinates iq [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The Lagrangian and Hamiltonian formalism for second-order derivatives have been studied by Ostrogradski [ 19] and the derivatives have been treated as coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, we can treat the derivatives iq D 1 \uf02d \uf061 and iq D\uf061 as coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Therefore, the Poisson brackets can be defined as \uf07b \uf07d i i i i i i i i q D B A B q D A q D B p A p B q D A B A \uf061 \uf061 \uf061 \uf061 \uf070 \uf070 \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf02d \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf02b \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf02d \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf0ba \uf02d \uf02d 1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (6) Here, the functions A and B are described in term of the canonical variables iq D 1 \uf02d \uf061 , iq D\uf061 , ip and i \uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, the generalized momenta ip and i \uf070 are conjugated to the generalized 4 coordinates iq D 1 \uf02d \uf061 and iq D\uf061 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, one can write the fundamental Poisson brackets as: \uf07b \uf07d ij j i p q D \uf064 \uf061 \uf0ba \uf02d , 1 and \uf07b \uf07d ij j iq D \uf064 \uf070 \uf061 \uf0ba , , \uf07b \uf07d \uf07b \uf07d \uf07b \uf07d \uf07b \uf07d i i j i j i j i p q D q D q D q D q D q D \uf070 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 , , 0 , , 1 1 1 \uf03d \uf03d \uf03d \uf03d \uf02d \uf02d \uf02d , where N j i ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 1 , \uf03d (7) Now, the fractional of the Hessian matrix is defined as j i ij q D q D L W \uf061 \uf061 2 2 2 \uf0b6 \uf0b6 \uf0b6 \uf03d (8) The fractional Lagrangian is called regular if it’s rank is N otherwise the Lagrangian is singular , R N \uf02d R < N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, we can define the generalized momenta i \uf070 corresponding to the generalized coordinates iq D\uf061 as: a a q D L \uf061 \uf070 2 \uf0b6 \uf0b6 \uf03d , R N a \uf02d \uf03d ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='1 (9) \uf06d \uf061 \uf06d \uf070 q D L 2 \uf0b6 \uf0b6 \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' N R N ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 1 \uf02b \uf02d \uf03d \uf06d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (10) Dirac showed in his formalism for investigating singular Lagrangian systems that the number of degrees of freedom can be reduced from N to N-R due to the constraints [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Since the rank of the Hessian matrix is N-R, we can solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (9) to obtain N-R accelerations a q D \uf061 2 in terms of a i i q D q D \uf070 \uf061 \uf061 , , 1 \uf02d and \uf06d \uf061q D2 as follows: ) , , , ( 2 1 2 \uf06d \uf061 \uf061 \uf061 \uf061 \uf070 q D q D q D w q D a i i a a \uf02d \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (11) Substituting (10) in (9), we can obtain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (12) ) , , , ( 1 a a i i p q D q D H \uf070 \uf070 \uf061 \uf061 \uf070 \uf06d \uf06d \uf02d \uf02d \uf03d 5 We can obtain a similar expression for the momenta \uf06d p and can be defined as: (13) Also, we can define the generalized momenta ip corresponding to the generalized coordinates iq D 1 \uf02d \uf061 as: \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf0b6 \uf0b6 \uf03d a a a q D L dt d q D L p \uf061 \uf061 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (14) \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0e7\uf0e7 \uf0e8 \uf0e6 \uf0b6 \uf0b6 \uf02d \uf0b6 \uf0b6 \uf03d \uf06d \uf061 \uf06d \uf061 \uf06d q D L dt d q D L p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (15) We can define the fractional Hamiltonian \uf06f H as \uf070 \uf06d \uf06d \uf061 \uf06d \uf06d \uf061 \uf061 \uf061 \uf061 \uf061 \uf070 H q D H q D W q D p W q D q D q D L H p a a a a a v v i 2 2 1 ) , , , ( \uf02d \uf02d \uf02b \uf02b \uf02d \uf03d \uf02d \uf06f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (16) R ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='., 1 \uf03d \uf06d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' N R a ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 1 \uf02b \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Equations (12) and (13) become ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (17) (18) Also, equations (17) and (18) represent primary constraints [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' A natural of singular Lagrangian indicates that the generalized momenta \uf06d p and \uf06d \uf070 are not independent of a p and a \uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, we can write the set of Hamilton-Jacobi partial differential equations as 0 , , , , ( 1 1 1 \uf03d \uf0f7\uf0f7 \uf0f8 \uf0f6 \uf0b6 \uf0b6 \uf02b \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf0b6 \uf0a2 \uf02d \uf02d \uf02d v a v a i i q D S q D S q D S q D S q D q D H \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf062 , (19) ,0 \uf03d \uf062 N R N ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 \uf02b \uf02d ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 1 a a i i p p q D q D H p \uf070 \uf061 \uf061 \uf06d \uf06d \uf02d \uf02d \uf03d 0 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 1 \uf03d \uf02b \uf03d \uf0a2 \uf02d p i i i i p H p p q D q D H \uf06d \uf06d \uf061 \uf061 \uf06d \uf070 0 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 1 \uf03d \uf02b \uf03d \uf0a2 \uf02d \uf070 \uf06d \uf06d \uf061 \uf061 \uf070 \uf06d \uf070 \uf070 H p q D q D H i i i i 6 Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=" the fractional Hamilton's principle function is written as ) ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 1 1 t q D q D q D q D S S a a \uf06d \uf061 \uf061 \uf06d \uf061 \uf061 \uf02d \uf02d \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' we define ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 a a q D S p \uf02d \uf0b6 \uf0b6 \uf03d \uf061 \uf06d \uf061 \uf06d q D S p 1 \uf02d \uf0b6 \uf0b6 \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' a a q D S \uf061 \uf070 \uf0b6 \uf0b6 \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' \uf06d \uf061 \uf06d \uf070 q D S \uf0b6 \uf0b6 \uf03d and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' t S p \uf0b6 \uf0b6 \uf03d \uf06f Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' the action function and the equations of motion in fractional form can be written as total differential equations as follows: ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 1 \uf06d \uf061 \uf070 \uf06d \uf06d \uf061 \uf06d \uf061 q dD p H q dD p H dt p H q dD a a p a a \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf03d \uf02d \uf02d \uf06f (20) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 \uf06d \uf061 \uf070 \uf06d \uf06d \uf061 \uf06d \uf061 \uf070 \uf070 \uf070 q dD H q dD H dt H q dD a a p a a \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf03d \uf02d \uf06f (21) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 1 1 1 \uf06d \uf061 \uf061 \uf070 \uf06d \uf06d \uf061 \uf061 \uf06d \uf061 q dD q D H q dD q D H dt q D H dp i i i i p \uf02d \uf02d \uf02d \uf02d \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf03d \uf02d \uf06f (22) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1 \uf06d \uf061 \uf061 \uf070 \uf06d \uf06d \uf061 \uf061 \uf06d \uf061 \uf070 q dD q D H q dD q D H dt q D H d i i i i p \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf03d \uf02d \uf02d \uf06f (23) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ) ( ) ( ) ( 1 \uf06d \uf061 \uf070 \uf06d \uf070 \uf06d \uf070 \uf06d \uf06d \uf061 \uf06d \uf06d \uf06d \uf070 \uf070 \uf070 \uf070 \uf070 \uf070 q dD H p H p H q dD H p H p H dt H p H p H dS a a a a a p a a p a p a a a a \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf02d \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf02d \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf0b6 \uf0a2 \uf0b6 \uf02b \uf02d \uf03d \uf02d \uf06f \uf06f \uf06f (24) If the total derivative of equation (19) is zero [3] 0 \uf03d \uf0a2\uf06f H d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (25) 0 \uf03d \uf0a2 p H d \uf06d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (26) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='0 \uf03d \uf0a2\uf070 \uf06d H d (27) This indicates that equations (20-24) are integrable, and the rank of Hessian matrix is R N \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Thus, the degrees of freedom are reduced from N to R N \uf02d , the constraints reduce the canonical phase space coordinates from } , , , { 1 i i i i q D p q D \uf070 \uf061 \uf061\uf02d to } , , , { 1 a a a a q D p q D \uf070 \uf061 \uf061\uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' we can ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='represent the path integral approach for singular systems in the fractional form as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='\uf0fa ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( \uf06f \uf06f \uf06f (28) N i ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='1 \uf03d , R N a \uf02d \uf03d ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='1 , N R N ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=', 1 \uf02b \uf02d \uf03d \uf06d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Example Second-Order Fractional Singular Lagrangian Let us consider the following mathematical singular Lagrangian with two primary first-class constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' \uf028 \uf029 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ) ( ) ( 2 1 2 2 1 3 1 3 3 2 3 2 2 2 2 2 q D q D q D q D q D q D q D q D L \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf02d \uf02d \uf02b \uf02b \uf02b \uf02b \uf03d (29) The corresponding generalized momenta, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (9, 10) and (14, 15) are 1 3 1 q D p \uf061 \uf02d \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (30) 2 3 2 1 2 q D q D p \uf061 \uf061 \uf02d \uf03d \uf02d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (31) p H q D p 3 3 1 3 \uf02d \uf03d \uf03d \uf02d \uf061 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (32) 1 2 1 q D \uf061 \uf070 \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (33) 2 2 2 q D \uf061 \uf070 \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (34) \uf070 \uf061 \uf070 3 3 3 H q D \uf02d \uf03d \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (35) Here, Equations (32) and (35) can be written as 0 3 1 3 3 \uf03d \uf02d \uf03d \uf0a2 \uf02d q D p H p \uf061 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (36) 8 0 3 3 3 \uf03d \uf02d \uf03d \uf0a2 q D H \uf061 \uf070 \uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (37) and represent as primary constraints [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The Hamiltonian 0 H is calculated as ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 2 1 ) ( 2 2 2 1 2 2 1 2 1 1 \uf070 \uf070 \uf061 \uf061 \uf061 \uf02b \uf02b \uf02d \uf02b \uf03d \uf02d q D q D p q D p H \uf06f (38) The corresponding set of HJPDEs, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (19), reads ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( 2 1 ) ( 2 2 2 1 2 2 1 2 1 1 \uf070 \uf070 \uf061 \uf061 \uf061 \uf02b \uf02b \uf02d \uf02b \uf02b \uf03d \uf02b \uf03d \uf0a2 \uf02d q D q D p q D p p H p H \uf06f \uf06f \uf06f \uf06f (39) 0 3 1 3 3 \uf03d \uf02d \uf03d \uf0a2 \uf02d q D p H p \uf061 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (40) 0 3 3 3 \uf03d \uf02d \uf03d \uf0a2 q D H \uf061 \uf070 \uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (41) Here, the Poisson brackets \uf07b \uf07d 0 , 3 \uf03d \uf0a2 \uf0a2 \uf06f H H p , \uf07b \uf07d 0 , 3 \uf03d \uf0a2 \uf0a2 \uf06f H H \uf070 and \uf07b \uf07d 0 , 3 3 \uf03d \uf0a2 \uf0a2 \uf070 H H p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' There are no secondary constraints, this means that these are first-class constraints [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' The equations of motion Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (20- 24) can be calculated as , 1 1 1 dt q D q dD \uf061 \uf061 \uf03d \uf02d (42) , 2 2 1 dt q D q dD \uf061 \uf061 \uf03d \uf02d (43) , 1 1 dt q dD \uf070 \uf061 \uf03d (44) , 2 2 dt q dD \uf070 \uf061 \uf03d (45) ,0 1 \uf03d \uf02ddp (46) , 2 2 dt q D dp \uf061 \uf02d \uf03d \uf02d (47) , 3 1 3 q dD dp \uf02d \uf02d \uf03d \uf02d \uf061 (48) , 1 1 dt p d \uf03d \uf02d \uf070 (49) , ) ( 2 1 2 2 dt q D p d \uf02d \uf02d \uf03d \uf02d \uf061 \uf070 (50) 9 , 3 3 q dD d \uf061 \uf070 \uf02d \uf03d \uf02d (51) 3 3 3 1 3 2 2 1 2 2 2 1 ) 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 2 1 ( q dD q dD p dt q D q D dS \uf061 \uf061 \uf061 \uf061 \uf070 \uf070 \uf070 \uf02b \uf02b \uf02b \uf02b \uf03d \uf02d \uf02d (52) 3 3 3 1 3 1 2 2 1 2 2 2 1 ) 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 2 1 ( q dD q D q dD q D dt q D q D dS \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf070 \uf070 \uf02b \uf02b \uf02b \uf02b \uf03d \uf02d \uf02d \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (53) 3 3 3 1 3 1 2 2 1 2 2 2 1 ) 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 2 1 ( q dD q D q dD q D dt q D q D S \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf070 \uf070 \uf02b \uf02b \uf02b \uf02b \uf03d \uf02d \uf02d \uf02d \uf0f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (54) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' By obtaining the fractional action function S ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' we can represent the path integral approach in fractional form as \uf0fa \uf0fb \uf0f9 \uf0ea \uf0eb \uf0e9 \uf02b \uf02b \uf02b \uf02b \uf03d \uf0f2 \uf0f2 \uf0f2 \uf0f2 \uf02d \uf02d \uf02d \uf02d \uf02d \uf02d \uf02d \uf02d 3 3 3 1 3 1 2 2 2 1 2 2 1 2 1 2 1 2 1 2 1 1 1 3 2 1 3 1 2 1 1 1 ) 2 2 ( exp ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' ( q dD q D q dD q D dt q D q D i d d dp dp q dD q dD q dD q dD t q D q D q D q D q D q D K \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf061 \uf070 \uf070 \uf070 \uf070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' (55) Author Contributions E H wrote the main manuscript text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Funding Funding information is not applicable/No funding was received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Availability of data and materials Data sharing not applicable to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Declarations Conflict of interest There is no conflict of interests with regards to the publication of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Conclusion In this work, we constructed a formalism for quantizing singular Lagrangian systems using path integral approach within fractional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' We wrote the equations of motion and action function in fractional form as total differential equations, besides the path integral approach is constructed within fractional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Then, we discussed a mathematical example to demonstrate our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 10 References [1] Dirac, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Generalized Hamiltonian Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Canadian Journal of Mathematical Physics, 2, 129-148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' [2] Dirac, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' [3] Guler, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 1992b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Canonical Formulation of Singular Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' IL Nuovo Cimento B, 107 (10), 1143-1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' [4] Muslih, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} +page_content=' Path Integral Formulation of Constrained Systems with Singular Higher- Order Lagrangians.' 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Petersbourg, 1 (1850) 385' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFKT4oBgHgl3EQfay5f/content/2301.11809v1.pdf'} diff --git a/HNE3T4oBgHgl3EQfuAs1/content/tmp_files/2301.04680v1.pdf.txt b/HNE3T4oBgHgl3EQfuAs1/content/tmp_files/2301.04680v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53638de95b140df483486995927acabf03b46bdd --- /dev/null +++ b/HNE3T4oBgHgl3EQfuAs1/content/tmp_files/2301.04680v1.pdf.txt @@ -0,0 +1,1764 @@ +Formation of Rocky Super-Earths From A Narrow Ring of Planetesimals +Konstantin Batygin1 & Alessandro Morbidelli2 +1Division of Geological and Planetary Sciences California Institute of Technology, Pasadena, CA 91125, +USA — kbatygin@gps.caltech.edu +2Laboratoire Lagrange, Universit´e Cote d’Azur, CNRS, Observatoire de la Cote d’Azur, Nice, France — +alessandro.morbidelli@oca.eu +The formation of super-Earths, the most +abundant planets in the Galaxy, remains elu- +sive. These planets have masses that typically +exceed that of the Earth by a factor of a few; +appear to be predominantly rocky, although +often surrounded by H/He atmospheres; and +frequently occur in multiples. Moreover, plan- +ets that encircle the same star tend to have +similar masses and radii, whereas those be- +longing to different systems exhibit remark- +able overall diversity. +Here, we advance a +theoretical picture for rocky planet formation +that satisfies the aforementioned constraints: +building upon recent work — which demon- +strates that planetesimals can form rapidly +at discrete locations in the disk — we pro- +pose that super-Earths originate inside rings +of silicate-rich planetesimals at approximately +∼1AU. Within the context of this picture, +we show that planets grow primarily through +pairwise collisions among rocky planetesimals, +until they achieve terminal masses that are +regulated by isolation and orbital migration. +We quantify our model with numerical sim- +ulations and demonstrate that our synthetic +planetary systems bear a close resemblance to +compact, multi-resonant progenitors of the ob- +served population of short-period extrasolar +planets. +Our results thus indicate that the +absence of short-period super-Earths within +the solar system can simply be attributed to +the comparatively low mass of the primordial +planetesimal ring within the protosolar neb- +ula. +Introduction. +It has long been known that the +genesis of planets begins through the coalescence +of solids within protoplanetary nebulae, and mod- +els of planet formation have traditionally assumed +that dust within circumstellar disks is smoothly dis- +tributed. Despite being common, this simplifying as- +sumption may be unfounded. +Several lines of evi- +dence have recently been marshaled in support of the +notion that rather than arising from a smooth gra- +dient of solids, planetesimal formation unfolds in a +small number of discrete rings [1, 2, 3, 4, 5, 6]. In +this vein, the work reported in ref. [3] proposes that +protoplanetary nebulae generally originate as decre- +tion disks that spread radially from tenths of an AU, +facilitating the condensation of outward-diffusing sil- +icate vapor into rocky dust grains at the disk’s pri- +mordial silicate sublimation-line. +Importantly, this +process naturally leads to the formation of rocky +planetesimals at a stellocentric distance comparable +to the Earth’s orbital radius (as well as the gener- +ation of more distant icy bodies close to Jupiter’s +present-day orbit) through gravito-hydrodynamic in- +stabilities [7, 8]. Such a model further yields a self- +consistent explanation for the isotopic dichotomy of +carbonaceous and non-carbonaceous iron meteorites, +as well as the physical origins of the solar system’s +broader architecture [3, 6]. +Within the framework of the aforementioned disk +model, the mass budget of silicate material that forms +at the rock-line is distinctively variable (Extended +Data Figure). That is, depending on the specific com- +bination of disk viscosity and metallicity, the cumu- +lative mass of rocky planetesimals entrained within +the r ∼ 1 AU silicate annulus can readily reach tens +of Earth masses (although we note that it can also +be null if the threshold for planetesimal formation +through gravitational collapse is not met). Moreover, +numerical modeling indicates that planetesimal for- +mation is expected to occur over a relatively short +temporal burst, such that dust is incorporated into +planetary building blocks over a timescale of ∼ 105 +years. +Adopting +the +ringed +planetesimal +formation +paradigm as a platform, a key goal of our work is +to consider the possibility that a typical system of +extrasolar super-Earths originates within such a ra- +dially confined annulus of rocky material. As we de- +1 +arXiv:2301.04680v1 [astro-ph.EP] 11 Jan 2023 + +Figure 1: +Schematic diagram of the planet formation +scenario considered in this work. Strong magnetic brak- +ing during the infall phase implies that the protoplane- +tary nebula originates as a decretion disk that viscously +spreads outward from a few tenths of an AU, carrying +minute dust grains to large stellocentric distances (top +panel). Beyond the condensation front, grains grow and +begin to drift inwards. Accordingly, a balance between +the radial outflow and sub-Keplerian azimuthal rotation +of the gas leads to the accumulation of rocky grains at +the silicate sublimation line, at an orbital distance of +r ∼ 1 AU (middle panel). The envisioned picture requires +the satisfaction of three criteria [3]: a small (sub-AU) cen- +trifugal radius for the infalling material, phase-transitions +of silicate species that facilitate significant changes in +characteristic particle radii across one or more sublima- +tion front(s), and a sufficiently quiescent disk for radial +particle pile-up and vertical settling to occur. While the +cumulative mass of the accrued dust ring is regulated by +both metallicity as well as the vigor of disk turbulence, +given nominal parameters, the silicate annulus can read- +ily reach a mass on the order of tens of Earth masses. +Gravito-hydrodynamic instabilities facilitate the conver- +sion of ∼ 1 mm dust into ∼ 100 km planetesimals, which +subsequently merge to generate multi-M⊕ objects. +As +planets deplete their local supply of solids and grow mas- +sive enough to experience substantial disk-driven migra- +tion, they exit the planetesimal feeding zone and undergo +orbital decay (bottom panel), which terminates when they +reach the inner edge of the protoplanetary disk. Impor- +tantly, when planets exit the planetesimal ring, their ac- +cretion stalls. +Thus, the terminal mass of super-Earth +type planets is approximately set by the balance between +accretion and migration timescales, yielding a natural +propensity towards intra-system uniformity. +scribe below, the process of planetary conglomera- +tion within a narrow ring of silicate-rich planetesi- +mals naturally yields a characteristic multi-M⊕ mass +scale of the resulting planets, and the simultaneous +operation of accretion and orbital migration regulates +the emergence of uniformity among the growing plan- +etary embryos (Fig. 1). +Results. +The starting point of our calculation cor- +responds to the epoch of large-scale planetesimal for- +mation within a protoplanetary disk. For definitive- +ness, here we adopt disk conditions derived from the +simulations reported in ref. [3], although we note that +for the purposes of our calculations, any ringed plan- +etesimal formation scenario is likely to lead to simi- +lar results. Our fiducial disk model is initialized with +a gas surface density of Σ0 = 2500 g/cm2 at 1 AU, +a corresponding peak dust surface density of Σ• = +500 g/cm2, and a dust grain radius of s• = 1 mm, con- +sistent with fragmentation-limited growth [9]. Ow- +ing primarily to viscous energy dissipation, the disk +maintains an appreciable aspect ratio of h/r ∼ 0.05 +throughout the planetesimal formation epoch. While +the gas surface density is taken to dissipate exponen- +tially with a time-constant of τdisk = 1.5 Myr, the +dust surface density decays much more rapidly, ow- +ing to the fact that pebbles get incorporated into a +Mring ∼ 20 M⊕ planetesimal swarm over a ∼ 105 year +timescale. The specific functional parameterizations +of these quantities are delineated in the Methods sec- +tion. +As clouds of dust within the r +∼ +1 AU sili- +cate ring consolidate into planetary building blocks +by means of gravitational collapse, their continued +growth can proceed through two distinct channels: +pairwise mergers among planetesimals and pebble ac- +cretion. +The efficiency of planetesimal accretion is +controlled by the extent to which gravitational focus- +ing can increase the collisional cross-section of pro- +toplanetary embryos. Pebble accretion, on the other +hand, depends critically on whether the capture of +dust proceeds in the 2D or 3D regimes — a deter- +mination that is sensitive to the characteristic size of +dust particles. Generically speaking, the process of +collisional fragmentation inhibits the growth of sili- +cate grains beyond the millimeter-scale within pro- +toplanetary disks, ensuring that even in relatively +quiescent nebulae, turbulent stirring can maintain +the dust sub-disk’s aspect ratio at an inflated level +[11]. Correspondingly, pebble accretion proceeds in +the comparatively inefficient 3D regime, contribut- +ing very little to the planetary conglomeration pro- +cess during the planetesimal formation epoch. +We +further find that leftover dust that is not incorpo- +2 + +protoplanetary disk +M infall of proto-solar material (efficient magnetic braking) +radial viscous spreading +9 +O(10 kyr) +Q +Ur +centrifugal radius +sub-Keplerian azimuthal flow +evolution +disk +silicate line +O(100 kyr) +silicate vapor outflow +collapse +inward drift of solids +grav. +model t = O : planetesimal formation +truncation radius +O(1 Myr) +migration +planetesimal accretionFigure 2: The formation sequence of a mass-uniform exoplanetary system. Over the course of the first 100,000 +years (panels A-E), D = 100 km super-planetesimals (gray, orange, red, green, and blue points, labeled according +to their inclinations) and lunar-mass planetary embryos (purple circles) – comprising Mring ≈ 20 M⊕ in total – +are gradually introduced into the simulation domain. These objects originate with eccentricities and inclinations +of ⟨e⟩ ∼ ⟨i⟩ ∼ 0.01, across a radial range spanned by the horizontal line shown in panel A. Growth of planetary +embryos is driven primarily by accretion of planetesimals, with aerodynamic drag and collisional damping facilitating +enhanced gravitational focusing (panels C-E). Injection of new material into the system terminates at the t = 105 +year mark (panel E), and over the course of the following few hundred thousand years, multi-Earth-mass planets +emerge, with the conglomeration process largely completed within the first 0.5 Myr (panel F, G). Over the course of +the remaining lifetime of the disk, the formed planets migrate inwards, locking into a mass-uniform multi-resonant +chain (panels H). Recent work [23, 10] has shown that tightly packed multi-resonant planetary configurations serve +as ideal initial conditions for reproducing both the period ratio distribution of observed extrasolar planets as well as +their inferred degree of mass-uniformity. +rated into planetary building blocks through gravito- +hydrodynamic instabilities, rapidly flows away from +the planetesimal ring as the nebula matures into an +accretion disk, and our estimates (see Methods sec- +tion 6) indicate that any auxiliary exterior flux of +pebbles plays a negligible role in driving the forma- +tion of rocky super-Earths (we confirm these analytic +expectations with numerical simulations below). +Analytical Estimates. +In contrast with the rel- +ative inefficiency of pebble accretion in the inner re- +gions of a protoplanetary disk, the efficacy of plan- +etesimal accretion within a narrow annulus of rocky +planetesimals is strongly enhanced. The reasons for +this are two-fold: first, by concentrating tens of Earth +masses of solids into a radially confined ring of plan- +etesimals, the rate of collisions among the constituent +bodies is strongly amplified. Second, the combined +action of aerodynamic drag and inelastic collisions +3 + +A + t = ５ kyr +E +t = 1oo kyr +0.04 +0.04 +e +0.02 +0.02 +planetesimal formation +0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 + = 25kyr +t= +250 kyr +B +t +2 ≤i<3 deg +0.04 +0.04 +3≤i< 4 deg +e +0.02 +0.02 +4≤ i< 5 deg +embryos +0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +50kyr +G +t = 500 kyr +C +t += +0.04 +0.04 +orbital migration +e +0.02 +0.02 +drag + collisions +0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +t= +75 kyr +H +t + = 3 Myr +D +0.04 +0.04 +2.06 Mg +1.94 Mg +0≤i<1 deg +e +2.54 Mg +0.02 +0.02 +1 ≤i<2 deg +2.04 Mg +0 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +a(AU) +a(AU)among planetesimals constitutes a fast-acting damp- +ing mechanism for the planetesimal velocity disper- +sion, magnifying the effect of gravitational focusing. +In this regime, the associated mass-accretion rate of +a planetary embryo can be deduced from a n − σ − v +relation, and the result is well-known [12, 14]: +˙M ∼ +Σpl π R2 Ω (1 + Θ), where Σpl ∼ Σ• is the planetes- +imal surface density, Ω is the orbital frequency, and +Θ = (vesc/⟨v⟩pl)2 is the Safronov number (i.e., the ra- +tio of the square of the escape velocity to the square of +the planetesimal velocity dispersion [13]). Moreover, +under the simplifying assumption of strong and time- +invariant gravitational focusing, it is straightforward +to show that a crude estimate for the timescale for an +Earth-mass body to emerge within the ring of rocky +planetesimals is given by T⊕ ∼ ¯ρ R⊕/(Σpl Ω Θ) [15], +where ¯ρ ∼ 3 g/cc is the embryo’s density. If we adopt +the fiducial parameters of our model and assume that +the escape velocity of the planetary embryo exceeds +the planetesimal velocity dispersion by a factor of a +few (corresponding to Θ ∼ 10), T⊕ can be as short as +∼ 105 years. +No matter the dominant growth mode, planetary +accretion cannot proceed without bounds. +For the +problem at hand, two distinct processes constitute +natural termination mechanisms for planetary con- +glomeration, the first being isolation. Isolation oc- +curs due to the depletion of planetesimals from the +local feeding zone of the embryo. The expression for +the isolation scale is easily obtained by equating the +cumulative mass of planetesimals within the feeding +zone (approximately two Hill radii within a heavily +dissipated disk) and the planetary mass itself, to yield +M ∼ 8 π3/2 Σ3/2 +pl r3/√3 M⋆. Given our fiducial pa- +rameters, the isolation mass within the planetesimal +ring evaluates to M ∼ 3 M⊕. +A second growth-limiting process that ensues in +our model is gas-driven orbital migration. +As a +growing planet becomes massive enough to raise +a substantial wake within the gaseous nebula, the +gravitational back-reaction of the wake upon the +planet drives an exchange of energy and angu- +lar momentum between the planet and the disk, +which in turn expels the planet from the planetes- +imal ring altogether. +Thus, within the framework +of our theoretical picture, an approximate equiva- +lence between the mass doubling timescale Tmass ∼ +3 M 1/3 ¯ρ2/3/(Σpl Ω Θ) and the migration timescale +[16] Tmig ∼ (4/Ω)(M⋆/M)(M⋆/Σ0 r2)(h/r)2 yields an +estimate for the mass of planets that are expected +to emerge from the rocky annulus of planetesimals. +Auspiciously, for the aforementioned nominal param- +eters, the planetary mass scale that comes out from +this relation also evaluates to M ∼ 3 M⊕. +Figure 3: Architectures of exoplanetary systems at time +of disk-dispersal, generated within the framework of our +model. The formation and evolution of the system high- +lighted with a light-blue-green rectangle is depicted in +Figure 2. +As the mass of the planetesimal ring is in- +creased from Mring = 5 M⊕ to 40 M⊕, both the number +of embryos that achieve the planetary mass-scale, as well +as their average mass itself increase. Within the numer- +ical model, radial migration is taken to smoothly termi- +nate at r = 0.5 AU across a characteristic length-scale +of ±0.1 AU (see Methods section 5). +This stellocentric +distance marked by a dotted line on the Figure, and also +represents the boundary between short-period progenitors +to the observed population of Super-Earths and low-mass +objects that remain stranded close to their formation site. +By and large, these (r < 0.5 AU) synthetic planetary sys- +tems adhere to a pattern of intra-system mass uniformity +with the normalized mass dispersion, DM/⟨M⟩, that sys- +tematically reaches values comparable to, or smaller than, +the observed value of (DM/⟨M⟩)data ≈ 0.48. Cumula- +tively, these results explain how planetary systems can +emerge with a broad diversity of masses while retaining +an unexpectedly high degree of self-uniformity. +Because the isolation and accretion-migration ter- +minal mass scales are similar, the process of plane- +tary conglomeration is unlikely to depend sensitively +4 + + ct, the path ⃗d has been taken with a velocity more than light’s and so cannot be in any +particle’s or light’s world line. In this case, the vector is named space-like and is outside the light cone. +A Closed Time-like Curve, or CTC for short, is a time-like line, in the above definition, with the same +starting and ending points. Since it is time-like, can be a particle’s world line. Any particle of the system owning +this kind of world line may return to a state of its past, or in other words, a coordinate of space and time that +has been before. +The possibility of the existence of CTCs was raised for the first time by a dutch mathematician, Willem Jacob +van Stockum. (van Stockum, 1938) After that, Kurt G¨odel introduced the G¨odel metric as a solution to Einstein’s +field equations, leading to express a universe containing CTCs. (G¨odel, 1949) Factually, Einstein formulated the +field equations, also known as EFE, within the general theory of relativity in (Einstein, 1916). Since then, +several solutions have been found for EFE, which are called metrics. These solutions tend to describe universes +that include exotic features; for instance, black holes in the Schwarzschild metric, traversable wormholes in the +Morris–Thorne metric, and CTCs in the G¨odel metric. +Intuitively, a space-time equipped with CTCs provides the possibility of traveling back in time. +Similar +to someone starting going rightward on the spherical earth and finally reaching a coordinate lefter than their +departure point, a particle of the system can enter the CTC and move forward on it and, by the passage of time, +since CTC is closed, eventually arrives at a time before its travel’s starting. (Smith, 2021) +1.1 +Paradoxes Come with CTC +At first sight, time travel causes several logical as well as intuitional paradoxes. In this section, we study the +definition, scenario, and possibly the given responses to some sorts of these paradoxes. Then, we discuss a similar +issue in section 2 and proceed with trying to respond to it. +1.1.1 +Consistency Paradoxes +A prominent sort of time travel paradox is the consistency paradox which happens by performing some changes +to the past. A convenient example of that is the grandfather paradox, which can be described as follow: +Imagine a person who travels back in time to one of their ancestors’ eras and kills their grandfather. +As a result, they will never be born, so they will not have time-traveled and killed their grandfather. +Consequently, they are born and travel to the past, and so on. (Aaronson, 2005) +There also exist a few other equivalent scenarios for the grandfather paradox, such as the story that Hawking +and Ellis claim: +Suppose that with a suitable rocket ship, a person travels in time to arrive before their departure. +They can alter any past events only if we assume they have free will, leading to stopping themselves +from setting out on their travel. Consequently, they will not travel in time, and therefore, nothing +will happen to prevent their time travel. (Hawking & Ellis, 1973) +Or the Hitler Paradox: +The first thing that may cross the minds of some humanitarians to do with the ability of time travel +is going back to some time before 1939 and killing Adolf Hitler to prevent the wage of World War +II. However, his murder, aside from the widespread impact that might have on the future, wipes the +reason3 for its own happening. Meaning that, without the tragedy of WWII, there is no motivation +for the time traveler to kill Hitler. (Brennan, 1997) +Basically, during all of these equivalent scenarios, an event a in the past influences an event b in the future, +for example traveling in time to the past, which then causes the occurrence of the event a. +1.1.2 +Fermi Paradox +One other type of contradiction is similar to the Fermi paradox, which can be stated as if traveling back in time +was possible, where would future time travelers be right now? (Jones, 1985) +Various answers are given to this paradox, such as time travel may be extremely expensive or dangerous (Smith, +2021) or reaching our era could be impossible by traveling in time for space-time might not be warped enough in +our time to allow the existence of closed time-like lines. (Hawking, 1999) +2Let ⃗d = (x, y, z). +3Unlike the other mentioned scenarios, in the Hitler Paradox, instead of physically preventing time travel, the reason +for it vanishes. +2 + +1.1.3 +Newcomb Paradox +A further contradiction that can be referred to regarding time travel is known as Newcomb’s paradox, in which +a game is played as follows: (Wolpert & Benford, 2013) +• Two players in the roles of “predictor” and “chooser” play the game with two boxes, namely A, containing +1000$, and B, which is empty. +• The chooser will select either both boxes A and B or just box B in their turn and will be paid all the +money in their chosen box(es). +• The predictor will do one of the following in their turn: +– If they foresee that the other player would select both boxes, they will leave box B empty. +– Otherwise, if they predict that the other player would select just box B, they will put 1’000’000$ in +it. +Figure 1: Newcomb’s game payoff matrix. +In game theory, based on the strategic dominance principle, the chooser should always choose both boxes A +and B, whereas using the expected utility principle and the fact that the predictor is “infallible”, the chooser +should always take box B. (Nozick, 1969) +In philosophy, it is believed that perfect prediction or time travel, which can be used as a tool for perfect +prediction, conflicts with free will; for the sake that it is not recognizable that the prediction is the result of choice +or vice versa. (Craig, 1988) +1.1.4 +Causality Loop +A causality loop is another paradox that might occur via time travel. It means that an event a in the past +causes4 an event b in the future, which is indeed the cause of a. Then, both events exist in space-time, while +their origin cannot be determined. (Lobo & Crawford, 2003) +1.1.5 +Knowledge Creation Paradox +The paradox that might happen by traveling in time and does not vanish by consistency methods since it is +not known as a logical contradiction at all is the knowledge creation paradox. To comprehend it, suppose +that someone travels back in time to reach G¨odel’s era and meets him before 1931, the publishing date of his +incompleteness theorem paper, where they dictate G¨odel the paper. As a result, he admires them and publishes +the paper as expected. Thus, it is said that every occurrence in the world, with and without their time travel, is +identical, and nothing paradoxical happens, excluding that neither G¨odel nor the time traveler genuinely thought +about and produced the contents of the paper. In other words, there is no original point of creation for the +incompleteness theorem, and knowledge has been created without anyone putting effort into it. +This non-intuitive feature of time travel, which is thought to be preserved in CTCs, is the foundation of +related results in (Aaronson, 2005) and (Aaronson, Bavarian, & Gueltrini, 2016). +They try to solve a hard +problem without allocating the desired amount of time or memory to it. +However, we argue that by considering a more precise formulation of the scenario, the universe is not entirely +the same in both visits from an outward observer’s point of view; in the first, G¨odel thinks about the incomplete- +ness theorem, while in the second visit, he communicates with a time traveler. Therefore, this scenario also can +be logically paradoxical. +Also, the aforementioned scenario can be seen as an instance of causality loops since it is not recognizable that +the time traveler induced G¨odel the incompleteness theorem or vice versa; they learned it from G¨odel. +Here, we should declare that in a universe containing CTCs, the whole universe, including all creatures, +indeliberately return to a time coordinate resulting in the universe being identical in any visit. +In contrast, +through time travel, just an individual travels to a specific moment. For more explanation, considering the story +4It is worth noticing the difference between the used terms influence and cause when defining the consistency paradoxes +and the causality loop, respectively. +3 + +Chooser +A+B +B +A+B +1000 +0 +Predictor +B +1'000'000 +1'001'000of the grandfather paradox, let us see the problem from the viewpoint of an observer out of space-time who does +not move on a CTC. Then for them, the world is not exactly as it was after the grandson’s time travel, since +in the first view of the coordinate of space-time, the grandson does not exist; however, in the second view, he +stands alongside his grandfather. Hence, arriving at an already-been moment via CTC is not equivalent to an +individual’s time travel. +To our best knowledge, the difference between these two concepts has not been discussed sufficiently, which +persuaded us to think about potential problems that might arise from CTCs. +1.2 +CTC Consistency +So far, various people with different approaches have tried to create conditions to make CTCs consistent and +eliminate their associated paradoxes. For instance, Novikov’s self-consistency principle explains that events +that alter the past occur with a probability equal to zero. +Therefore, by excluding all the self-inconsistent +happenings, time travel paradoxes vanish. (Friedman et al., 1990) Here, we explain the method that Deutsch +proposed in (Deutsch, 1991) and work based on it. +In (Deutsch, 1991), he studied the physical effects of the existence of CTCs with a quantum computational ap- +proach and modeled the computations using CTCs. Furthermore, assuming that classical physics has a minimum +approximate consistency near closed time-like curves, he showed that chronology violation might place conflicting +constraints on inputs that result in paradoxes in the classical case. Even though, in the quantum case, all these +paradoxes are avoidable. +At first, Deutsch generally claims that every computational network in which there is a closed path in space- +time for the information can be converted into a simplified standard computational network, which with every +set of inputs, generates the same outputs as the original network. Also, in this equivalent network, bits interact +only in gates, where operations are performed, too. The n bits on closed time-like paths first go to an ambiguous +future for all gates. Then go back to the past of all gates with a negative delay and, finally, resume their original +paths. If the required time for passing all gates in the computational network was T, the bits delay time, or +equivalently the time each bit travels to the past, is −T. At the same time, m bits from a definite past enter the +network as input, creating the output by communication with n CTC bits, and going to an unambiguous future. +In addition, Deutsch expresses that for CTCs to be consistent, the evolutionary operator of each gate must +have a fixed point. Hence, this point would be the stable state of the information on closed time-like paths. He +also shows that such a fixed point always exists in the quantum case, although it may not exist in the classical +case. +1.3 +TMCTC Computational Model +In 2016, Scott Aaronson, based on Deutschian CTC, proposed the Turing machine equipped with CTC, named +TMCTC, in the classical and QTMCTC in the quantum case. +In the following, we will discuss the classical +computational model. +The TMCTC has two different types of memory registers (Turing machine’s tapes): +RCR: Registers that respect the chronology, coming from a known past and going to an unambiguous +future. Note that the input of the model will be on these registers. +RCTC: Registers round on the CTC. +(a) Tape +(b) A general scheme (Aaronson & Watrous, 2009) +Figure 2: The TMCTC computational model +Similar to a classical Turing machine, we assume that both tapes have infinitely many cells, though we can +use a finite number of them in each Turing program. Hence we can show the Turing machine’ information with +a pair of binary strings (x, y), such that x is the content of RCR and y is the content of RCTC. +Moreover, for each input x of the machine, there is an infinite dimensional stochastic matrix Sx, which maps every +binary string y ∈ {0, 1}∗ to a binary string Sx(y) ∈ {0, 1}∗ with a defined probability. In fact, this stochastic +matrix is a Markov chain, and a fixed-point for the operator of each machine is equivalent to a stationary +distribution for its corresponding Markov chain. +4 + +RCR +1 +0 +# +0 +1 +# +0 +0 +0 +RCTCAnswer +不个 +c +RCTC +RCR +01.4 +Proof of TMCTC-Computability of Halting Problem +Aaronson et al. have introduced a TMCTC program for solving the halting problem in (Aaronson et al., 2016). +This program takes a ⟨P⟩, the description of a Turing machine P, on RCR as input and determines whether P +without any inputs will eventually halt or not. +For doing this, we consider σt as the configuration of Turing machine P within t steps. Thus the state σ0 for P() +is obvious. Also, σt+1 is simply attainable from σt by running one more step of P, for every t. Additionally, for +every arbitrary string y, and by knowing ⟨P⟩, it is easy to specify whether there is a t for which y = σt or not. +Furthermore, we call σt a halting history if it demonstrates that P halts in the tth step of running; otherwise, +we name it a non-halting history. +At this time, the TMCTC, by taking ⟨P⟩ on the RCR, writes an arbitrary string y on the RCTC, and we define the +function S⟨P ⟩(y) as follows: +1. If there existed a t such that y = σt: +1.1. If σt was a halting history, then S⟨P ⟩(y) = y, and it would output 1 on the RCR. +1.2. Otherwise, S⟨P ⟩(y) = σt+1 with probability 1 +2, and S⟨P ⟩(y) = σ0 with probability 1 +2. +2. Otherwise, S⟨P ⟩(y) = σ0. +Now, if P() halts, there is a t, for which σt is a halting history, and y = σt is the only fixed-point of the +operator S⟨P ⟩(y). As a result, TMCTC will output 1, meaning that P() will halt. +In contrast, if P() never halts, the geometric distribution over steps is the only fixed point of the operator S⟨P ⟩. +In other words, P(σi) = ( 1 +2)i+1, so the TMCTC will never halt. +2 +The Problem with the Proof +In this section, we want to bring up some doubts, as a result of the physical constraints of our universe, about +the aforementioned proof. +To begin with, let us study CTC in a physical context. Having this in mind, consider a universe containing +CTCs and therefore, the ability to violate chronology. Suppose that a particle of it, namely x, which is born +at the moment t0, goes toward the future5 until reaching the moment t−1, a time before x’s birth. Then, if x +continues its motion on the CTC and reaches the moment t0 without any damages, assuming that all the other +particles in the first visit of t0 by x are identical with the second turn, x will be born again. Thus, there will be +two copies of x in the universe. Consequently, both x’s move toward the future; so that, when they arrive at the +moment t0 and x is reborn, there are three copies of it in the world. Likewise, this leads to having infinitely many +x’s in the world which seems impossible in our universe. Hence, we argue that x must be destroyed at some point +between t−1 and t0, which will be restated as the strong axiom in section 3.1. +Figure 3: The motion of particle x on a CTC. +The same problem can be discussed for a Turing machine equipped with CTC running the program of the +previous section. Suppose that in our physical world, the TMCTC is created and starts its computation at moment +t0 with a ⟨P⟩ on its RCR, and an arbitrary y on its RCTC. If by the passage of time, the machine returns to the +moment t0, it will be recreated and therefore according to the argument we made above, there will be infinitely +many of it in the world. Thus, by reaching a moment t−1 < t0, the TMCTC should be destroyed before t0, which +will be restated as the weak axiom in section 3.1, and, as a result, is not able to use any information it has +acquired from the future. Meaning that y is always the arbitrary string and will not get any closer to a halting +history. More precisely, if we trace S⟨P ⟩(y), we have: +1. If there existed a t such that y = σt: +5Actually, x does not move in time; instead, time naturally passes for x, and since time-like lines are close in space-time, +at some point, by moving in the same direction as the future light cone, the t component decreases for x’s coordinate in +space-time. +5 + +x +x +t_11.1. If σt was a halting history, then y will remain on the RCTC, and RCR would output 1. Basically, this +is the only case that the program responds correctly without any usage of CTCs. However, there is +no guarantee that it always happens. +1.2. If σt was a non-halting history, either another step of the Turing machine is supposed to be appended to +y, or it ought to be changed to the trivial configuration, σ0, until the upcoming visit of t0. Nevertheless, +by the destruction of the Turing machine before t0, this progress will be lost, and the TMCTC will +always start its computation with y on the RCTC. +2. Otherwise, the trivial configuration is supposed to be written on the RCTC before the subsequent visit of +t0, which, similar to the previous case, will not be accessible. +Moreover, there are some ambiguities about CTCs in the physical universe that have not been addressed during +the proof. For instance, all definitions are about microscopic particles; how are they extended for macroscopic +particles such as Turing machines? Further, time is considered just like space, which seems not reasonable. +3 +Our Proposed Solution +We aim to study the problem brought up in the previous section from two different points of view. Firstly, we +try to propose a solution by considering the rules of classical physics over the universe, after which we will look +at the problem outside the current world. +3.1 +Solving the Problem in Our Current Universe +In section 2, we raised a problem, describing that the lasting of particles for a whole round on a CTC results in +having an infinite number of each particle in the universe. It seems like the issue does not match our physical +intuition, even though it appears to be mathematically consistent. In other words, we assume that the problem +only exists in our physical world, while the mathematical world is a possible and untroubled platform for TMCTC. +As a consequence of the argument we made in the previous section, the two following axioms can be stated for a +classical universe containing CTCs: +Strong axiom: No particle survives a full round of movement on a CTC and will be destroyed before +returning to its starting point in space-time. +Weak axiom: Every Turing machine rounding on a CTC, will be destroyed before returning to its +starting point in space-time. +Now premising at least the weak axiom, we will solve the problem by transferring data between Turing +machines. +We should remark that, according to the definitions in the first section, moving in the positive or negative direction +of time mean movement toward the future or past, respectively. +Suppose a TMCTC, namely M, starts its calculation at time t0, moves in positive time until time t1, and since +then, continues moving in negative time, reaches t2 ≤ t0 and moves again in positive time to reach t1 again. In +this case, according to the weak axiom, M will be destroyed at time t3 when t2 ≤ t3 ≤ t1 and M is moving in +negative time, or t′ +3 when t2 ≤ t′ +3 ≤ t0 and M is moving in positive time. +Now to solve the problem, suppose that another TMCTC, namely M ′, can be placed in M’s path before reaching +t3 or t′ +3, which we call the data transferring hypothesis. Thus, the calculated output of M until that moment +can be used as input of M ′. Then, M ′ will move from t3 or t′ +3 to t0, when it gives its data as input of M, without +any process or alteration. +Within this approach, particles and Turing machines will be destroyed during an entire round movement on CTC; +nevertheless, data can travel in time and remains stable. Thus, M can use the information gathered by moving +on a CTC, and so given claims and proofs of TMCTC will be valid. +(a) +(b) +Figure 4: Data transferring between Turing machines M and M ′. +Hence, in a universe where data transferring was possible, as Aaronson claimed, not only PCTC = PSAPCECTC +but the halting problem would be computable. +6 + ++ +in' +out' +W +M +in +out +to +t2 +in' +out' +M +M' +in +out +Tout +1nin' +out' +M' +M ++ +in +out +Iout +to +in +"3 +in' +W +out' +W +td +in +out3.1.1 +Prerequisites for the Data Transferring Hypothesis +We demonstrate the necessary conditions of the data transferring hypothesis via the following conversation: +A: A prerequisite of the data transferring hypothesis is the existence of Turing machines throughout +time; since otherwise, in the first round of the CTC, there was no Turing machine in space-time until +1936, when Turing machines were invented. However, in the subsequent rounds, there should exist at +least one Turing machine at every moment, including before 1936, to establish the data transferring +hypothesis. This leads to visiting dissimilar universes in different cycles on the CTC. +Figure 5: The different cycles the data transferring hypothesis requires. The red lines show the time that +Turing machines exist which is not the same in the first and second rounds. +B: Due to the fact that CTC is a time-like line, is it necessary to be unique, start from the origin of +time, and continue to a far future? +For instance, can we assume that every particle is able to own its particular CTC and let the cor- +responding CTC to a Turing machine be a small loop, passing only 2022? In other words, while +the first-ever Turing machine in the world rounds on a CTC and according to the weak axiom, is +destroyed before reaching 1936, another Turing machine rounds on a CTC passing only 2022, over +the entire path of which Turing machines always exist. Moreover, some particles may not round on +a CTC. +Figure 6: Various CTCs for different particles. +A: In this case, how might the communication of different particles be? +B: We can say that if two particles on a CTC round interacted with each other, they would also +interact on all other CTC rounds. Like Novikov’s self-consistency principle which says that only +those particles can travel back in time that does not change the past. +A: Consider a piece of stone, half of which rounds on a 20 million-year length CTC and the other +half rounds on a 2 million-year length CTC. Then what would happen to this stone? +Similarly, what if we consider these two CTCs for two separate stones? +B: Hence, having a single CTC for all particles in the universe seems reasonable. However, is it +necessary for the CTC to return to a moment before 1936, the origin of Turing machines? +A: Assuming CTC does not return a specific moment is a strong hypothesis. Therefore, It is reason- +able to discuss other models in which the starting and ending points are not necessarily similar. +(a) Loop +(b) Loop∞ +Figure 7: Chronology-violating models rather than CTC. +By the above discussion, the data transferring hypothesis also seems like not possible in our universe. So, we +should discuss possible situations that a universe containing CTCs may have in the next section. +7 + +TM +TM +1936 +1936 +CTC rounds +1st round +2nd round +3rd round +Outside observer +0 +1936 +T +2T-1936 2T +3T +4T +5T +6T+3.1.2 +Possible Situations +In this section, we want to discuss almost all possible cases for a universe equipped with CTC. For each of these, +we will discuss problems that can appear and try to propose solutions for them. It should be noted that there +might be other cases we have missed, and we would be glad if you informed us of any you have reached. +The weak axiom without further conditions +Assuming there is no concentration on the cycle on which +the CTC rounds, nor any specific condition for the strong axiom, we only know that each particle in space-time +will be destroyed before returning to its birth moment. +Hence, as we discussed in the previous section, the +data transferring hypothesis needs Turing machines to be existed throughout time, while we know that it is not +satisfied in our current universe. Thus, although the data transferring hypothesis would be a helpful claim in +possible universes, it does not work in our world. +Returning to an approximately equivalent universe +In this case, suppose that particles moving on +a CTC return to a universe approximately equivalent to the one in the previous cycle instead of an identical +universe, meaning that some items of the universe can be excepted to be different in two rounds on the CTC; +such as the existence of Turing machines. In other words, in the first-ever pass of space-time, Turing machines +were invented in 1936, before which there was no Turing machine in the world. However, in all following cycles +resulting from CTCs, Turing machines exist throughout time. +It should be remarked that according to the strong axiom we inferred from the problem discussed in section 2, no +TMCTC remains alive in a whole cycle of CTC. Rather, the concept of Turing machines, with necessarily different +instances, is well-known at all times. +Thus, the prerequisites for the data transferring hypothesis will be held, and therefore, according to what we +explained above, the deduced results about complexity and computability classes of TMCTC are valid. +The weak axiom in the last possible moment +Suppose that each particle on a CTC can move safely +and freely in positive and negative time directions until it returns just before its birth moment when it is destroyed +and immediately recreated. Therefore, not only holds the strong axiom and there is no more than one version of +any particle in the universe, but from a third person’s point of view, who is out of space-time, all creatures are +always alive. +However, in this case, the problem remains for Turing machines, since by destruction of a Turing machine, all +of its information will be missed and will not be accessible after its recreation. Thus, it seems like the Turing +machine’s obtained data has been erased and cannot be used in the next cycle on CTC. +Transferring data between various Turing Machines +Implementing the data transferring hypothesis +amongst more than two Turing machines would be another situation that comes to mind for solving the problem. +However, still requires in every moment of the cycle, at least one Turing machine exists in order to carry the +information. Therefore, it does not address the issue, and the problem still remains. +Transferring data between different time directions +We have already defined that the movement of +particles on the CTC in the positive or negative direction of time depends on whether they are moving toward the +future or the past. Suppose that the data transferring hypothesis is possible between a Turing machine moving +in the positive direction and one in the negative direction, meaning that a Turing machine M1, which in moment +t0 goes toward the past, can transfer its information to a Turing machine M2, which goes toward the future in +moment t0. Hence, the existence of Turing machines at all times is no more required for the data transferring +hypothesis. However, it should be discussed whether such communication between particles with different time +directions is possible. For instance, due to movement in different time directions, the two Turing machines can +touch each other just in a second, after which they have no access to each other, and therefore, transferring an +arbitrary amount of information in a second must be possible, which seems not. This case indeed can solve the +problem; nevertheless, it is unlikely to be possible. +Figure 8: Data transferring between Turing machines M1, moving to the past, and M2, moving toward +the future, in t0. +8 + +in2 +M2 +out2 ++ +to +M1 +out1 +in1After discussing all cases we came into, we request readers notify the authors if they attained any other +reasonable scenario. +4 +Conclusion +In this paper, after studying the definition of closed time-like curves and the fundamental difference revisiting the +past within them has with the initial idea of time travel, we considered the proof of computability of the halting +problem by TMCTC provided in (Aaronson et al., 2016). It raised the physical objection discussed in section 2 as +the weak and strong axioms, explaining that no particle of the universe survives a full round on a CTC, leading +to the inability of TMCTC to solve the halting problem. +Then, we tried to address this issue using the data transferring hypothesis, which basically utilizes another TMCTC +as a medium for storing information over a cycle of CTC. The data transferring hypothesis also required other +seemingly infeasible conditions, such as the existence of Turing machines throughout time. +Finally, we reviewed all the possible physical scenarios as far as we could think of for a universe containing CTCs +in the last section, albeit there might be additional scenarios that we welcome being acquainted with. +References +Aaronson, S. (2005). Guest column: NP-complete problems and physical reality. ACM Sigact News, +36(1), 30-52. +Aaronson, S., Bavarian, M., & Gueltrini, G. (2016). Computability theory of closed timelike curves. +arXiv preprint arXiv:1609.05507. +Aaronson, S., & Watrous, J. (2009). Closed timelike curves make quantum and classical computing +equivalent. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, +465(2102), 631-647. +Brennan, J. H. (1997). Time travel: A new perspective. Llewellyn Publications. +Craig, W. L. (1988). Tachyons, time travel, and divine omniscience. The Journal of Philosophy, 85(3), +135-150. +Deutsch, D. +(1991). +Quantum mechanics near closed timelike lines. +Physical Review D, 44(10), +3197–3218. +Einstein, A. (1905). On the electrodynamics of moving bodies. Annalen der physik, 17(10), 891-921. +Einstein, A. (1916). The foundation of the general theory of relativity. Annalen der physik, 49, 769-822. +Friedman, J., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., & Yurtsever, +U. (1990). Cauchy problem in spacetimes with closed timelike curves. Physical Review D, 42(6), +1915. +G¨odel, K. (1949). An example of a new type of cosmological solutions of einstein’s field equations of +gravitation. Reviews of Modern Physics, 21(3), 447–450. +Hawking, S. W. +(1999). +Space and time warps (public lecture). +https://www.hawking.org.uk/ +in-words/lectures/space-and-time-warps: The Stephen Hawking Estate. +Hawking, S. W., & Ellis, G. F. (1973). The large scale structure of space-time. Cambridge, England: +Cambridge University Press. +Jones, E. M. (1985). ”where is everybody?” an account of fermi’s question. Los Alamos National Lab., +NM (USA). +Lobo, F., & Crawford, P. (2003). Time, closed timelike curves and causality. In The nature of time: +Geometry, physics and perception (p. 289-296). Springer. +Nozick, R. (1969). Newcomb’s problem and two principles of choice. In N. Rescher (Ed.), Essays in +honor of carl g. hempel: A tribute on the occasion of his sixty-fifth birthday (Vol. 24, p. 114–146). +Dordrecht: Springer Netherlands. +Smith, N. J. (2021). Time travel. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy (Fall 2021 +ed.). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/ +fall2021/entries/time-travel/. +van Stockum, W. J. (1938). Ix.—the gravitational field of a distribution of particles rotating about an +axis of symmetry. Proceedings of the Royal Society of Edinburgh, 57, 135-154. +Wolpert, D. H., & Benford, G. (2013). The lesson of newcomb’s paradox. Synthese, 190(9), 1637–1646. +9 + diff --git a/KNFJT4oBgHgl3EQfwy3r/content/tmp_files/load_file.txt b/KNFJT4oBgHgl3EQfwy3r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec3592b1479f5e6ef9ff205384a73e703f73da5a --- /dev/null +++ b/KNFJT4oBgHgl3EQfwy3r/content/tmp_files/load_file.txt @@ -0,0 +1,411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf,len=410 +page_content='Turing Machines Equipped with CTC in Physical Universes Sara Babaee K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1∗, Farzad Didehvar1† 1Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran ∗Email: sarababaei@aut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='ir †Email: didehvar@aut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='ir Abstract We study the paradoxical aspects of closed time-like curves and their impact on the theory of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' After introducing the TMCTC, a classical Turing machine benefiting CTCs for backward time travel, Aaronson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' proved that P = PSPACE and the ∆2 sets, such as the halting problem, are computable within this computational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Our critical view is the physical consistency of this model, which leads to proposing the strong axiom, explaining that every particle rounding on a CTC will be destroyed before returning to its starting time, and the weak axiom, describing the same notion, particularly for Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' We claim that in a universe containing CTCs, the two axioms must be true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' otherwise, there will be an infinite number of any particle rounding on a CTC in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' An immediate result of the weak axiom is the incapability of Turing machines to convey information for a full round on a CTC, leading to the proposed TMCTC programs for the aforementioned corollaries failing to function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' We suggest our solution for this problem as the data transferring hypothesis, which applies another TMCTC as a means for storing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A prerequisite for it is the existence of the concept of Turing machines throughout time, which makes it appear infeasible in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, we discuss possible physical conditions that can be held for a universe containing CTCs and conclude that if returning to an approximately equivalent universe by a CTC was conceivable, the above corollaries would be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Keywords: Turing machine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' closed time-like curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' time travel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' strong axiom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' weak axiom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' data transferring hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1 Introduction Roughly speaking, space-time is a four-dimensional continuous coordinate system that combines the three spatial dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=', Euclidean space) with one-dimensional time in such a way that the four axes are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For instance, time passes slower in higher velocities according to special relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Einstein, 1905) Thus, space-time consists of points, which we name events, that can be demonstrated as (x, y, z, t) and are used to show the coordinates of particles of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Also, the path that is taken by a particle in space-time is called its world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Physical phenomena can be explained in variant types of space-times, such as flat, like Minkowski, or curved space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In each of them, numerous properties may appear, such as chronology-violating, meaning that an event a preceding an event b might occur after b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Various unfamiliar and somehow unusual features, like wormholes and singularities, tend to emerge in space-time owning this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Deutsch, 1991) An example of the chronology-violating space-times we want to study in this paper is a universe containing CTCs, which entails curved space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' To realize what exactly a CTC is, we should first review some concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Let us consider space-time as a three- dimensional coordinate system in which a plane graded in the ct1 unit represents space, and the orthogonal axis on the plane graded in the t unit displays the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Now, if we assume a ray of light on the origin of space-time, since it moves with velocity c, in the next time unit, t, it will have taken a ct distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, in time t, the light will be at any point on the circle with origin (0, 0, t) and radius ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Due to the fact that time and space are continuous, the possible paths the light ray can take illustrate a cone with an apex at the origin called the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' This cone states the future or positive direction of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Similarly, the possible paths that light has taken to reach the origin of space-time also form another cone, which expresses the past or negative direction of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1c is the universal constant for speed of light in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='11632v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='CC] 27 Jan 2023 Any vector v = (x, y, z, t) = (⃗d, t)2 in space-time, demonstrates a movement between two points A and B with spatial difference ⃗d in time t and based on the relation between ⃗d and t is divided into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If |⃗d| < ct, the path ⃗d has been taken with a velocity less than light’s and so can be in a particle’s world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In this case, the vector is named time-like and is inside the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If |⃗d| = ct, the path ⃗d has been taken with the velocity of light and so cannot be in any particle’s world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In this case, the vector is named light-like and is on the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Finally, if |⃗d| > ct, the path ⃗d has been taken with a velocity more than light’s and so cannot be in any particle’s or light’s world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In this case, the vector is named space-like and is outside the light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A Closed Time-like Curve, or CTC for short, is a time-like line, in the above definition, with the same starting and ending points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Since it is time-like, can be a particle’s world line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Any particle of the system owning this kind of world line may return to a state of its past, or in other words, a coordinate of space and time that has been before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The possibility of the existence of CTCs was raised for the first time by a dutch mathematician, Willem Jacob van Stockum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (van Stockum, 1938) After that, Kurt G¨odel introduced the G¨odel metric as a solution to Einstein’s field equations, leading to express a universe containing CTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (G¨odel, 1949) Factually, Einstein formulated the field equations, also known as EFE, within the general theory of relativity in (Einstein, 1916).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Since then, several solutions have been found for EFE, which are called metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' These solutions tend to describe universes that include exotic features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' for instance, black holes in the Schwarzschild metric, traversable wormholes in the Morris–Thorne metric, and CTCs in the G¨odel metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Intuitively, a space-time equipped with CTCs provides the possibility of traveling back in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Similar to someone starting going rightward on the spherical earth and finally reaching a coordinate lefter than their departure point, a particle of the system can enter the CTC and move forward on it and, by the passage of time, since CTC is closed, eventually arrives at a time before its travel’s starting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Smith, 2021) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1 Paradoxes Come with CTC At first sight, time travel causes several logical as well as intuitional paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In this section, we study the definition, scenario, and possibly the given responses to some sorts of these paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, we discuss a similar issue in section 2 and proceed with trying to respond to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1 Consistency Paradoxes A prominent sort of time travel paradox is the consistency paradox which happens by performing some changes to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A convenient example of that is the grandfather paradox, which can be described as follow: Imagine a person who travels back in time to one of their ancestors’ eras and kills their grandfather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' As a result, they will never be born, so they will not have time-traveled and killed their grandfather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Consequently, they are born and travel to the past, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Aaronson, 2005) There also exist a few other equivalent scenarios for the grandfather paradox, such as the story that Hawking and Ellis claim: Suppose that with a suitable rocket ship, a person travels in time to arrive before their departure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' They can alter any past events only if we assume they have free will, leading to stopping themselves from setting out on their travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Consequently, they will not travel in time, and therefore, nothing will happen to prevent their time travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Hawking & Ellis, 1973) Or the Hitler Paradox: The first thing that may cross the minds of some humanitarians to do with the ability of time travel is going back to some time before 1939 and killing Adolf Hitler to prevent the wage of World War II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, his murder, aside from the widespread impact that might have on the future, wipes the reason3 for its own happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Meaning that, without the tragedy of WWII, there is no motivation for the time traveler to kill Hitler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Brennan, 1997) Basically, during all of these equivalent scenarios, an event a in the past influences an event b in the future, for example traveling in time to the past, which then causes the occurrence of the event a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='2 Fermi Paradox One other type of contradiction is similar to the Fermi paradox, which can be stated as if traveling back in time was possible, where would future time travelers be right now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Jones, 1985) Various answers are given to this paradox, such as time travel may be extremely expensive or dangerous (Smith, 2021) or reaching our era could be impossible by traveling in time for space-time might not be warped enough in our time to allow the existence of closed time-like lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Hawking, 1999) 2Let ⃗d = (x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 3Unlike the other mentioned scenarios, in the Hitler Paradox, instead of physically preventing time travel, the reason for it vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='3 Newcomb Paradox A further contradiction that can be referred to regarding time travel is known as Newcomb’s paradox, in which a game is played as follows: (Wolpert & Benford, 2013) Two players in the roles of “predictor” and “chooser” play the game with two boxes, namely A, containing 1000$, and B, which is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The chooser will select either both boxes A and B or just box B in their turn and will be paid all the money in their chosen box(es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The predictor will do one of the following in their turn: – If they foresee that the other player would select both boxes, they will leave box B empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' – Otherwise, if they predict that the other player would select just box B, they will put 1’000’000$ in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Figure 1: Newcomb’s game payoff matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In game theory, based on the strategic dominance principle, the chooser should always choose both boxes A and B, whereas using the expected utility principle and the fact that the predictor is “infallible”, the chooser should always take box B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Nozick, 1969) In philosophy, it is believed that perfect prediction or time travel, which can be used as a tool for perfect prediction, conflicts with free will;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' for the sake that it is not recognizable that the prediction is the result of choice or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Craig, 1988) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='4 Causality Loop A causality loop is another paradox that might occur via time travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' It means that an event a in the past causes4 an event b in the future, which is indeed the cause of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, both events exist in space-time, while their origin cannot be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Lobo & Crawford, 2003) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='5 Knowledge Creation Paradox The paradox that might happen by traveling in time and does not vanish by consistency methods since it is not known as a logical contradiction at all is the knowledge creation paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' To comprehend it, suppose that someone travels back in time to reach G¨odel’s era and meets him before 1931, the publishing date of his incompleteness theorem paper, where they dictate G¨odel the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' As a result, he admires them and publishes the paper as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, it is said that every occurrence in the world, with and without their time travel, is identical, and nothing paradoxical happens, excluding that neither G¨odel nor the time traveler genuinely thought about and produced the contents of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In other words, there is no original point of creation for the incompleteness theorem, and knowledge has been created without anyone putting effort into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' This non-intuitive feature of time travel, which is thought to be preserved in CTCs, is the foundation of related results in (Aaronson, 2005) and (Aaronson, Bavarian, & Gueltrini, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' They try to solve a hard problem without allocating the desired amount of time or memory to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, we argue that by considering a more precise formulation of the scenario, the universe is not entirely the same in both visits from an outward observer’s point of view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' in the first, G¨odel thinks about the incomplete- ness theorem, while in the second visit, he communicates with a time traveler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Therefore, this scenario also can be logically paradoxical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Also, the aforementioned scenario can be seen as an instance of causality loops since it is not recognizable that the time traveler induced G¨odel the incompleteness theorem or vice versa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' they learned it from G¨odel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Here, we should declare that in a universe containing CTCs, the whole universe, including all creatures, indeliberately return to a time coordinate resulting in the universe being identical in any visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In contrast, through time travel, just an individual travels to a specific moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For more explanation, considering the story 4It is worth noticing the difference between the used terms influence and cause when defining the consistency paradoxes and the causality loop, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=" 3 Chooser A+B B A+B 1000 0 Predictor B 1'000'000 1'001'000of the grandfather paradox, let us see the problem from the viewpoint of an observer out of space-time who does not move on a CTC." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then for them, the world is not exactly as it was after the grandson’s time travel, since in the first view of the coordinate of space-time, the grandson does not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' however, in the second view, he stands alongside his grandfather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, arriving at an already-been moment via CTC is not equivalent to an individual’s time travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' To our best knowledge, the difference between these two concepts has not been discussed sufficiently, which persuaded us to think about potential problems that might arise from CTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='2 CTC Consistency So far, various people with different approaches have tried to create conditions to make CTCs consistent and eliminate their associated paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For instance, Novikov’s self-consistency principle explains that events that alter the past occur with a probability equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Therefore, by excluding all the self-inconsistent happenings, time travel paradoxes vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (Friedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=', 1990) Here, we explain the method that Deutsch proposed in (Deutsch, 1991) and work based on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In (Deutsch, 1991), he studied the physical effects of the existence of CTCs with a quantum computational ap- proach and modeled the computations using CTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Furthermore, assuming that classical physics has a minimum approximate consistency near closed time-like curves, he showed that chronology violation might place conflicting constraints on inputs that result in paradoxes in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Even though, in the quantum case, all these paradoxes are avoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' At first, Deutsch generally claims that every computational network in which there is a closed path in space- time for the information can be converted into a simplified standard computational network, which with every set of inputs, generates the same outputs as the original network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Also, in this equivalent network, bits interact only in gates, where operations are performed, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The n bits on closed time-like paths first go to an ambiguous future for all gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then go back to the past of all gates with a negative delay and, finally, resume their original paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If the required time for passing all gates in the computational network was T, the bits delay time, or equivalently the time each bit travels to the past, is −T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' At the same time, m bits from a definite past enter the network as input, creating the output by communication with n CTC bits, and going to an unambiguous future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In addition, Deutsch expresses that for CTCs to be consistent, the evolutionary operator of each gate must have a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, this point would be the stable state of the information on closed time-like paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' He also shows that such a fixed point always exists in the quantum case, although it may not exist in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='3 TMCTC Computational Model In 2016, Scott Aaronson, based on Deutschian CTC, proposed the Turing machine equipped with CTC, named TMCTC, in the classical and QTMCTC in the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In the following, we will discuss the classical computational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The TMCTC has two different types of memory registers (Turing machine’s tapes): RCR: Registers that respect the chronology, coming from a known past and going to an unambiguous future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Note that the input of the model will be on these registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' RCTC: Registers round on the CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (a) Tape (b) A general scheme (Aaronson & Watrous, 2009) Figure 2: The TMCTC computational model Similar to a classical Turing machine, we assume that both tapes have infinitely many cells, though we can use a finite number of them in each Turing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence we can show the Turing machine’ information with a pair of binary strings (x, y), such that x is the content of RCR and y is the content of RCTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Moreover, for each input x of the machine, there is an infinite dimensional stochastic matrix Sx, which maps every binary string y ∈ {0, 1}∗ to a binary string Sx(y) ∈ {0, 1}∗ with a defined probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In fact, this stochastic matrix is a Markov chain, and a fixed-point for the operator of each machine is equivalent to a stationary distribution for its corresponding Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 4 RCR 1 0 # 0 1 # 0 0 0 RCTCAnswer 不个 c RCTC RCR 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='4 Proof of TMCTC-Computability of Halting Problem Aaronson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' have introduced a TMCTC program for solving the halting problem in (Aaronson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' This program takes a ⟨P⟩, the description of a Turing machine P, on RCR as input and determines whether P without any inputs will eventually halt or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For doing this, we consider σt as the configuration of Turing machine P within t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus the state σ0 for P() is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Also, σt+1 is simply attainable from σt by running one more step of P, for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Additionally, for every arbitrary string y, and by knowing ⟨P⟩, it is easy to specify whether there is a t for which y = σt or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Furthermore, we call σt a halting history if it demonstrates that P halts in the tth step of running;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' otherwise, we name it a non-halting history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' At this time, the TMCTC, by taking ⟨P⟩ on the RCR, writes an arbitrary string y on the RCTC, and we define the function S⟨P ⟩(y) as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If there existed a t such that y = σt: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If σt was a halting history, then S⟨P ⟩(y) = y, and it would output 1 on the RCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Otherwise, S⟨P ⟩(y) = σt+1 with probability 1 2, and S⟨P ⟩(y) = σ0 with probability 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Otherwise, S⟨P ⟩(y) = σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Now, if P() halts, there is a t, for which σt is a halting history, and y = σt is the only fixed-point of the operator S⟨P ⟩(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' As a result, TMCTC will output 1, meaning that P() will halt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In contrast, if P() never halts, the geometric distribution over steps is the only fixed point of the operator S⟨P ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In other words, P(σi) = ( 1 2)i+1, so the TMCTC will never halt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 2 The Problem with the Proof In this section, we want to bring up some doubts, as a result of the physical constraints of our universe, about the aforementioned proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' To begin with, let us study CTC in a physical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Having this in mind, consider a universe containing CTCs and therefore, the ability to violate chronology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Suppose that a particle of it, namely x, which is born at the moment t0, goes toward the future5 until reaching the moment t−1, a time before x’s birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, if x continues its motion on the CTC and reaches the moment t0 without any damages, assuming that all the other particles in the first visit of t0 by x are identical with the second turn, x will be born again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, there will be two copies of x in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Consequently, both x’s move toward the future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' so that, when they arrive at the moment t0 and x is reborn, there are three copies of it in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Likewise, this leads to having infinitely many x’s in the world which seems impossible in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, we argue that x must be destroyed at some point between t−1 and t0, which will be restated as the strong axiom in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Figure 3: The motion of particle x on a CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The same problem can be discussed for a Turing machine equipped with CTC running the program of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Suppose that in our physical world, the TMCTC is created and starts its computation at moment t0 with a ⟨P⟩ on its RCR, and an arbitrary y on its RCTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If by the passage of time, the machine returns to the moment t0, it will be recreated and therefore according to the argument we made above, there will be infinitely many of it in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, by reaching a moment t−1 < t0, the TMCTC should be destroyed before t0, which will be restated as the weak axiom in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1, and, as a result, is not able to use any information it has acquired from the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Meaning that y is always the arbitrary string and will not get any closer to a halting history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' More precisely, if we trace S⟨P ⟩(y), we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If there existed a t such that y = σt: 5Actually, x does not move in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' instead, time naturally passes for x, and since time-like lines are close in space-time, at some point, by moving in the same direction as the future light cone, the t component decreases for x’s coordinate in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 5 x x t_11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If σt was a halting history, then y will remain on the RCTC, and RCR would output 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Basically, this is the only case that the program responds correctly without any usage of CTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, there is no guarantee that it always happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' If σt was a non-halting history, either another step of the Turing machine is supposed to be appended to y, or it ought to be changed to the trivial configuration, σ0, until the upcoming visit of t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Nevertheless, by the destruction of the Turing machine before t0, this progress will be lost, and the TMCTC will always start its computation with y on the RCTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Otherwise, the trivial configuration is supposed to be written on the RCTC before the subsequent visit of t0, which, similar to the previous case, will not be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Moreover, there are some ambiguities about CTCs in the physical universe that have not been addressed during the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For instance, all definitions are about microscopic particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' how are they extended for macroscopic particles such as Turing machines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Further, time is considered just like space, which seems not reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 3 Our Proposed Solution We aim to study the problem brought up in the previous section from two different points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Firstly, we try to propose a solution by considering the rules of classical physics over the universe, after which we will look at the problem outside the current world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1 Solving the Problem in Our Current Universe In section 2, we raised a problem, describing that the lasting of particles for a whole round on a CTC results in having an infinite number of each particle in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' It seems like the issue does not match our physical intuition, even though it appears to be mathematically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In other words, we assume that the problem only exists in our physical world, while the mathematical world is a possible and untroubled platform for TMCTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' As a consequence of the argument we made in the previous section, the two following axioms can be stated for a classical universe containing CTCs: Strong axiom: No particle survives a full round of movement on a CTC and will be destroyed before returning to its starting point in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Weak axiom: Every Turing machine rounding on a CTC, will be destroyed before returning to its starting point in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Now premising at least the weak axiom, we will solve the problem by transferring data between Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' We should remark that, according to the definitions in the first section, moving in the positive or negative direction of time mean movement toward the future or past, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Suppose a TMCTC, namely M, starts its calculation at time t0, moves in positive time until time t1, and since then, continues moving in negative time, reaches t2 ≤ t0 and moves again in positive time to reach t1 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In this case, according to the weak axiom, M will be destroyed at time t3 when t2 ≤ t3 ≤ t1 and M is moving in negative time, or t′ 3 when t2 ≤ t′ 3 ≤ t0 and M is moving in positive time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Now to solve the problem, suppose that another TMCTC, namely M ′, can be placed in M’s path before reaching t3 or t′ 3, which we call the data transferring hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, the calculated output of M until that moment can be used as input of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, M ′ will move from t3 or t′ 3 to t0, when it gives its data as input of M, without any process or alteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Within this approach, particles and Turing machines will be destroyed during an entire round movement on CTC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' nevertheless, data can travel in time and remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, M can use the information gathered by moving on a CTC, and so given claims and proofs of TMCTC will be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (a) (b) Figure 4: Data transferring between Turing machines M and M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, in a universe where data transferring was possible, as Aaronson claimed, not only PCTC = PSAPCECTC but the halting problem would be computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 6 + in\' out\' W M in out to t2 in\' out\' M M\' in out Tout 1nin\' out\' M\' M + in out Iout to in "3 in\' W out\' W td in out3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1 Prerequisites for the Data Transferring Hypothesis We demonstrate the necessary conditions of the data transferring hypothesis via the following conversation: A: A prerequisite of the data transferring hypothesis is the existence of Turing machines throughout time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' since otherwise, in the first round of the CTC, there was no Turing machine in space-time until 1936, when Turing machines were invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, in the subsequent rounds, there should exist at least one Turing machine at every moment, including before 1936, to establish the data transferring hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' This leads to visiting dissimilar universes in different cycles on the CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Figure 5: The different cycles the data transferring hypothesis requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The red lines show the time that Turing machines exist which is not the same in the first and second rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' B: Due to the fact that CTC is a time-like line, is it necessary to be unique, start from the origin of time, and continue to a far future?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For instance, can we assume that every particle is able to own its particular CTC and let the cor- responding CTC to a Turing machine be a small loop, passing only 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In other words, while the first-ever Turing machine in the world rounds on a CTC and according to the weak axiom, is destroyed before reaching 1936, another Turing machine rounds on a CTC passing only 2022, over the entire path of which Turing machines always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Moreover, some particles may not round on a CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Figure 6: Various CTCs for different particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A: In this case, how might the communication of different particles be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' B: We can say that if two particles on a CTC round interacted with each other, they would also interact on all other CTC rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Like Novikov’s self-consistency principle which says that only those particles can travel back in time that does not change the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A: Consider a piece of stone, half of which rounds on a 20 million-year length CTC and the other half rounds on a 2 million-year length CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then what would happen to this stone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Similarly, what if we consider these two CTCs for two separate stones?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' B: Hence, having a single CTC for all particles in the universe seems reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, is it necessary for the CTC to return to a moment before 1936, the origin of Turing machines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' A: Assuming CTC does not return a specific moment is a strong hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Therefore, It is reason- able to discuss other models in which the starting and ending points are not necessarily similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' (a) Loop (b) Loop∞ Figure 7: Chronology-violating models rather than CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' By the above discussion, the data transferring hypothesis also seems like not possible in our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' So, we should discuss possible situations that a universe containing CTCs may have in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 7 TM TM 1936 1936 CTC rounds 1st round 2nd round 3rd round Outside observer 0 1936 T 2T-1936 2T 3T 4T 5T 6T+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content='2 Possible Situations In this section, we want to discuss almost all possible cases for a universe equipped with CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For each of these, we will discuss problems that can appear and try to propose solutions for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' It should be noted that there might be other cases we have missed, and we would be glad if you informed us of any you have reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The weak axiom without further conditions Assuming there is no concentration on the cycle on which the CTC rounds, nor any specific condition for the strong axiom, we only know that each particle in space-time will be destroyed before returning to its birth moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, as we discussed in the previous section, the data transferring hypothesis needs Turing machines to be existed throughout time, while we know that it is not satisfied in our current universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, although the data transferring hypothesis would be a helpful claim in possible universes, it does not work in our world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Returning to an approximately equivalent universe In this case, suppose that particles moving on a CTC return to a universe approximately equivalent to the one in the previous cycle instead of an identical universe, meaning that some items of the universe can be excepted to be different in two rounds on the CTC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' such as the existence of Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' In other words, in the first-ever pass of space-time, Turing machines were invented in 1936, before which there was no Turing machine in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, in all following cycles resulting from CTCs, Turing machines exist throughout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' It should be remarked that according to the strong axiom we inferred from the problem discussed in section 2, no TMCTC remains alive in a whole cycle of CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Rather, the concept of Turing machines, with necessarily different instances, is well-known at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, the prerequisites for the data transferring hypothesis will be held, and therefore, according to what we explained above, the deduced results about complexity and computability classes of TMCTC are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The weak axiom in the last possible moment Suppose that each particle on a CTC can move safely and freely in positive and negative time directions until it returns just before its birth moment when it is destroyed and immediately recreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Therefore, not only holds the strong axiom and there is no more than one version of any particle in the universe, but from a third person’s point of view, who is out of space-time, all creatures are always alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, in this case, the problem remains for Turing machines, since by destruction of a Turing machine, all of its information will be missed and will not be accessible after its recreation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Thus, it seems like the Turing machine’s obtained data has been erased and cannot be used in the next cycle on CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Transferring data between various Turing Machines Implementing the data transferring hypothesis amongst more than two Turing machines would be another situation that comes to mind for solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, still requires in every moment of the cycle, at least one Turing machine exists in order to carry the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Therefore, it does not address the issue, and the problem still remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Transferring data between different time directions We have already defined that the movement of particles on the CTC in the positive or negative direction of time depends on whether they are moving toward the future or the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Suppose that the data transferring hypothesis is possible between a Turing machine moving in the positive direction and one in the negative direction, meaning that a Turing machine M1, which in moment t0 goes toward the past, can transfer its information to a Turing machine M2, which goes toward the future in moment t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Hence, the existence of Turing machines at all times is no more required for the data transferring hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' However, it should be discussed whether such communication between particles with different time directions is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' For instance, due to movement in different time directions, the two Turing machines can touch each other just in a second, after which they have no access to each other, and therefore, transferring an arbitrary amount of information in a second must be possible, which seems not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' This case indeed can solve the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' nevertheless, it is unlikely to be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Figure 8: Data transferring between Turing machines M1, moving to the past, and M2, moving toward the future, in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 8 in2 M2 out2 + to M1 out1 in1After discussing all cases we came into, we request readers notify the authors if they attained any other reasonable scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 4 Conclusion In this paper, after studying the definition of closed time-like curves and the fundamental difference revisiting the past within them has with the initial idea of time travel, we considered the proof of computability of the halting problem by TMCTC provided in (Aaronson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' It raised the physical objection discussed in section 2 as the weak and strong axioms, explaining that no particle of the universe survives a full round on a CTC, leading to the inability of TMCTC to solve the halting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Then, we tried to address this issue using the data transferring hypothesis, which basically utilizes another TMCTC as a medium for storing information over a cycle of CTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The data transferring hypothesis also required other seemingly infeasible conditions, such as the existence of Turing machines throughout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Finally, we reviewed all the possible physical scenarios as far as we could think of for a universe containing CTCs in the last section, albeit there might be additional scenarios that we welcome 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+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' The lesson of newcomb’s paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' Synthese, 190(9), 1637–1646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNFJT4oBgHgl3EQfwy3r/content/2301.11632v1.pdf'} diff --git a/M9E4T4oBgHgl3EQf9A55/content/tmp_files/2301.05352v1.pdf.txt b/M9E4T4oBgHgl3EQf9A55/content/tmp_files/2301.05352v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e44ee3b868b606b8b10ed5091b7e1b81b3305108 --- /dev/null +++ b/M9E4T4oBgHgl3EQf9A55/content/tmp_files/2301.05352v1.pdf.txt @@ -0,0 +1,2338 @@ +CONCENTRATION IN GOSSIP OPINION DYNAMICS +OVER RANDOM GRAPHS ∗ +YU XING† AND KARL H. JOHANSSON† +Abstract. We study concentration phenomena in gossip opinion dynamics over random graphs. +In the model, a network is generated from a random graph with independent edges, and agents +interact pairwise randomly over the network. During the process, regular agents average the opinions +of themselves and their neighbors as updates, whereas stubborn agents do not change opinions. To +approximate the original process, we introduce gossip dynamics over an averaged graph, obtained by +averaging all possible networks generated from the random graph. Using concentration inequalities, +we derive high-probability bounds for the difference between the expected final opinions of the regular +agents in the original gossip process and those in the dynamics over the averaged graph. The result +shows that the expected final opinions for most networks generated from the random graph are close +to the final opinions of the averaged graph, which are convex combinations of stubborn-agent states +with weights depending on link probability between regular and stubborn agents. We further show +how such concentration can help study the effect of network topology on the expected final opinions +in two cases, using matrix perturbation theory: (i) When the influence of stubborn agents is large, +the original expected final opinions polarize. (ii) When the influence of stubborn agents is small, the +expected final opinions are close to each other. With the help of concentration inequalities of Markov +chains, we obtain high-probability bounds for the difference between the time average of agent states +in the original process and the expected final opinions of the averaged graph. As an application, we +apply the results to gossip dynamics over stochastic block models, which generate networks with +community structure, and derive quantitative characterization of the opinion profile. +Key words. opinion dynamics, multiagent systems, social networks, random graphs, concentra- +tion, community structure, stochastic block models +MSC codes. 93A14, 91D30, 93E15, 60F10 +1. Introduction. Social opinion dynamics study how interactions over networks +shape individual opinion evolution and have various applications [43, 58]. The last two +decades have witnessed great developments in the study of opinion dynamics, and nu- +merous mathematical approaches have been applied to the modeling and analysis of +such dynamics [3, 44]. An open problem is how to analyze the influence of specific +network structures on the opinion evolution process [44]. Many structures can be +modeled by random graph models [11, 35], but how to combine random graph theory +with the study of opinion dynamics needs further investigation. For example, commu- +nity structure describes the property that subgroups of agents are connected densely +with each other but loosely with other subgroups [26]. Such a property can often +be observed in reality, and can be modeled by stochastic block models (SBM) [1]. It +is well-known that random graphs enjoy concentration properties [18, 52]. That is, +the adjacency matrix of a randomly generated graph is close to its averaged version +with high probability. It is possible to characterize large-scale opinion evolution us- +ing concentration properties. Such results may link microscopic update of agents to +macroscopic behavior of a system [25, 44], give quantitative predictions for real opin- +ion evolution [27], and provide design insights for community detection algorithms +based on state observations [46, 55]. Novel concentration results for gossip opinion +dynamics over random networks are derived in this paper. +∗This work was funded by the Knut & Alice Wallenberg Foundation and Swedish Research Coun- +cil. +†Division of Decision and Control Systems, School of Electrical Engineering and Computer Sci- +ence, KTH Royal Institute of Technology, and Digital Futures, Stockholm, Sweden. (yuxing2@kth.se, +kallej@kth.se). +1 +arXiv:2301.05352v1 [eess.SY] 13 Jan 2023 + +2 +Y. XING AND K. H. JOHANSSON +1.1. Related Work. Individual opinions represent personal attitudes towards +some topics, events, or other persons [27], and can be modeled by scalar or vector +quantities. Opinion dynamics describe how opinions evolve through interpersonal in- +teractions [3, 16, 44] and continuous-state models are considered in this paper. The +simplest dynamics, the French–DeGroot (FD) model [21], shows how a group comes +to an agreement. In the model, agents update according to the average of their neigh- +bors’ opinions. Extensions of this model to time-varying cases have been extensively +studied [9, 15]. The gossip model is a random counterpart of the FD model, in which +agents randomly interact with each other. This update rule captures the haphazard +characteristic of social interactions. Despite its simplicity, the model can exhibit var- +ious behavior: Consensus has been studied in [12, 23], and opinion fluctuations and +disagreement emerge when there are stubborn agents who never change opinions [2]. +The Friedkin–Johnsen (FJ) model [28] is another generalization of the FD model. +It allows agents to be affected by their initial opinions, and generates long-term dis- +agreement. Unlike the previous models, the Hegselmann–Krause (HK) model [30], the +Deffuant–Weisbuch (DW) model [20], and their variations [10, 51, 59] explore how ho- +mophily influence shapes the opinion evolution. In these models, agents interact only +with those who hold beliefs similar to them, and hence tend to form clusters. Other +models [37, 47] consider negative or antagonistic interactions, which may enlarge opin- +ion difference, and end in polarization. +In addition to interpersonal influences, exogenous influences also play crucial roles +in opinion formation. Exogenous influences, such as opinion leaders, social media, +and random events, are ubiquitous in real social dynamics. It has been shown that +stubborn agents can fully determine the discussion outcome of the FD model [44]. +In the gossip model with stubborn agents, there are persistent fluctuations and long- +term disagreement [2, 45]. However, if the network is highly fluid, then regular (non- +stubborn) agents can have similar expected final opinions [2]. In contrast, for regular +agents forming two communities connected to different stubborn agents, their final +positions polarize if the influence of stubborn agents is large [19]. The current paper +revisits this classic model and show how to characterize the process in more detail +with the help of random graphs. The authors in [6] study the Tayler model and +theoretically characterize the opinion profile for the case with one or two stubborn +agents. Another phenomenon induced by exogenous influences is that random noise +can drive agents in the HK model out of clustering states and help the model reach a +quasi-consensus [50]. +Real networks often consist of numerous agents. To study large-scale group behav- +ior, researchers have proposed macroscopic models in contrast to the earlier discussed +agent-based models. Such models investigate the evolution of agent-state distributions +and can be applied to mobility modeling [53] and mean-field games [4, 7]. Eulerian ap- +proaches were introduced into the control literature for analyzing bounded confidence +models [14, 33, 38]. The paper [40] shows how to apply Eulerian approaches to spa- +tially distributed ordinary differential equations, to get partial differential equations +capturing behavior of large-scale dynamics. Graphon theory has been used recently +for modeling heterogeneous large-scale networks, and [8, 13] study the convergence of +Euler approximations of mean-field games. Random graph theory is another frame- +work for large-scale network modeling [11, 35]. Random graph models [5, 22] repro- +duce features of real networks. Most properties of random graphs (e.g., connectivity +and power-law degree distributions) hold with high probability when the network +size is large [11]. Dynamics over networks with community structure are of particular +interest. There have been papers studying how the community structure influences + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +3 +opinion evolution, mainly via mean-field approximations and simulation, such as the +DW model [24, 29], the Sznajd model [48], and voter models [41, 42]. +1.2. Contribution. In this paper we study concentration in the gossip model +over random graphs. First, we show that the expected final opinions of regular agents +concentrate around the opinions in another gossip model over an averaged graph, +which is obtained by averaging all possible networks generated from the random graph +(Theorem 4.3). The final opinions of the averaged graph are weighted averages of +stubborn-agent states, with the weights depending on link probability between regu- +lar and stubborn agents. The result states that the difference between the two opinion +vector can be bounded by a quantity depending on the maximum and minimum ex- +pected degrees of the random graph and stubborn-agent states, with high probability. +Next, using matrix perturbation theory, we study the effect of the link probability +on the expected final opinions of the averaged graph (Theorem 4.7): (i) When the +influence of stubborn agents is large, regular agents connected to stubborn agents +with positive probability have expected final opinions that are weighted averages of +stubborn-agent states. The weights depend only on this link probability. Agents not +connected to stubborn agents have expected final opinions that are averages of their +neighbors’ opinions. (ii) When the influence of stubborn agents is small, the expected +final opinions have similar values. These results for the averaged graph combined with +Theorem 4.3 yield the same conclusions for the expected final opinions of the random +graph (Theorem 4.9). We then show that the difference between the time average of +agent states and the expected final opinions of the averaged graph can be bounded +by error depending on time, network size, and random graph parameters, with prob- +ability vanishing as time and the network size grow to infinity (Theorem 4.11). We +apply these general results to the gossip model over an SBM, and thereby theoretically +characterize opinion evolution over networks with community structure. The example +generalizes our early work [56], in which a two-community SBM is studied. We also +discuss how to generalize the results to other types of exogenous influences besides +stubborn agents, such as noise. +We find that, unlike classic concentration results for adjacency matrices [11, 18], +the concentration of the expected final opinions depends not only on the maximum ex- +pected degree of the random graph but also on the minimum expected number of edges +between a regular agent and the stubborn. Given random graph models, the obtained +results can characterize large-scale opinion evolution. Different from convergence and +stability analysis [7, 8, 13, 14, 38], the current paper studies how network topology +affects the concentration of expected final opinions and when such patterns appear +during the evolution, by providing high-probability bounds for finite networks. In par- +ticular, Theorems 4.3, 4.7, and 4.9 develop a unified framework for analyzing the effect +of stubborn-agent influence and network topology, as they generalize the polarization +result in [19], complement the consensus result in [2], and quantitatively characterize +opinion evolution over networks with community structure (see Section 4.3). +Because random graphs are widely used in modeling real networks [11, 35], the +current framework makes it possible to provide both qualitative and quantitative +description for real opinion evolution. More precisely, given a real network, we can first +establish random graph models from network properties, determine qualitative results +for the evolution (e.g., whether polarization or consensus would happen), and then +give high-probability bounds for the prediction. The obtained correspondence between +community structure and agent states can also inspire development of community +detection methods based on state observations [46, 55]. Suppose that the network + +4 +Y. XING AND K. H. JOHANSSON +is unknown but several samples of an opinion trajectory are available. It is possible +to recover the agent community labels by clustering agent states. Developing such a +community detection algorithm is not done in this paper, but some further discussion +on the problem is provided in Section 4.4. +1.3. Outline. The paper is organized as follows. We define the considered model +in Section 2 and formulate the problem in Section 3. Section 4 provides the main +results. Section 5 presents numerical experiments and Section 6 concludes the paper. +Proofs are provided in the Appendix. +Notation. Denote the n-dimensional Euclidean space by Rn, the set of n×m real +matrices by Rn×m, the set of nonnegative integers by N, and N+ = N \ {0}. Denote +the natural logarithm by log x, x ∈ R. Let 1n be the all-one vector with dimension +n, e(n) +i +be the n-dimensional unit vector with i-th entry being one, In be the n × n +identity matrix, and 0m,n be the m × n all-zero matrix. Denote the Euclidean norm +of a vector and the spectral norm of a matrix by ∥ · ∥, and the maximum absolute +column (row) sum norm of a matrix by ∥ · ∥1 (∥ · ∥∞). For x ∈ Rn, denote its i- +th entry by xi, and for A ∈ Rn×n, denote its (i, j)-th entry by aij or [A]ij. Let +ρ(A) be the spectral radius of a square matrix A. For symmetric A ∈ Rn×n, denote +its eigenvalue by λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A), and let λmin(A) := λ1(A) and +λmax(A) := λn(A). By diag(A1, . . . , Ak) denote the diagonal or block diagonal matrix +with A1, . . . , Ak ∈ Rm×n on the diagonal. The cardinality of a set S is |S|. P{A} +is the probability of an event A and E{X} is the expectation of a random vector X. +An event A happens almost surely (a.s.) if P{A} = 1. For a sequence of event An, we +say An happens with high probability if P{An} → 1 as n → ∞. For two sequences of +real numbers, f(n) and g(n) > 0, n ∈ N, we write f(n) = O(g(n)) if |f(n)| ≤ Cg(n) +for all n ∈ N and some C > 0, f(n) = o(g(n)) if |f(n)|/g(n) → 0, and f(n) ∼ g(n) +if f(n)/g(n) → 1 as n → ∞. If f(n) > 0 as well, n ∈ N, write f(n) = ω(g(n)) +if g(n) = o(f(n)), f(n) = Ω(g(n)) if g(n) = O(f(n)), and f(n) = Θ(g(n)) if both +f(n) = O(g(n)) and f(n) = Ω(g(n)) hold. Occasionally we use subscripts to highlight +the dependence of these relations on n (e.g., f(n) = on(g(n))). For x, y ∈ R, let +x∨y := max{x, y} and x∧y := min{x, y}. An undirected graph G = (V, E, A) has the +agent set V, the edge set E, and the adjacency matrix A = [aij] with aij = 1 (aij = 0) +if {i, j} ∈ E ({i, j} ̸∈ E). The degree of i ∈ V is di = � +j∈V aij. +2. Preliminaries. In this section, we discuss the two crucial parts of the con- +sidered opinion dynamics over random graphs. Section 2.1 defines random graphs, +whereas Section 2.2 introduces the gossip model and states its key properties. We de- +fine modified random graphs with stubborn agents in Section 2.3, and the considered +gossip dynamics over random graphs in Section 2.4. +2.1. Random Graphs. In this subsection, we define random graphs, which +characterize properties of real networks. A commonly-used random graph model as- +sumes that edges in a network are generated independently [11, 18]. +Definition 2.1 (Random graph). Let n ∈ N+ be the number of agents, and the +symmetric matrix Ψ = [ψij] ∈ [0, 1]n×n be the link probability matrix. The random +graph RG(n, Ψ) generates an undirected graph G = (V, E, A) by letting {i, j} ∈ E with +probability ψij independent of other agent pairs, where |V| = n and i, j ∈ V. +The preceding definition is general and includes many classic examples. +Example 2.2. +(i) When ψij = ψ ∈ [0, 1], i, j ∈ V, the random graph is the Erd˝os–R´enyi (ER) + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +5 +model [22]. In this case agents are connected to each other homogeneously. +(ii) Let w = (w1, . . . , wn)T ∈ Rn with wi ≥ 0 and maxi w2 +i < � +k wk, and ψij = +wiwj/(� +k wk). Then the random graph generates networks with the expected degree +sequence w [17], and can be used for describing power-law degree distributions of real +networks [5]. +(iii) Assume that the agent set V has K ∈ N+ disjoint subsets called communities, +V1, . . . , VK, and denote the community label of i ∈ Vk by Ci = k, 1 ≤ k ≤ K. Let n = +[n1, . . . , nK]T be the size vector of the communities, where nk := |Vk| and 1Tn = n, +and the symmetric matrix Π = [πij] ∈ [0, 1]K×K be the link probability matrix between +communities. Then SBM(K, n, Π) is a random graph with link probability ψij = πCiCj, +i ̸= j, and ψii = 0. The SBM, studied in community detection [1], intuitively shows +how a community structure forms. The size of communities can also be random (see +Remark 3 of [1]). In the SBM, agents from each community are homogeneous. More +general models such as the degree-corrected SBM [26] can include heterogeneity. +2.2. Gossip Model with Stubborn Agents. In this subsection, we introduce +the gossip model with stubborn agents, which forms the second part of the considered +dynamics, and summarize some basic properties of the gossip model. +A gossip model with stubborn agents (we call it “the gossip model” hereafter +for short) is a random process evolving over a graph G = (V, E, A). The agent set +V = Vr ∪ Vs (disjoint) contains regular and stubborn agents. Set Vr = {1, . . . , nr} and +Vs = {1+nr, . . . , ns+nr}, with |V| = n = nr+ns, where nr (ns) is the number of regular +(stubborn) agents, and n is the network size. A regular agent i has opinion Xi(t) ∈ R +at time t ∈ N. A stubborn agent j has opinion z(s) +j , which does not change, representing +persistent influence of a reluctant agent or an information source. Stacking the states, +denote the state vector of regular agents at time t by X(t) ∈ Rnr and that of stubborn +agents by z(s) ∈ Rns (with slightly abuse of notation, we use z(s) +j , instead of z(s) +j−r0n, to +represent the state of j). At each time t an edge is selected, and the two corresponding +agents interact. The selection is modeled by an interaction probability matrix W = +[wij] ∈ Rn×n, where wij = wji = aij/α and α = �n +i=1 +�n +j=i+1 aij is the number +of edges. An edge {i, j} is selected with probability wij, independently of previous +update. Such a selection process describes random daily encounters in social networks. +If both i and j are regular, then Xi(t + 1) = Xj(t + 1) = (Xi(t) + Xj(t))/2. If one of +the agents is stubborn, say j, then i updates as Xi(t + 1) = (Xi(t) + z(s) +j )/2, and j +does not update. Other agents do not update at t. The update rule can be written as +X(t + 1) = Q(t)X(t) + R(t)z(s), +(2.1) +with {[Q(t) R(t)]} a sequence of i.i.d. random matrices such that with probability wij +[Q(t), R(t)] = +� +[Inr − 1 +2(e(nr) +i +− e(nr) +j +)(e(nr) +i +− e(nr) +j +)T, 0nr,ns], +if i, j ∈ Vr, +[Inr − 1 +2e(nr) +i +(e(nr) +i +)T, +1 +2e(nr) +i +(e(ns) +j +)T], +if i ∈ Vr, j ∈ Vs, +(2.2) +where we use e(ns) +j +to represent e(ns) +j−r0n for j ∈ Vs, again for notation simplicity. +Denote the expected interaction matrices by ¯Q := EG{Q(t)} and ¯R := EG{R(t)}, +where the subscript G highlights that the averaging depends on W, and hence on the +graph G. The following well-known results [2] (for rigorous analysis in the discrete- +time version, see e.g., [45, 55]) indicate that the expected final opinions depend on +the expected interaction matrices and the stubborn-agent states. +Proposition 2.3 (Stability and limit theorems). Suppose that G is connected and +has at least one stubborn agent. The following results hold for the gossip model (2.1). + +6 +Y. XING AND K. H. JOHANSSON +(i) The model has a unique stationary distribution π with mean x, and X(t) +converges in distribution to π, as t → ∞. +(ii) The expectation of the regular-agent state vector converges to x, namely, +x = lim +t→∞ EG{X(t)} = (I − ¯Q)−1 ¯Rz(s). +(2.3) +(iii) Denote S(t) := 1 +t +�t−1 +i=0 X(i). Then limt→∞ S(t) = x a.s. +The results show that, although agent states may not reach a consensus or converge +to a fixed value (instead, they may fluctuate a.s. [2]), they converge in distribution +to a stationary distribution. Also, the state time average S(t) converges to x. This +vector x characterizes the average final positions of regular agents. +2.3. Random Graphs with Stubborn Agents. To study the interplay be- +tween network structure and stubborn agents, we introduce random graphs with stub- +born agents. We modify Definition 2.1 as follows. +Definition 2.4 (Random graph with stubborn agents, RG-S). +Let nr ∈ N+ +be the number of regular agents, ns ∈ N+ the number of stubborn agents, and n = +nr + ns. Let the symmetric matrix Ψ(r) = [ψ(r) +ij ] ∈ [0, 1]nr×nr be the link probability +matrix between the regular, and Ψ(s) = [ψ(s) +ij ] ∈ [0, 1]nr×ns be the link probability +matrix between the regular and the stubborn. The random graph with stubborn agents +RG-S(nr, ns, Ψ(r), Ψ(s)) generates a graph G = (V, E, A) with |V| = n according to the +following rule: (i) A graph on the regular agents is generated from RG(nr, Ψ(r)). (ii) +For i ∈ Vr and j ∈ Vs, {i, j} ∈ E with probability ψ(s) +i,j−nr, independent of other edges. +The RG-S includes stubborn agents in the network, and the link probability matrix +Ψ(s) captures the influence of the stubborn on regular agents. Denote the portion of +regular agents by r0 := nr/n ∈ (0, 1), and that of the stubborn by s0 := ns/n ∈ (0, 1). +These portions can be functions of n, instead of constants. +2.4. Gossip Dynamics over Random Graphs. The previous subsections de- +fined the random graph models and the random gossip dynamics. To analyze opinion +dynamics evolving over random graphs, we bring these two models together. Assume +that a graph G is generated from an RG-S and then fixed. Over this graph, the gossip +takes place. See Figure 1 for an illustration. From now on, by the gossip model, we +mean the gossip dynamics evolving over a graph G generated from an RG-S: +XG,n(t + 1) = Q(t)XG,n(t) + R(t)z(s), +(2.4) +where XG,n(t) is the state vector and the superscripts G and n highlight the depen- +dence of the process on G. Here [Q(t) R(t)] has the expression given in (2.2). +For the gossip model, we know from Section 2.2 that the expected final opinion +vector exists and is unique, if the graph G is connected. Denote this vector by +xG,n := lim +t→∞ EG{XG,n(t)} = (I − ¯Q)−1 ¯Rz(s). +(2.5) +Here we use the superscripts G and n to indicate that x depends on the sampled graph +G and the network size n. Note that ¯Q and ¯R are now conditional expectations. +To study behavior of the gossip model, we use a reference system without network +randomness. By averaging all possible graphs G = (V, E, A) generated from the RG-S, +we obtain a graph ¯G = (V, ¯E, E{A}), where E{A} is the weighted adjacency matrix. +Define a gossip model over this averaged graph ¯G as follows. + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +7 +Fig. 1. Illustration of the gossip model over an RG-S. On the left side of the figure, a network +is generated and then fixed. Circles and squares represent regular and stubborn agents, respectively. +On the right, the gossip model evolves over the generated network. An edge is selected at a time. +Definition 2.5 (Gossip model over averaged graph). +Consider a random graph RG-S(nr, ns, Ψ(r), Ψ(s)) and the averaged graph ¯G = +(V, ¯E, E{A}) obtained from averaging all graphs generated from the RG-S. The gossip +model over the averaged graph is the following model that evolves over ¯G. +X∗,n(t + 1) = Q(t)X∗,n(t) + R(t)z(s), +(2.6) +where X∗,n(t) is the state vector and [Q(t) R(t)], which depends on ¯G, is given in (2.2). +Assume that the averaged graph is connected. This gossip process also has expected +final opinions. For the gossip model over the averaged graph ¯G, its interaction prob- +ability matrix is W = E{A}/E{α}. Let ¯Q := E ¯G{Q(t)} and ¯R := E ¯G{R(t)}. The +expected final opinions of the gossip model over the averaged graph is +x∗,n := lim +t→∞ E ¯G{X∗,n(t)} = (I − ¯Q)−1 ¯Rz(s). +(2.7) +3. Problem Formulation. This section formulates the considered problems. +The first problem that we consider is when the expected final opinions xG,n con- +centrate around the expected final opinions of the averaged graph x∗,n: +Problem 1. Given an RG-S and the gossip model (2.4), provide high probability +bounds for ∥xG,n − x∗,n∥. +It is well-known that random graph models have concentration properties [18, 52]. +That is, under certain degree conditions, the adjacency matrix A of the randomly gen- +erated graph G is close to its averaged version E{A} with high probability. Problem 1 +arises naturally from this observation. It is solved by Theorem 4.3 in Section 4.1. +The second problem is to provide conditions resulting in the polarization or con- +sensus of xG,n: +Problem 2. Given an RG-S and the gossip model (2.4), provide conditions for +(i) the entries of xG,n are close to the stubborn agents’ states, +(ii) the entries of xG,n are close to each other. +This problem concerns how network topology and stubborn agents shape the +profile of the expected final opinions xG,n. Note that E{A} has a simpler form than +A, so it is easier to characterize x∗,n (Theorem 4.7). Then using the solution to +Problem 1, we are able to answer Problem 2 in Theorem 4.9. +Finally, we derive bounds for the difference between the state time average SG,n(t) +:= (�t−1 +i=0 XG,n(i))/t and the expected final opinions of the averaged graph x∗,n: +Problem 3. Given an RG-S and the gossip model (2.4), provide high probability +bounds for ∥SG,n(t) − x∗,n∥. + +0 +0 +0 +口 +C +RG-S +0 +口 +0 +口 +0 +口 +t=2 +t=3 +t=4 +X9,n(t+ 1) = Q(t)X9,n(t) + R(t)≥(s) +G=(V,,A)8 +Y. XING AND K. H. JOHANSSON +This problem is important because only agent states can be observed in practice, +rather than the expected states considered in Problems 1 and 2. From Proposition 2.3 +we know that it is possible to use the state time average to estimate the expected +opinions. Hence a natural question would be when the time average becomes close +to the expected final opinions. Studying this problem can help us understand how +network topology and stubborn agents affect transient behavior of the process. The +result is given by Theorem 4.11 in Section 4.2. +4. Main Results. In this section, we first study the expected final opinions +of the gossip model, by comparing it with those of the averaged graph. We then +investigate how the state time average behaves. After that we apply the results to +gossip over an SBM-S. A discussion on further extensions concludes the section. +4.1. Concentration of Expected Final Opinions. In this section, we study +properties of the expected final opinions xG,n. Theorem 4.3 shows that the differ- +ence ∥xG,n − x∗,n∥ can be bounded by a vanishing term with high probability. Next, +Theorem 4.7 examines how the profile of x∗,n depends on network topology and stub- +born agents. Finally, we characterize the profile of xG,n in Theorem 4.9 by combining +Theorems 4.3 and 4.7. +First we introduce some notations. Given a graph G = (V, E, A), for i ∈ V, +by d(r) +i +denote the number of edges connecting i and the regular, and by d(s) +i +the +number of edges between i and the stubborn. So di = d(r) +i ++ d(s) +i . Note that the +weighted adjacency matrix E{A} contains crucial information of a random graph. +Let ∆r := maxi∈Vr{E{di}} be the maximum expected degree of regular agents, and +∆s = ∆sr := maxi∈Vs{E{di}} be the maximum expected degree of the stubborn. In +addition, denote ∆rr := maxi∈Vr{E{d(r) +i }} and ∆rs := maxi∈Vr{E{d(s) +i }}. Similarly, +let δr := mini∈Vr{E{di}}, δs = δsr := mini∈Vs{E{di}}, δrr := mini∈Vr{E{d(r) +i }}, and +δrs := mini∈Vr{E{d(s) +i }} be the corresponding minimum expected degrees. +From (2.2) and W = A/α, where W is the interaction probability matrix and α +is the number of edges in a graph G, it holds that ¯Q = EG{Q(t)} = Ir0n − ¯ +M/(2α), +¯R = EG{R(t)} = ¯U/(2α), where +¯ +M := +� +����� +d1 +−a12 +. . . +−a1,r0n +−a21 +d2 +... +... +... +−ar0n−1,r0n +−ar0n,1 +. . . +−ar0n,r0n−1 +dr0n +� +����� +, ¯U := +� +�� +a1,r0n+1 +. . . +a1n +... +... +ar0n,r0n+1 +. . . +ar0n,n +� +�� . +Denote the expected interaction matrices of the gossip model over the averaged graph +by ¯Q = Ir0n − E{ ¯ +M}/(2E{α}) and ¯R = E{ ¯U}/(2E{α}) = Ψ(s)/(2E{α}). +The following assumptions are used in the analysis. +Assumption 4.1. Assume that the following conditions hold. +(i.1) δrs > 8 log n. +(i.2) λ1(E{ ¯ +M}) > 4√∆r log n, ∆r ≥ log n, and ∆rs ∨ ∆sr ≥ log n. +(ii) Both the gossip model over the random graph and the gossip model over the +averaged graph have the same stubborn-agent states z(s). +Remark 4.2. The conditions (i.1) and (i.2) are parallel. Only one of them is needed +for main results. The condition (i.2) is more general because δrs can be zero (i.e., +some regular agents are not connected to any stubborn agents). Note that the last +two conditions in (i.2) cannot imply the first one: consider E{d(r) +i } = (log n)2 and + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +9 +E{d(s) +i } = 9 log n, i ∈ Vr. Then λ1(E{ ¯ +M}) = 9 log n < √∆r log n for large n. How- +ever, (i.1) holds in this case. The lower bounds for δrs, ∆r, and ∆rs ∨ ∆sr can be +relaxed to Ω(log n). This bound is necessary for graph connectivity with high prob- +ability [1, 18]. Although (i.1) assumes that there exists positive probability that a +regular agent is connected to some stubborn agents, this regular agent need not be +connected to any stubborn agents in a sampled graph. +□ +Next we state the first main theorem. It concerns the concentration of xG,n and +provides a high-probability bound for the difference between xG,n and x∗,n. +Theorem 4.3 (Concentration of expected final opinions). +For xG,n and x∗,n given in (2.5) and (2.7), respectively, the following results hold. +(i) Under Assumption 4.1 (i.1) and (ii), it holds that +P{xG,n exists, and ∥xG,n − x∗,n∥ ≤ εx,n} ≥ 1 − ηx,n = 1 − o(1), +(4.1) +where +εx,n = 4 +�� +(∆rs ∨ ∆sr) log n +δrs ++ 2√∆r log n∥Ψ(s)∥ +δ2rs +� +∥z(s)∥, +ηx,n = r0n1− +δrs +8 log n + 2(1 + r0)n− 1 +5 + 2n− 2 +3 . +(ii) Under Assumption 4.1 (i.2) and (ii), (4.1) holds with +εx,n = 2 +� +� +(∆rs ∨ ∆sr) log n +λ1(E{ ¯ +M}) − 4√∆r log n + +2√∆r log n∥Ψ(s)∥ +λ1(E{ ¯ +M})(λ1(E{ ¯ +M}) − 4√∆r log n) +� +∥z(s)∥, +ηx,n = 2(1 + r0)n− 1 +5 + 2n− 1 +8 . +Proof. See Appendix B. +Remark 4.4. The theorem indicates that the difference ∥xG,n − x∗,n∥ can be +bounded by the norm of stubborn-agent states z(s) multiplied by a quantity depend- +ing on the agent degrees, with probability that depends on the network size and the +portion of the regular (and also on the smallest expected number of edges linking a +regular agent and the stubborn (δrs) in (i)). A lower bound of λ1(E{ ¯ +M}) can be found +in [36], which is related to “bottleneck” of the network. Note that (2.5) and (2.7) give +a trivial bound ∥xG,n − x∗,n∥ = O(∥z(s)∥). The bound is nontrivial (i.e., o(∥z(s)∥)), +if δrs or λ1(E{ ¯ +M}) is large enough. A sufficient condition in the δrs > 0 case is +δrs = ω( +� +(∆r log n)1/2(∆rs ∨ ∆sr)), from ∥Ψ(s)∥ ≤ ∥Ψ(s)∥1 ∨ ∥Ψ(s)∥∞ ≤ ∆rs ∨ ∆sr. +That is, the influence of the stubborn on any regular agent has to be large enough. +A direct consequence of (4.1) is that the opinion mean 1T +nrxG,n/nr is close to its ex- +pected version 1T +nrx∗,n/nr with difference o(1), if |z(s) +j | ≤ cx for a positive constant +cx. In classic concentration results for adjacency matrices [18, 34], the upper bounds +only contain maximum expected degrees. Our results include the minimum expected +degree δrs (or λ1(E{ ¯ +M})), for the presence of stubborn agents. The logarithm term +in the upper bounds may be removed under more careful analysis [34], which is left +to future work. +□ +Theorem 4.3 indicates that xG,n is close to x∗,n. A consequence is that xG,n +i +is +close to x∗,n +i +for most i ∈ V, as given by the following proposition. + +10 +Y. XING AND K. H. JOHANSSON +Proposition 4.5. For ε∗ > 0 and x#, x+ ∈ Rm, denote Vε∗ := {i ∈ V : |x# +i − +x+ +i | > ε∗}. If P{∥x# − x+∥ ≤ ε+} ≥ 1 − η+,n, then it holds that +P +� +|Vε∗| ≤ ε2 ++ +ε2∗ +� +≥ 1 − η+. +Remark 4.6. Set x# = xG,n, x+ = x∗,n, ε+ = εx,n, and η+ = ηx,n. Then +|Vε∗|/nr ≤ ε2 +x,n/(nrε2 +∗) with high probability. Under conditions for ε2 +x,n = o(∥z(s)∥), +for example δrs = ω( +� +(∆r log n)1/2(∆rs ∨ ∆sr)) in Remark 4.4, if |z(s) +j | ≤ cx for a +constant cx > 0, then |Vε∗|/nr = o(1). That is, xG,n +i +is very close to its expected +version x∗,n +i +for all agents except a small part o(nr), with high probability. In this way +it is possible to approximate the profile of xG,n by using x∗,n point-wisely. +□ +Another advantage of relating xG,n to a reference version x∗,n is that we can +obtain more properties of xG,n, which hold for almost all G, by studying x∗,n. To +show this, we proceed to study the profile of x∗,n. Let +¯L := +� +��� +d(r) +1 +−a12 +. . . +−a1,r0n +... +... +−ar0n,1 +. . . +−ar0n,r0n−1 +d(r) +r0n +� +��� +be the Laplacian of the graph on regular agents. The following proposition concerns +the profile of x∗,n under different cases of stubborn-agent influences. +Theorem 4.7 (Profile of x∗,n). Suppose that the averaged graph ¯G is connected. +The following results hold for x∗,n given in (2.7). +(i) (Bound for large stubborn-agent influence) If δrs > 0, then +∥x∗,n − x†,n∥ ≤ ε†,n, +(4.2) +with x†,n = (diag(Ψ(s)1s0n))−1Ψ(s)z(s) and ε†,n = 2∆rr∥Ψ(s)∥∥z(s)∥/δ2 +rs. +If δrs = 0 but λ1(E{ ¯ +M}) > 0 (hence (E{ ¯ +M})−1 exists). Assume that E{d(s) +i } > 0 +for 1 ≤ i ≤ n1 < r0n and E{d(s) +i } = 0 for n1 + 1 ≤ i ≤ n1 + n2 = r0n. Denote +E{ ¯ +M} =: +� ˆ +M (11) +ˆ +M (12) +ˆ +M (21) +ˆ +M (22) +� +, (E{ ¯ +M})−1 =: +� ˜ +M (11) +˜ +M (12) +˜ +M (21) +˜ +M (22) +� +, Ψ(s) =: +� +Ψ(s) ++ +0n2,s0n +� +, +where +ˆ +M (11), ˜ +M (11) ∈ Rn1×n1, +ˆ +M (21), ˜ +M (21) ∈ Rn2×n1, and Ψ(s) ++ +∈ Rn1×s0n, and +δ+ +rs = min1≤i≤n1{E{d(s) +i }}. If δ+ +rs > ∥ ˆ +M (21)∥2/λ1( ˆ +M (22)), then (4.2) holds with +x†,n = +� +(diag(Ψ(s) ++ 1s0n))−1 +˜ +M (21) +� +Ψ(s) ++ z(s), +ε†,n = 1 +δ+ +rs +� +∥ ˆ +M (21)∥2/λ1( ˆ +M (22)) +δ+ +rs − ∥ ˆ +M (21)∥2/λ1( ˆ +M (22)) ++ 2 max1≤i≤n1{E{d(r) +i }} +δ+ +rs +� +∥Ψ(s)∥∥z(s)∥. +(ii) (Bound for small stubborn-agent influence) +If λ1(E{ ¯ +M}) > 0 and λ2(E{¯L}) > 2∆rs > 0, then (4.2) holds with +x†,n = +1T +r0nΨ(s)z(s) +r0nλ1(E{ ¯ +M})1r0n, +ε†,n = +2 +λ2(E{¯L}) − 2∆rs +� +∆rs +λ1(E{ ¯ +M}) + 1 +� +∥Ψ(s)∥∥z(s)∥. + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +11 +Proof. See Appendix C. +Remark 4.8. The first part of (i) indicates that, if the minimum influence of +the stubborn (δrs) is much larger than the link strength between regular agents +(∆rr), then x∗,n +i +is almost a weighted average of stubborn-agent states. The weights +depend only on the link probability between i and the stubborn. That is, polar- +ization may emerge. The second part of (i) shows that agents not connected to +the stubborn have expected final opinions that are also weighted averages of stub- +born states. But the weights, given by ˜ +M (21), depend highly on network topology. If +δ(21) +rr +:= min1+n1≤i≤r0n{E{� +1≤j≤n1 aij}} > 0, then ∥ ˆ +M (12)∥2/λ1( ˆ +M (22)) has an up- +per bound (∥ ˆ +M (12)∥1∨∥ ˆ +M (12)∥∞)2/δ(21) +rr +. The bound depends on how agents with and +without stubborn neighbors are connected, similar to ∆rs∨∆sr and δrs in Theorem 4.3. +In contrast, (ii) shows that, if the influence of stubborn agents (∆rs) is smaller +than the link strength between regular agents (λ2(E{¯L})), most entries of x∗,n are +close to a weighted average of stubborn states. In other words, the expected opin- +ion vector is almost a consensus vector. Note that this conclusion does not contra- +dict that the consensus state is an average of regular initial states when there is no +stubborn agent [12, 23]. This is because we also need the concentration condition +δrs ∨ λ1(E{ ¯ +M}) > 0 to obtain the profile of xG,n (see Theorem 4.9). +□ +Consequently, by Theorems 4.3 and 4.7 and the triangle inequality, we are able +to get the following result, characterizing the profile of xG,n. +Theorem 4.9 (Profile of xG,n). Under the conditions of Theorems 4.3 and 4.7, +the following holds for xG,n given in (2.5) +P{∥xG,n − x†,n∥ ≤ εx,n + ε†,n} ≥ 1 − ηx,n, +where εx,n and ηx,n are given in Theorem 4.3, and x†,n and ε†,n in Theorem 4.7. +Remark 4.10. The theorem provides high-probability bounds for the difference +between the expected final opinions and a specific vector (the polarization or consensus +vector given in Theorem 4.7). The bound consists of two parts: the deviation of the +expected final opinions from the averaged version, and the difference between the +latter and the specific vector. This result quantitatively characterizes how the expected +final opinions are far away from typical large-scale behavior. Polarization is given by +Theorem 4.7 (i), generalizing [19] which studies a weighted graph. Theorem 4.7 (ii) +covers the consensus case, investigated also in [2], showing that the expected final +opinions achieve a consensus when the influence of stubborn agents is small. Moreover, +we can also characterize the case in the middle by using Theorem 4.3. In that case, +neither polarization nor consensus emerge. Instead, the expected final opinions exhibit +much diversity [25, 27]. It is possible to obtain concentration of opinion variances as +in [2], by solving the stationary covariance matrix and analyzing its concentration +(one way to analyze the concentration is similar to [54] but we omit the details.) +□ +4.2. Concentration of State Time Average. In this subsection we study the +concentration of state time average SG,n(t) = (�t−1 +i=0 XG,n(i))/t. We show when the +time average is close to the average version of expected final opinions x∗,n. The pre- +vious subsection studies how to bound the difference ∥xG,n − x∗,n∥. Proposition 2.3 +indicates that the time average SG,n(t) should be close to xG,n when t is large enough. +By bounding ∥SG,n(t)−xG,n∥ and the triangle inequality, we have the following result. + +12 +Y. XING AND K. H. JOHANSSON +Theorem 4.11 (Concentration of state time average). +Suppose that maxi∈Vr{|Xi(0)|} ∨ maxj∈Vs{|z(s) +j |} ≤ cx for some cx > 0. Then the +following hold for x∗,n and SG,n(t). +(i) Under Assumption 4.1 (i.1) and (ii), for εS,n > 0, t > 2¯s∗/εS,n, it holds that +P{∥SG,n(t) − x∗,n∥ ≤ √r0nεS,n + εx,n} ≥ 1 − ηS,n,t − ηS,n, +(4.3) +where εx,n is given in Theorem 4.3 (i), and +¯s∗ = 12√r0ncxE{α} +δrs +, ηS,n,t = 2r0n exp +� +− (tεS,n − 2¯s∗)2 +2t(¯s∗)2 +� +, +ηS,n = r0n1− +δrs +8 log n + 2(1 + r0)n− 1 +5 + 2n− 2 +3 = on(1), +(ii) Under Assumption 4.1 (i.2) and (ii), (4.3) holds for εS,n > 0, t > 2¯s∗/εS,n +with εx,n given in Theorem 4.3 (ii) and +¯s∗ = +6√r0ncxE{α} +λ1(E{ ¯ +M}) − 4√∆r log n, ηS,n,t = 2r0n exp +� +− (tεS,n − 2¯s∗)2 +2t(¯s∗)2 +� +, +ηS,n = 2(1 + r0)n− 1 +5 + 2n− 1 +8 = on(1). +Proof. See Appendix D. +Remark 4.12. The theorem provides high-probability bounds for the different be- +tween the state time average and the averaged version of the expected final opin- +ions. The results capture the dependence of the bounds and the probability on agent +degrees, stubborn-agent states, and time, and they thus establish concentration for +almost all graphs and all large enough time steps. The error εS,n controls the con- +centration of SG,n(t) around xG,n. Note that εx,n has a term ∥z(s)∥ = O(√r0n), so +εS,n can be set to be εx,n/∥z(s)∥ so that √r0nεS,n = o(√r0n). The probability ηS,n +depends on the network size (and also on δrs in (i)). The terms εx,n and ηS,n do not +vanish for fixed n, even if t approaches infinity, which captures the effect of the net- +work size on the concentration. When n is large, the concentration probability depends +mostly on ηS,n,t. In the δrs > 0 case, assume that δrs = ω(√∆r log n), and then set +εS,n = εx,n/∥z(s)∥ = O(√∆r log n/δrs) = on(1). When t = Ω(n3∆r), ηS,n,t = O(n−c) +with some c > 0. That is, the difference ∥SG,n(t) − x∗,n∥ = o(√r0n) with high proba- +bility. Then Proposition 4.5 ensures entry-wise concentration of SG,n(t). This example +indicates that the time for concentration also depends on the network size, although +the time lower bound may not be tight. The larger the network is, the longer we +need to wait. Since E{X(t)} is hard to obtain in practice, computing the time average +SG,n(t) helps us know the expected final positions. +□ +4.3. Concentration in Gossip over an SBM. In this subsection, we present +an example to show applications of the main results. We assume that the network +on the regular is an SBM with three equal-sized communities. Conditions for general +SBMs can be obtained similarly. +Denote the k-th community of the regular by Vrk with |Vrk| = nr1, 1 ≤ k ≤ 3, so +Vr = ∪3 +k=1Vrk. Sort the agents as follows: Vrk = {1 + (k − 1)nr1, . . . , knr1}, 1 ≤ k ≤ 3. +Assume that there are 2ns1 stubborn agents. So n = 3nr1 + 2ns1. The SBM(3, n, Π) +with n = nr113 has a link probability matrix with πkk = p1 and πkl = p2, 1 ≤ k ̸= +l ≤ 3, where p1 := (log n)β1/n, p2 := (log n)β2/n and β1 ≥ β2 ≥ 1. Let the link + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +13 +probability matrix between the regular and the stubborn be +Ψ(s) = +� +� +p31nr1,ns1 +0nr1,ns1 +c(s) +21 p31nr1,ns1 +c(s) +22 p31nr1,ns1 +0nr1,ns1 +p31nr1,ns1 +� +� , +where p3 = (log n)γ/n with γ ≥ 1 and c(s) +21 , c(s) +22 ∈ {0, 1}. In a graph generated +by this RG-S, there are three regular communities and two stubborn communities +Vsm = {3nr + (m − 1)ns1 + 1, . . . , 3nr + mns1}, m = 1, 2. Regular agents have the +same intra- and inter-community link probabilities. Also, agents in Vr1 (resp. Vr3) +have the same probability linking to Vs1 (resp. Vs2) and no edges to Vs2 (resp. Vs1). +Whether Vr2 is directly influenced by stubborn agents depends on c(s) +21 and c(s) +22 . Such +a network model describes a commonly-studied social network that consists of two +leader subgroups with opposite opinions, their follower subgroups, and another fol- +lower subgroup probably with fewer edges connecting to the leader groups [25]. The +condition β1, β2, γ ≥ 1 implies that the expected degree of an agent is Ω(log n). Hence +the SBM is connected with high probability [1]. +Suppose that nr1, ns1 = Θ(n). Then ∆r = Θ((log n)β1∨γ) and ∆rs, ∆sr, ∥Ψ(s)∥ = +Θ((log n)γ). If c(s) +21 = c(s) +22 = 1, then δrs = Θ((log n)γ). If c(s) +21 = c(s) +22 = 0, then +λ1(E{ ¯ +M}) = Ω((log n)β2∧γ) from Theorem 1 of [36]. In what follows we study only +the δrs > 0 case. The conditions in the δrs = 0 case can be obtained similarly. +From Theorem 4.3, ∥xG,n − x∗,n∥ ≤ O(∥z(s)∥(log n)−γ+[(β1∨γ)+1]/2) = o(∥z(s)∥) +if 2γ − (β1 ∨ γ) > 1. It can be shown that there exist χk ∈ R, 1 ≤ k ≤ 3, such that +x∗,n +i += χk for i ∈ Vrk [55]. When γ > β1, Theorem 4.7 (i) ensures that χ1 is close to the +average opinion of Vs1 (denoted by ς1), χ3 close to the average opinion of Vs2 (denoted +by ς3), and χ2 is not far away from the average opinion of all stubborn agents (denoted +by ς2). In contrast, Theorem 4.7 (ii) states that χk, 1 ≤ k ≤ 3, are close to the average +of stubborn-agent opinions (denoted by ¯x), if λ2(E{¯L}) = Θ((log n)β2) > (log n)γ, +ensured by β2 > γ. From the above discussion, Proposition 4.5, and Theorem 4.9, we +have the following conclusions: (i) When γ > β1 > 1 (the influence of the stubborn is +large), for a small ε∗ > 0, the number of agents in Vrk such that |xG,n +i +− ςk| ≤ ε∗ is +nr1 − O(nr((log n)1−γ ∨ (log n)2(β1−γ))) = nr1(1 − o(1)) with high probability . The +expected final opinions thus polarize. (ii) When β2 > γ (the influence of stubborn +agents is small) and 2γ − β1 > 1, the number of agents such that |xG,n +i +− ¯x| ≤ ε∗ is +3nr1 − O(nr((log n)1+β1−2γ ∨ (log n)2(γ−β2))) = 3nr1(1 − o(1)) with high probability. +The expected final opinions are close to a consensus state, even if the network has +a community structure. (iii) When β1 ≥ γ ≥ β2 (the influence of stubborn agents is +moderate) and 2γ − β1 > 1, the number of agents in Vrk such that |xG,n +i +− χk| ≤ ε∗ is +nr1 − O(nr(log n)1+β1−2γ) = nr1(1 − o(1)) with high probability. In other words, +the expected final opinions have three clusters (χk need not be ςk). Now we turn +to the state time average (Theorem 4.11). Note that E{α} = Θ(n(log n)β1∨γ), so +¯s∗ = Θ(n3/2(log n)(β1∨γ)−γ). Set εS,n = εx,n/∥z(s)∥ = Θ((log n)−γ+[(β1∨γ)+1]/2). Then +when 2γ−(β1∨γ) > 1, for time t = Ω(n3(log n)β1∨γ), the state time average SG,n(t) is +close to x∗,n with error O(√nr(log n)−γ+[(β1∨γ)+1]/2) = o(√nr) with high probability +for almost all G. +4.4. Discussions. In this subsection, we discussion several extensions of the +considered problem. +First, Theorem 4.9 has a strong connection to community detection for dynamical +processes, which studies how to partition agents based on only state observations [46, + +14 +Y. XING AND K. H. JOHANSSON +55]. In [55] we demonstrate that, by using Polyak averaging and clustering techniques, +it is possible to recover the community structure based on agent states, for the gossip +model over a weighted graph. Theorem 4.9 and the example given in Section 4.3 +guarantee that such methods can still perform well for the model over SBMs, and +with high probability the communities can be recovered (i.e., exact recovery [1]). +Secondly, the average weight of the model is assumed to be 1/2 in (2.1), but +similar concentration results with minor modifications can be derived if the weight +is q ∈ (0, 1] (i.e., Xi(t + 1) = (1 − q)Xi(t) + qXj(t)). Furthermore, the analysis can +include the case where agents are affected by random noise, for example, Xi(t + 1) = +(1 − q1 − q2)Xi(t) + q1Xj(t) + q2Yi(t) with q1, q2 ∈ (0, 1) and stationary Yi(t). It is +possible to obtain parallel results by considering E{Yi(t)} as stubborn-agent states. +Lastly, Theorem 4.11 shows that the state time average is close to the expected +final opinions of the averaged graph after time t = Ω(n3∆r). In the SBM exam- +ple (Section 4.3), such concentration holds from t = Ω(n3(log n)β1∨γ). For a gossip +model over a weighted graph, [57] shows that agent states in the same community +concentrate around the initial average opinion of that community in the time inter- +val (Θ(n log n), Θ(n(log n)β1−β2). It is possible to extend this result to the SBM case +(the details are omitted due to space limit). These two results together describe how +the model evolves during finite periods, indicating that we can characterize not only +asymptotic behavior of the model but also transient behavior in much more detail, +by using random graph modeling. +5. Simulation. In this section we present numerical simulations to illustrate the +obtained theoretical results. +We use the SBM studied in Section 4.3 to generate the network model. There +are three regular communities and two stubborn communities. Let the size of regular +communities be nr1 = 25, 250, 2500, and the size of stubborn communities be ns1 = +0.2nr1. The link probability matrix of the random graph is given in Section 4.3, and +we set β1 = 3 and β2 = 2.5. We consider three cases γ = 4, 3, 2.4, corresponding to +the cases with large, moderate, and small influence of stubborn agents, respectively. +We first study the case where c(s) +21 = c(s) +22 = 1 (that is, the community Vr2 has +edges connected to both stubborn agent communities). For each nr1 and γ, a network +is generated and then fixed. The stubborn agents in the first stubborn community +have states generated independently and uniformly from (0.9, 1) and those in the +second stubborn community have states from (0, 0.1). The expected final opinions +are computed according to (2.5). Figure 2 shows that, as the network size increases, +the large influence of stubborn agents boosts polarization (i.e., agents in different +communities move away from each other), whereas the expected opinions become +closer when the influence of stubborn agents is small. In the moderate influence case, +the expected opinions concentrate around their averaged counterparts. +Now we examine how the edges between regular and stubborn agents influence +the profile of the expected final opinions. Consider three cases: (i) c(s) +21 = c(s) +22 = 1, (ii) +c(s) +21 = 1 and c(s) +22 = 0, (iii) c(s) +21 = c(s) +22 = 0. In the first case, agents in Vr2 are connected +to both stubborn communities with positive probability. In the second case, they +are only connected to Vs1 with positive probability. In the final case, they are not +connected to any stubborn agents. We set nr1 = 250 and γ = 4 and generate xG,n +the same as earlier. Figure 3 shows that, in the first case, because Vr2 is influenced +by both stubborn communities, the agents in this community ends in a neutral place. +However, in the second case, agents in Vr2 have states close to Vs1 because only links +connecting Vr2 and Vs1 exist. In the third case, the expected final opinions of Vr2 is + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +15 +nr1 = 25 +nr1 = 250 +nr1 = 2500 +(a) Large influence (γ = 4). +nr1 = 25 +nr1 = 250 +nr1 = 2500 +(b) Moderate influence (γ = 3). +nr1 = 25 +nr1 = 250 +nr1 = 2500 +(c) Small influence (γ = 2.4). +Fig. 2. The profile of the expected final opinions under different stubborn influence. The dashed +lines represent the three distinct values of x∗,n corresponding to the communities (see Section 4.3). +(a) c(s) +21 = c(s) +22 = 1. +(b) c(s) +21 = 1 and c(s) +22 = 0. +(c) c(s) +21 = c(s) +22 = 0. +Fig. 3. The profile of the expected final opinions xG,n under different values of c(s) +21 and c(s) +21 . +similar to the first case, but this similarity results from that Vr2 have the same number +of edges linking to Vr1 and Vr3, rather than to the stubborn communities. +To illustrate the concentration of the state time average SG,n(t), we run the gossip +model with nr1 = 250 (thus n = 850), γ = 4, 3, 2.5, respectively, and c(s) +21 = c(s) +22 = 0. +The states of regular agents are generated uniformly from (0, 1). Figure 4 presents the +histogram of SG,n(t) in the three γ cases with t = 2 × 104, 5 × 104. We can see that + +0.3 +I0.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +I +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +StatesL +10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +0.2 +Frequency +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +1 +0.05 +1 +1 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +- +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +0.2 +Frequency +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.3 +II0.25 +0.2 +Frequency +0.15 +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.30.25 +Frequency +0.2 +0.15 +0.1 +0.05 +0 +Il +0 +0.2 +0.4 +0.6 +0.8 +States116 +Y. XING AND K. H. JOHANSSON +t = 2 × 104 +t = 5 × 104 +(a) γ = 4. +t = 2 × 104 +t = 5 × 104 +(b) γ = 3. +t = 2 × 104 +t = 5 × 104 +(c) γ = 2.4. +Fig. 4. The profile of the state time average SG,n(t) with t = 2 × 104 and 5 × 104. +the profile of SG,n(t) is similar to the profile of the expected final opinions shown in +Figure 2 and concentration actually appears much earlier than the predicted bound +in Remark 4.12. Note that these time steps are not too large because agents interact +less than 60 times on average, and interact only a few times with each neighbor. +6. Conclusion. In this paper, we studied concentration in gossip opinion dy- +namics over random graphs. High-probability bounds were derived for the concentra- +tion of the expected final opinions. We further showed how such concentration can +help study the effect of network topology on the expected final opinions. With the help +of concentration inequalities for Markov chains, we obtained concentration bounds for +the state time average. We then applied the results to the gossip dynamics over an +SBM, and obtained quantitative characterization of the opinion profile. Future work +includes investigating sharp concentration thresholds for the gossip and other models. +Appendix A. Auxiliary Concentration Results. +In this section, we present auxiliary concentration lemmas from which the main +results given in the paper are obtained. These lemmas are consequences of the follow- +ing standard conclusions in high-dimensional probability theory and matrix analysis. +Lemma A.1 (The Chernoff inequality, Theorems 4.4 and 4.5 of [39]). Suppose that +X1, . . . , Xn are independent Bernoulli random variables such that P{Xi = 1} = pi = +1 − P{Xi = 0}. Let X := �n +i=1 Xi and µ := E{X} = �n +i=1 pi. Then for 0 < δ < 1, +P{X ≥ (1 + δ)µ} ≤ e−µδ2/3, P{X ≤ (1 − δ)µ} ≤ e−µδ2/2. +(A.1) +Lemma A.2 (The matrix Bernstein inequality, Theorem 5.4.1 and Exercise 5.4.15 +of [52]). Suppose that Y1, . . . , YN ∈ Rn×n are independent zero-mean random matri- +ces, and are such that ∥Yi∥ ≤ K a.s., 1 ≤ i ≤ N. Then for a ≥ 0, it holds that +P +����� +N +� +i=1 +Yi +���� ≥ a +� +≤ 2n exp +� +−a2/2 +σ2 + Ka/3 +� +, +(A.2) + +0.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +States10.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +Statesdhil +10.20.15 +Frequency +0.1 +0.05 +0 +0 +0.2 +0.4 +0.6 +0.8 +Statesdhil +1CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +17 +where σ2 = ∥ �N +i=1 E{Y 2 +i }∥. If Y1, . . . , YN ∈ Rm×n are independent, mean zero, and +such that ∥Yi∥ ≤ K a.s. Then for all a ≥ 0, it holds that +P +����� +N +� +i=1 +Yi +���� ≥ a +� +≤ 2(m + n) exp +� +−a2/2 +σ2 + Ka/3 +� +, +(A.3) +where σ2 = max{∥ �N +i=1 E{Y T +i Yi}∥, ∥ �N +i=1 E{YiY T +i }∥}. +Lemma A.3. For A, B ∈ Rn×n, if A and B are symmetric, then the Weyl in- +equality holds (Theorem 4.3.1 and (6.3.4.1) of [32]): +max +1≤i≤n |λi(A) − λi(B)| ≤ ∥A − B∥. +(A.4) +If A and B are invertible, then ((5.8.1) of [32]) +∥A−1 − B−1∥ ≤ ∥A−1∥∥B−1∥∥A − B∥. +(A.5) +First, we derive a concentration bound for the matrix ¯ +M. +Lemma A.4 (Concentration of +¯ +M). +Suppose that ∆r ≥ log n. Then P{∥ ¯ +M − +E{ ¯ +M}∥ ≤ εM,n} ≥ 1−ηM,n = 1−o(1), where εM,n = 4√∆r log n and ηM,n = 2r0n− 1 +5 . +Proof. Decompose ¯ +M − E{ ¯ +M} = �r0n +i=1 +�n +j=i+1 Yij, where Yij = (aij − E{aij}) +(Eii + Ejj − Eij − Eji), 1 ≤ i < j ≤ r0n, and Yij = (aij − E{aij})Eii, 1 ≤ i ≤ +r0n < j ≤ n. Here Eij = eieT +j , 1 ≤ i, j ≤ n. Hence E{Yij} = 0, Var(Yij) = 2(pij − +p2 +ij)(Eii + Ejj − Eij − Eji) for 1 ≤ i < j ≤ r0n, and Var(Yij) = (pij − p2 +ij)Eii for +1 ≤ i ≤ r0n < j ≤ n, where pij := E{aij}. Denote ¯Y := �r0n +i=1 +�n +j=i+1 Var(Yij), so +v2 = ∥ ¯Y ∥ ≤ 4 max1≤i≤r0n{�n +j=1 pij} = 4 max1≤i≤r0n{E{di}} = 4∆r. +From (A.2), for a > 0, since ∥Yij∥ ≤ 2, +P{∥ ¯ +M − E{ ¯ +M}∥ > a} ≤ 2r0n exp +� +−a2 +4(2∆r + a/3) +� +. +Set a = 4√∆r log n, and from the assumption ∆r ≥ log n we have that +P{∥ ¯ +M − E{ ¯ +M}∥ > 4 +� +∆r log n} ≤ 2r0n exp +� +−4∆r log n +2∆r + 4√∆r log n/3 +� +≤ 2r0n exp +�−4 log n +2 + 4/3 +� += 2r0n− 1 +5 . +As a consequence of the preceding lemma, we can estimate the deviation between +the inverse of ¯ +M and E{ ¯ +M}. +Corollary A.5 (Concentration of ¯ +M −1). +(i) If δrs > 8 log n, then it holds that +P{∥ ¯ +M −1 − (E{ ¯ +M})−1∥ ≤ ε′ +M,n} ≥ 1 − η′ +M,n = 1 − o(1), +(A.6) +where ε′ +M,n = 2εM,n/δ2 +rs, η′ +M,n = r0n1−δrs/(8 log n)+ηM,n, and εM,n and ηM,n are given +in Lemma A.4. +(ii) If λ1(E{ ¯ +M}) > εM,n and ∆r ≥ log n, then (A.6) holds with ε′ +M,n = εM,n/ +[λ1(E{ ¯ +M})(λ1(E{ ¯ +M}) − εM,n)] and η′ +M,n = ηM,n. + +18 +Y. XING AND K. H. JOHANSSON +Proof. Note that E{[ ¯ +M]ii} = E{di}, and E{[ ¯ +Mij]} = −E{aij}. So by the Ger- +shgorin circle theorem, λmin(E{ ¯ +M}) ≥ min1≤i≤r0n{E{di} − E{d(r) +i }} = min1≤i≤r0n +{E{d(s) +i }} = δrs. Thus, for symmetric E{ ¯ +M}, (E{ ¯ +M})−1 exists when δrs > 0. Hence, +∥(E{ ¯ +M})−1∥ = +1 +λmin(E{ ¯ +M}) ≤ 1 +δrs +. +(A.7) +Similarly, from the Gershgorin circle theorem, it follows that λmin( ¯ +M) ≥ min1≤i≤r0n +{di − d(r) +i } = min1≤i≤r0n{d(s) +i }. Using (A.1) with δ = 1/2, we obtain that +P +� +min +1≤i≤r0n{d(s) +i } > 1 +2δrs +� +≥ 1 − P +� r0n +� +i=1 +� +d(s) +i +≤ 1 +2E{d(s) +i } +�� +≥ 1 − r0ne−δrs/8 = 1 − r0n1−δrs/(8 log n). +As a result, with probability at least 1 − r0n1−δrs/(8 log n), +∥ ¯ +M −1∥ = +1 +λmin( ¯ +M) ≤ 2 +δrs +. +(A.8) +Therefore, from (A.5), with probability at least 1 − r0n1−δrs/(8 log n) − ηM,n, +∥ ¯ +M −1 − (E{ ¯ +M})−1∥ ≤ ∥ ¯ +M −1∥∥(E{ ¯ +M})−1∥∥ ¯ +M − E{ ¯ +M}∥ ≤ 2εM,n +δ2rs +. +To show (ii), note from (A.4) that with probability at least 1 − ηM,n +|λmin( ¯ +M) − λmin(E{ ¯ +M})| ≤ εM,n, +(A.9) +so λmin( ¯ +M) ≥ λmin(E{ ¯ +M}) − εM,n > 0 when λmin(E{ ¯ +M}) > εM,n. Again from (A.5), +∥ ¯ +M −1 − (E{ ¯ +M})−1∥ ≤ ∥ ¯ +M −1∥∥(E{ ¯ +M})−1∥∥ ¯ +M − E{ ¯ +M}∥ += +1 +λmin( ¯ +M) +1 +λmin(E{ ¯ +M})∥ ¯ +M − E{ ¯ +M}∥ +≤ +εM,n +λmin(E{ ¯ +M})(λmin(E{ ¯ +M}) − εM,n), +with probability at least 1 − ηM,n, when ∆r ≥ log n. +Similar to ¯ +M, we can obtain concentration of ¯U. +Lemma A.6 (Concentration of ¯U). Suppose that ∆rs ∨∆sr ≥ log n. Then P{∥ ¯U − +Ψ(s)∥ ≤ εU,n} ≥ 1 − ηU,n = 1 − o(1), where εU,n = 2 +� +(∆rs ∨ ∆sr) log n and ηU,n = +2n−1/5. +Proof. Decompose ¯U−E{ ¯U} = �r0n +i=1 +�n +j=r0n+1 Y ′ +ij, where Y ′ +ij = (aij−E{aij})e(r) +i +(e(s) +j )T. Here e(r) +i +:= e(r0n) +i +and e(s) +j +:= e(s0n) +j−r0n. Hence, ∥Y ′ +ij∥ = |aij−E{aij}|∥e(r) +i (e(s) +j )T∥ +≤ ∥e(r) +i (e(s) +j )T∥ = 1. Now note that +¯Y ′ := +r0n +� +i=1 +n +� +j=r0n+1 +E{(Y ′ +ij)TY ′ +ij} += +r0n +� +i=1 +n +� +j=r0n+1 +E{(aij − E{aij})2}e(s) +j (e(r) +i )Te(r) +i (e(s) +j )T + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +19 += diag +� r0n +� +i=1 +(pi,r0n+1 − p2 +i,r0n+1), . . . , +r0n +� +i=1 +(pi,n − p2 +i,n) +� +, +where pij = E{aij}, so +∥ ¯Y ′∥ ≤ +max +r0n+1≤j≤n +� r0n +� +i=1 +(pij − p2 +ij) +� +≤ +max +r0n+1≤j≤n +� r0n +� +i=1 +pij +� += ∆sr. +Similarly, let ¯Y ′′ := �r0n +i=1 +�n +j=r0n+1 E{Y ′ +ij(Y ′ +ij)T}, and then we have that +∥ ¯Y ′′∥ ≤ +���� diag +� +n +� +j=r0n+1 +(p1j − p2 +1j), . . . , +n +� +j=r0n+1 +(pr0n,j − p2 +r0n,j) +����� +≤ +max +1≤i≤r0n +� +n +� +j=r0n+1 +(pij − p2 +ij) +� +≤ ∆rs. +Let σ2 = ∆rs ∨ ∆sr and K = 1, and set a = 2 +� +(∆rs ∨ ∆sr) log n. From (A.3), +P{∥ ¯U − E{ ¯U}∥ > 2 +� +(∆rs ∨ ∆sr) log n} +≤ 2(s0n + r0n) exp +� +−2(∆rs ∨ ∆sr) log n +(∆rs ∨ ∆sr) + 2 +� +(∆rs ∨ ∆sr) log n/3 +� += 2n exp +� +−2 log n +1 + 2 +� +(log n)/(∆rs ∨ ∆sr)/3 +� +≤ 2n− 1 +5 . +The preceding concentration bounds are useful in analyzing the difference ∥xG,n− +x∗,n∥ = ∥(I − ¯Q)−1 ¯Rz(s) − (I − ¯Q)−1 ¯Rz(s)∥ = ∥[ ¯ +M −1 ¯U − (E{ ¯ +M})−1Ψ(s)]z(s)∥. But +to make sure that xG,n is well-defined, we study ¯Q and α in the following lemma. +Lemma A.7 (Bound of α and ρ( ¯Q)). +(i) Suppose that δrs > 8 log n. Then it holds that +P{[ρ( ¯Q) ≤ εQ,n < 1] ∩ [α ≥ E{α}/2 > 0]} ≥ 1 − ηQ,n = 1 − o(1), +(A.10) +where εQ,n = 1 − δrs/(6E{α}), and ηQ,n = r0n1−δrs/(8 log n) + 2n−2/3. +(ii) Suppose that λ1(E{ ¯ +M}) > εM,n and ∆r ≥ log n. Then (A.10) holds with +εQ,n = 1 − (λ1(E{ ¯ +M}) − εM,n)/(3E{α}) and ηQ,n = ηM,n + 2n−1/8, where εM,n and +ηM,n are given in Lemma A.4. +Proof. Applying (A.1) with δ = 1/2 yields that +P +� +α − E{α} ≤ −1 +2E{α} +� +≤ e− E{α} +8 . +(A.11) +When E{α} ≥ δrs > 8 log n > 0, e−E{α}/8 ≤ e− log n = n−1. If E{α} ≥ ∆r ≥ log n > 0, +e−E{α}/8 ≤ n−1/8. Hence, α ≥ E{α}/2 > 0 with high probability. +Note that I − ¯Q = E{ ¯ +M}/(2E{α}), so (I − ¯Q)−1 exists under conditions of +either (i) or (ii). Since ¯Q = I − ¯ +M/(2α) is symmetric and positive semi-definite, to +show ρ( ¯Q) = λmax( ¯Q) < 1, it suffices to provide a lower bound for λ1( ¯ +M/(2α)). +First we derive a bound under δrs > 8 log n. From (A.8), we know that P{λ1( ¯ +M) > +δrs/2} ≥ 1 − r0n1−δrs/(8 log n). In addition, applying (A.1) with δ = 1/2 yields that +P +� 1 +2α ≤ +1 +3E{α} +� +≤ P +� +α − E{α} ≥ 1 +2E{α} +� +≤ e− E{α} +12 +≤ n− 2 +3 , +(A.12) + +20 +Y. XING AND K. H. JOHANSSON +so ρ( ¯Q) ≤ 1−δrs/(6E{α}) holds with probability at least 1−r0n1−δrs/(8 log n) −n−2/3. +Thus (i) is proved. Combining (A.9), (A.11), and (A.12) yields (ii). +Appendix B. Proof of Theorem 4.3. +From Lemma A.7, (I − ¯Q)−1 exists +with high probability and (I − ¯Q)−1 exists under either Assumption 4.1 (i.1) or (i.2). +In either case, xG,n and x∗,n are well-defined. If I − ¯Q is singular, define xG,n := ∞. +For finite xG,n, it holds that +∥xG,n − x∗,n∥ = ∥(I − ¯Q)−1 ¯Rz(s) − (I − ¯Q)−1 ¯Rz(s)∥ += +���� +�� ¯ +M +2α +�−1 ¯U +2α − +� E{ ¯ +M} +2E{α} +�−1 Ψ(s) +2E{α} +� +z(s) +���� += ∥[ ¯ +M −1 ¯U − (E{ ¯ +M})−1Ψ(s)]z(s)∥ += ∥{ ¯ +M −1( ¯U − Ψ(s)) + [ ¯ +M −1 − (E{ ¯ +M})−1]Ψ(s)}z(s)∥ +≤ (∥ ¯ +M −1∥∥ ¯U − Ψ(s)∥ + ∥ ¯ +M −1 − (E{ ¯ +M})−1∥∥Ψ(s)∥)∥z(s)∥. +From (A.8), Lemma A.6, Corollary A.5 (i), and Lemma A.7 (i), it holds that +∥xG,n − x∗,n∥ ≤ +� 2 +δrs +εU,n + ε′ +M,n∥Ψ(s)∥ +� +∥z(s)∥ +≤ 4 +�� +(∆rs ∨ ∆rs) log n +δrs ++ 2√∆r log n∥Ψ(s)∥ +δ2rs +� +∥z(s)∥, +(B.1) +P{xG,n exists, and (B.1) holds} ≥ 1 − ηU,n − η′ +M,n − 2n− 2 +3 += 1 − r0n1− +δrs +8 log n − 2(1 + r0)n− 1 +5 − 2n− 2 +3 . +In this way, we prove (i) of the theorem. The second part follows from (A.9), Lem- +ma A.6, Corollary A.5 (ii), and Lemma A.7 (ii). +Appendix C. Proof of Theorem 4.7. +Denote ¯S := diag(d(s) +1 , . . . , d(s) +rn ) = +¯ +M − ¯L. Since ¯G is connected, (I − ¯Q)−1 exists. To prove the first part of (i), note that +(E{ ¯S})−1 exists when δrs > 0, so +∥x∗,n − (E{ ¯S})−1Ψ(s)z(s)∥ +≤ ∥(E{ ¯ +M})−1 − (E{ ¯S})−1∥∥Ψ(s)∥∥z(s)∥ +≤ ∥(E{ ¯ +M})−1∥∥(E{ ¯S})−1∥∥E{ ¯ +M} − E{ ¯S}∥∥Ψ(s)∥∥z(s)∥ +(From (A.5)) +≤ 1 +δrs +1 +δrs +∥E{¯L}∥∥Ψ(s)∥∥z(s)∥ ≤ 2∆rr +δ2rs +∥Ψ(s)∥∥z(s)∥. +(From (A.7)) +For the second part of (i), note that +x∗,n − x†,n = +� +[ ˜ +M (11) − (diag(Ψ(s) ++ 1s0n))−1]Ψ(s) ++ z(s) +0 +� +, +so it suffices to bound ∥[ ˜ +M (11) −(diag(Ψ(s) ++ 1s0n))−1]Ψ(s) ++ z(s)∥ = ∥x∗,n −x†,n∥. Since G +is connected, ˆ +M (21) is non-zero. Thus there exist rows of ˆ +M (22) with strictly dominant +diagonals, which implies that ˆ +M (22) is invertible. From the inverse formula of block +matrices [31], it follows that ˜ +M (11) = [ ˆ +M (11) − ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)]−1. Hence, +∥ ˜ +M (11) − ( ˆ +M (11))−1∥ + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +21 += ∥[ ˆ +M (11) − ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)]−1 − ( ˆ +M (11))−1∥ +≤ ∥[ ˆ +M (11) − ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)]−1∥∥( ˆ +M (11))−1∥∥ ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)∥ +(C.1) += +∥ ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)∥ +λ1( ˆ +M (11))λ1( ˆ +M (11) − ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)) +≤ +∥ ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)∥ +λ1( ˆ +M (11))(λ1( ˆ +M (11)) − ∥ ˆ +M (12)( ˆ +M (22))−1 ˆ +M (21)∥) +(C.2) +≤ +∥ ˆ +M (21)∥2/λ1( ˆ +M (22)) +λ1( ˆ +M (11))(λ1( ˆ +M (11)) − ∥ ˆ +M (21)∥2/λ1( ˆ +M (22))) +, +(C.3) +where (C.1) follows from (A.5) and (C.2) from (A.4). Similarly we obtain that +∥( ˆ +M (11))−1 − (diag(Ψ(s) ++ 1s0n))−1∥ ≤ ∥ ˆ +M (11) − diag(Ψ(s) ++ 1s0n)∥ +λ1( ˆ +M (11))δ+ +rs +. +(C.4) +The Gershgorin theorem yields that λ1( ˆ +M (11)) ≥ δ+ +rs and ∥ ˆ +M (11) − diag(Ψ(s) ++ 1s0n)∥ ≤ +2 max1≤i≤n1{E{d(r) +i }}, so the conclusion follows from (C.3) and (C.4). +Now we show (ii). The assumption λ2(E{¯L}) > 2∆rs > 0 ensures that the eigen- +value λ1(E{ ¯ +M}) is simple. So by ξ we denote a unit eigenvector corresponding to the +eigenvalue λ1(E{ ¯ +M}). Since E{ ¯ +M} is symmetric, it has orthogonal unit eigenvalues +w(2), . . . , w(r0n) corresponding to its eigenvalues λ2(E{ ¯ +M}) ≤ · · · ≤ λr0n(E{ ¯ +M}). +Also ξ, w(2), . . . , w(r0n) form a basis of Rr0n, and ξξT + �r0n +j=2 w(j)(w(j))T = Ir0n. So +����(E{ ¯ +M})−1Ψ(s)z(s) − +1 +r0nλ1(E{ ¯ +M})1r0n1T +r0nΨ(s)z(s) +���� += +����(E{ ¯ +M})−1 +� +ξξT + +r0n +� +j=2 +w(j)(w(j))T +� +Ψ(s)z(s) − +1 +r0nλ1(E{ ¯ +M})1r0n1T +r0nΨ(s)z(s) +���� +≤ +����(E{ ¯ +M})−1ξξTΨ(s)z(s) − +1 +r0nλ1(E{ ¯ +M})1r0n1T +r0nΨ(s)z(s) +���� ++ +����(E{ ¯ +M})−1 +� r0n +� +j=2 +w(j)(w(j))T +� +Ψ(s)z(s) +���� =: (I) + (II). +Note that (E{ ¯ +M})−1ξ = ξ/λ1(E{ ¯ +M}), so +(I) = +���� +1 +λ1(E{ ¯ +M})ξξTΨ(s)z(s) − +1 +r0nλ1(E{ ¯ +M})1r0n1T +r0nΨ(s)z(s) +���� +≤ +1 +λ1(E{ ¯ +M}) +����ξξT − +1 +r0n1r0n1T +r0n +����∥Ψ(s)z(s)∥ +≤ +2∥Ψ(s)∥∥z(s)∥∆rs +λ1(E{ ¯ +M})(λ2(E{¯L}) − 2∆rs), +where the last inequality is obtained from the following lemma with A = E{ ¯ +M}, +B = E{¯L}, and ζ = (λ2(E{¯L}) − 2∆rs)/2, which is a consequence of Theorem 5.5 in +Chapter I and Theorem 3.6 in Chapter V of [49]. +Lemma C.1. Let A, B ∈ Rn×n be symmetric, and µ with corresponding unit +eigenvector u (resp. ν with unit eigenvector v) be a simple eigenvalue of A (resp. + +22 +Y. XING AND K. H. JOHANSSON +B). Denote r = Av − νv. If there exists ζ > 0 such that the eigenvalues of A except +µ lie outside the interval [ν − ζ, ν + ζ], then +∥uuT − vvT∥ ≤ ∥r∥ +ζ += ∥Av − Bv∥ +ζ +≤ ∥A − B∥ +ζ +. +For (II), it holds that +(II) = +���� +r0n +� +j=2 +1 +λj(E{ ¯ +M})w(j)(w(j))TΨ(s)z(s) +���� += +� +� +� +� +r0n +� +j=2 +[(w(j))TΨ(s)z(s)]2 +λ2 +j(E{ ¯ +M}) +≤ +1 +λ2(E{ ¯ +M}) +� +� +� +� +r0n +� +j=2 +[(w(j))TΨ(s)z(s)]2 += +1 +λ2(E{ ¯ +M}) +���� +r0n +� +j=2 +w(j)(w(j))TΨ(s)z(s) +���� += ∥(I − ξξT)Ψ(s)z(s)∥ +λ2(E{ ¯ +M}) +≤ 2∥Ψ(s)∥∥z(s)∥ +λ2(E{ ¯ +M}) . +The conclusion follows from (A.4) and then combining (I) and (II). +Appendix D. Proof of Theorem 4.11. +Since we have derived a bound for ∥xG,n − x∗,n∥ in Theorem 4.3, it suffices to +bound the term ∥SG,n(t)−xG,n∥. To this end, we introduce the following concentration +inequality (Lemma 1 of [55]) for the state time average of a Markov chain. +Lemma D.1 (Concentration of state time average). +Consider a discrete-time +Markov chain {X(t)} taking values on a compact state space X and having a unique +stationary distribution π. For a function f : X → R and ι := +� +X f(x)π(dx), denote +g(x) := �∞ +t=0 E{f(X(t))−ι|X(0) = x} and ∥g∥s := sup{|g(x)| : x ∈ X}. If ∥g∥s < ∞, +then it holds for Sf(t) := 1 +t +�t−1 +i=0 f(X(i)), ε > 0, and t > 2∥g∥s/ε that +P{|Sf(t) − ι| ≥ ε} ≤ 2 exp +� +− (tε − 2∥g∥s)2 +2t∥g∥2s +� +. +Conditioned on a graph G, ρ( ¯Q) < 1 ensures that the gossip model has a well-defined +unique stationary distribution with mean xG,n. This result follows from a standard +argument for gossip models (see [2, 45, 55]). Lemma A.7 ensures that ρ( ¯Q) < 1 holds +with probability at least 1 − ηQ,n, where the probability is over the randomness of G. +Now we derive a bound for ∥SG,n(t) − xG,n∥ given a graph G such that ρ( ¯Q) < 1. +Let fi(x) = xi, ∀x ∈ Rr0n, 1 ≤ i ≤ r0n, and Lemma D.1 ensures that +PG{|SG,n +i +(t) − xG,n +i +| ≥ ε} ≤ 2 exp +� +− (tε − 2∥gG,n +i +∥s)2 +2t∥gG,n +i +∥2s +� +(D.1) +holds for all ε > 0 and t > 2∥gG,n +i +∥s/ε, where gG,n +i +is the i-th component of GG,n(x) = +�∞ +t=0 EG{X(t) − xG,n|X(0) = x}, x ∈ Rr0n. Note that ∥gG,n +i +∥s < ∞ because for all + +CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS +23 +x ∈ X = [−cx, cx]r0n +∥GG,n(x)∥ ≤ +∞ +� +t=0 +���� ¯Qtx − +∞ +� +i=t +¯Qi ¯Rz(s) +���� += +∞ +� +t=0 +���� ¯Qt +� +x − +∞ +� +i=0 +¯Qi ¯Rz(s) +����� +≤ +∞ +� +t=0 +∥ ¯Q∥t∥x − xG,n∥ += ∥x − xG,n∥ +1 − ρ(Q) +≤ 2√r0ncx +1 − ρ( ¯Q) =: sG,n +∗ +. +Hence ∥gG,n +i +∥s = supx∈X {|GG,n +i +(x)|} ≤ supx∈X {∥GG,n(x)∥} ≤ sG,n +∗ +. Therefore, +from (D.1) it follows that for all ε > 0 and t > 2sG,n +∗ +/ε +PG{∥SG,n(t) − xG,n∥ ≥ √r0nε} ≤ PG{∃i ∈ Vr, |SG,n +i +(t) − xG,n +i +| ≥ ε} +≤ 2r0n exp +� +− (tε − 2sG,n +∗ +)2 +2t(sG,n +∗ +)2 +� +, +Lemma A.7 implies that +P +� +exp +� +− (tε − 2sG,n +∗ +)2 +2t(sG,n +∗ +)2 +� +≤ exp +� +− (tε − 2¯s∗)2 +2t(¯s∗)2 +�� +≥ 1 − ηQ,n, +where if δrs > 8 log n then ¯s∗ = 12√r0ncxE{α}/δrs and ηQ,n = r0n1−δrs/(8 log n) + +2n−2/3, and if λ1(E{ ¯ +M}) > εM,n and ∆r ≥ log n then ¯s∗ = 6√r0ncxE{α}/(λ1(E{ ¯ +M}) +−εM,n) and ηQ,n = ηM,n + 2n−1/8. Denoting S1 = {ρ( ¯Q) ≤ εQ,n}, by the law of total +probability we have that +P{∥SG,n(t) − xG,n∥ ≥ √r0nε} += P{∥SG,n(t) − xG,n∥ ≥ √r0nε|S1}P{S1} + P{∥SG,n(t) − xG,n∥ ≥ √r0nε|Sc +1}P{Sc +1} +≤ 2r0n exp +� +− (tε − 2¯s∗)2 +2t(¯s∗)2 +� +P{S1} + P{Sc +1} ≤ 2r0n exp +� +− (tε − 2¯s∗)2 +2t(¯s∗)2 +� ++ ηQ,n. +Therefore, the conclusion follows from the above bound and Theorem 4.3. +REFERENCES +[1] E. 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Hu, Multiagent opinion dynamics of bounded confidence with +nonlocal aggregative interaction, SIAM Journal on Control and Optimization, 55 (2017), +pp. 2543–2573. + diff --git a/M9E4T4oBgHgl3EQf9A55/content/tmp_files/load_file.txt b/M9E4T4oBgHgl3EQf9A55/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbc398335aee152399e351b40297d1ab403aee24 --- /dev/null +++ b/M9E4T4oBgHgl3EQf9A55/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf,len=1478 +page_content='CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS ∗ YU XING† AND KARL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We study concentration phenomena in gossip opinion dynamics over random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the model, a network is generated from a random graph with independent edges, and agents interact pairwise randomly over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' During the process, regular agents average the opinions of themselves and their neighbors as updates, whereas stubborn agents do not change opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To approximate the original process, we introduce gossip dynamics over an averaged graph, obtained by averaging all possible networks generated from the random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Using concentration inequalities, we derive high-probability bounds for the difference between the expected final opinions of the regular agents in the original gossip process and those in the dynamics over the averaged graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The result shows that the expected final opinions for most networks generated from the random graph are close to the final opinions of the averaged graph, which are convex combinations of stubborn-agent states with weights depending on link probability between regular and stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We further show how such concentration can help study the effect of network topology on the expected final opinions in two cases, using matrix perturbation theory: (i) When the influence of stubborn agents is large, the original expected final opinions polarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) When the influence of stubborn agents is small, the expected final opinions are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' With the help of concentration inequalities of Markov chains, we obtain high-probability bounds for the difference between the time average of agent states in the original process and the expected final opinions of the averaged graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' As an application, we apply the results to gossip dynamics over stochastic block models, which generate networks with community structure, and derive quantitative characterization of the opinion profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' opinion dynamics, multiagent systems, social networks, random graphs, concentra- tion, community structure, stochastic block models MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 93A14, 91D30, 93E15, 60F10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Social opinion dynamics study how interactions over networks shape individual opinion evolution and have various applications [43, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The last two decades have witnessed great developments in the study of opinion dynamics, and nu- merous mathematical approaches have been applied to the modeling and analysis of such dynamics [3, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' An open problem is how to analyze the influence of specific network structures on the opinion evolution process [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Many structures can be modeled by random graph models [11, 35], but how to combine random graph theory with the study of opinion dynamics needs further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For example, commu- nity structure describes the property that subgroups of agents are connected densely with each other but loosely with other subgroups [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Such a property can often be observed in reality, and can be modeled by stochastic block models (SBM) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is well-known that random graphs enjoy concentration properties [18, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, the adjacency matrix of a randomly generated graph is close to its averaged version with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is possible to characterize large-scale opinion evolution us- ing concentration properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Such results may link microscopic update of agents to macroscopic behavior of a system [25, 44], give quantitative predictions for real opin- ion evolution [27], and provide design insights for community detection algorithms based on state observations [46, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Novel concentration results for gossip opinion dynamics over random networks are derived in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' ∗This work was funded by the Knut & Alice Wallenberg Foundation and Swedish Research Coun- cil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' †Division of Decision and Control Systems, School of Electrical Engineering and Computer Sci- ence, KTH Royal Institute of Technology, and Digital Futures, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (yuxing2@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='se, kallej@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05352v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='SY] 13 Jan 2023 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Individual opinions represent personal attitudes towards some topics, events, or other persons [27], and can be modeled by scalar or vector quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Opinion dynamics describe how opinions evolve through interpersonal in- teractions [3, 16, 44] and continuous-state models are considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The simplest dynamics, the French–DeGroot (FD) model [21], shows how a group comes to an agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the model, agents update according to the average of their neigh- bors’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Extensions of this model to time-varying cases have been extensively studied [9, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The gossip model is a random counterpart of the FD model, in which agents randomly interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This update rule captures the haphazard characteristic of social interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Despite its simplicity, the model can exhibit var- ious behavior: Consensus has been studied in [12, 23], and opinion fluctuations and disagreement emerge when there are stubborn agents who never change opinions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The Friedkin–Johnsen (FJ) model [28] is another generalization of the FD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It allows agents to be affected by their initial opinions, and generates long-term dis- agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Unlike the previous models, the Hegselmann–Krause (HK) model [30], the Deffuant–Weisbuch (DW) model [20], and their variations [10, 51, 59] explore how ho- mophily influence shapes the opinion evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In these models, agents interact only with those who hold beliefs similar to them, and hence tend to form clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Other models [37, 47] consider negative or antagonistic interactions, which may enlarge opin- ion difference, and end in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In addition to interpersonal influences, exogenous influences also play crucial roles in opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Exogenous influences, such as opinion leaders, social media, and random events, are ubiquitous in real social dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It has been shown that stubborn agents can fully determine the discussion outcome of the FD model [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the gossip model with stubborn agents, there are persistent fluctuations and long- term disagreement [2, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' However, if the network is highly fluid, then regular (non- stubborn) agents can have similar expected final opinions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In contrast, for regular agents forming two communities connected to different stubborn agents, their final positions polarize if the influence of stubborn agents is large [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The current paper revisits this classic model and show how to characterize the process in more detail with the help of random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The authors in [6] study the Tayler model and theoretically characterize the opinion profile for the case with one or two stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Another phenomenon induced by exogenous influences is that random noise can drive agents in the HK model out of clustering states and help the model reach a quasi-consensus [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Real networks often consist of numerous agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To study large-scale group behav- ior, researchers have proposed macroscopic models in contrast to the earlier discussed agent-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Such models investigate the evolution of agent-state distributions and can be applied to mobility modeling [53] and mean-field games [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Eulerian ap- proaches were introduced into the control literature for analyzing bounded confidence models [14, 33, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The paper [40] shows how to apply Eulerian approaches to spa- tially distributed ordinary differential equations, to get partial differential equations capturing behavior of large-scale dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Graphon theory has been used recently for modeling heterogeneous large-scale networks, and [8, 13] study the convergence of Euler approximations of mean-field games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Random graph theory is another frame- work for large-scale network modeling [11, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Random graph models [5, 22] repro- duce features of real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Most properties of random graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', connectivity and power-law degree distributions) hold with high probability when the network size is large [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Dynamics over networks with community structure are of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' There have been papers studying how the community structure influences CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 3 opinion evolution, mainly via mean-field approximations and simulation, such as the DW model [24, 29], the Sznajd model [48], and voter models [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this paper we study concentration in the gossip model over random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' First, we show that the expected final opinions of regular agents concentrate around the opinions in another gossip model over an averaged graph, which is obtained by averaging all possible networks generated from the random graph (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The final opinions of the averaged graph are weighted averages of stubborn-agent states, with the weights depending on link probability between regu- lar and stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The result states that the difference between the two opinion vector can be bounded by a quantity depending on the maximum and minimum ex- pected degrees of the random graph and stubborn-agent states, with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Next, using matrix perturbation theory, we study the effect of the link probability on the expected final opinions of the averaged graph (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7): (i) When the influence of stubborn agents is large, regular agents connected to stubborn agents with positive probability have expected final opinions that are weighted averages of stubborn-agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The weights depend only on this link probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Agents not connected to stubborn agents have expected final opinions that are averages of their neighbors’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) When the influence of stubborn agents is small, the expected final opinions have similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' These results for the averaged graph combined with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 yield the same conclusions for the expected final opinions of the random graph (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We then show that the difference between the time average of agent states and the expected final opinions of the averaged graph can be bounded by error depending on time, network size, and random graph parameters, with prob- ability vanishing as time and the network size grow to infinity (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We apply these general results to the gossip model over an SBM, and thereby theoretically characterize opinion evolution over networks with community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The example generalizes our early work [56], in which a two-community SBM is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We also discuss how to generalize the results to other types of exogenous influences besides stubborn agents, such as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We find that, unlike classic concentration results for adjacency matrices [11, 18], the concentration of the expected final opinions depends not only on the maximum ex- pected degree of the random graph but also on the minimum expected number of edges between a regular agent and the stubborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Given random graph models, the obtained results can characterize large-scale opinion evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Different from convergence and stability analysis [7, 8, 13, 14, 38], the current paper studies how network topology affects the concentration of expected final opinions and when such patterns appear during the evolution, by providing high-probability bounds for finite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In par- ticular, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9 develop a unified framework for analyzing the effect of stubborn-agent influence and network topology, as they generalize the polarization result in [19], complement the consensus result in [2], and quantitatively characterize opinion evolution over networks with community structure (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Because random graphs are widely used in modeling real networks [11, 35], the current framework makes it possible to provide both qualitative and quantitative description for real opinion evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' More precisely, given a real network, we can first establish random graph models from network properties, determine qualitative results for the evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', whether polarization or consensus would happen), and then give high-probability bounds for the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The obtained correspondence between community structure and agent states can also inspire development of community detection methods based on state observations [46, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that the network 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON is unknown but several samples of an opinion trajectory are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is possible to recover the agent community labels by clustering agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Developing such a community detection algorithm is not done in this paper, but some further discussion on the problem is provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We define the considered model in Section 2 and formulate the problem in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Section 4 provides the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Section 5 presents numerical experiments and Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proofs are provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the n-dimensional Euclidean space by Rn, the set of n×m real matrices by Rn×m, the set of nonnegative integers by N, and N+ = N \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the natural logarithm by log x, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let 1n be the all-one vector with dimension n, e(n) i be the n-dimensional unit vector with i-th entry being one, In be the n × n identity matrix, and 0m,n be the m × n all-zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the Euclidean norm of a vector and the spectral norm of a matrix by ∥ · ∥, and the maximum absolute column (row) sum norm of a matrix by ∥ · ∥1 (∥ · ∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For x ∈ Rn, denote its i- th entry by xi, and for A ∈ Rn×n, denote its (i, j)-th entry by aij or [A]ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let ρ(A) be the spectral radius of a square matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For symmetric A ∈ Rn×n, denote its eigenvalue by λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A), and let λmin(A) := λ1(A) and λmax(A) := λn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' By diag(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , Ak) denote the diagonal or block diagonal matrix with A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , Ak ∈ Rm×n on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The cardinality of a set S is |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' P{A} is the probability of an event A and E{X} is the expectation of a random vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' An event A happens almost surely (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=') if P{A} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For a sequence of event An, we say An happens with high probability if P{An} → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For two sequences of real numbers, f(n) and g(n) > 0, n ∈ N, we write f(n) = O(g(n)) if |f(n)| ≤ Cg(n) for all n ∈ N and some C > 0, f(n) = o(g(n)) if |f(n)|/g(n) → 0, and f(n) ∼ g(n) if f(n)/g(n) → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If f(n) > 0 as well, n ∈ N, write f(n) = ω(g(n)) if g(n) = o(f(n)), f(n) = Ω(g(n)) if g(n) = O(f(n)), and f(n) = Θ(g(n)) if both f(n) = O(g(n)) and f(n) = Ω(g(n)) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Occasionally we use subscripts to highlight the dependence of these relations on n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', f(n) = on(g(n))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For x, y ∈ R, let x∨y := max{x, y} and x∧y := min{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' An undirected graph G = (V, E, A) has the agent set V, the edge set E, and the adjacency matrix A = [aij] with aij = 1 (aij = 0) if {i, j} ∈ E ({i, j} ̸∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The degree of i ∈ V is di = � j∈V aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this section, we discuss the two crucial parts of the con- sidered opinion dynamics over random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 defines random graphs, whereas Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 introduces the gossip model and states its key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We de- fine modified random graphs with stubborn agents in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, and the considered gossip dynamics over random graphs in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Random Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this subsection, we define random graphs, which characterize properties of real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A commonly-used random graph model as- sumes that edges in a network are generated independently [11, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (Random graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let n ∈ N+ be the number of agents, and the symmetric matrix Ψ = [ψij] ∈ [0, 1]n×n be the link probability matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The random graph RG(n, Ψ) generates an undirected graph G = (V, E, A) by letting {i, j} ∈ E with probability ψij independent of other agent pairs, where |V| = n and i, j ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The preceding definition is general and includes many classic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) When ψij = ψ ∈ [0, 1], i, j ∈ V, the random graph is the Erd˝os–R´enyi (ER) CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 5 model [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this case agents are connected to each other homogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) Let w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , wn)T ∈ Rn with wi ≥ 0 and maxi w2 i < � k wk, and ψij = wiwj/(� k wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then the random graph generates networks with the expected degree sequence w [17], and can be used for describing power-law degree distributions of real networks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (iii) Assume that the agent set V has K ∈ N+ disjoint subsets called communities, V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , VK, and denote the community label of i ∈ Vk by Ci = k, 1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let n = [n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , nK]T be the size vector of the communities, where nk := |Vk| and 1Tn = n, and the symmetric matrix Π = [πij] ∈ [0, 1]K×K be the link probability matrix between communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then SBM(K, n, Π) is a random graph with link probability ψij = πCiCj, i ̸= j, and ψii = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The SBM, studied in community detection [1], intuitively shows how a community structure forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The size of communities can also be random (see Remark 3 of [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the SBM, agents from each community are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' More general models such as the degree-corrected SBM [26] can include heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Gossip Model with Stubborn Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this subsection, we introduce the gossip model with stubborn agents, which forms the second part of the considered dynamics, and summarize some basic properties of the gossip model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A gossip model with stubborn agents (we call it “the gossip model” hereafter for short) is a random process evolving over a graph G = (V, E, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The agent set V = Vr ∪ Vs (disjoint) contains regular and stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Set Vr = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , nr} and Vs = {1+nr, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , ns+nr}, with |V| = n = nr+ns, where nr (ns) is the number of regular (stubborn) agents, and n is the network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A regular agent i has opinion Xi(t) ∈ R at time t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A stubborn agent j has opinion z(s) j , which does not change, representing persistent influence of a reluctant agent or an information source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Stacking the states, denote the state vector of regular agents at time t by X(t) ∈ Rnr and that of stubborn agents by z(s) ∈ Rns (with slightly abuse of notation, we use z(s) j , instead of z(s) j−r0n, to represent the state of j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' At each time t an edge is selected, and the two corresponding agents interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The selection is modeled by an interaction probability matrix W = [wij] ∈ Rn×n, where wij = wji = aij/α and α = �n i=1 �n j=i+1 aij is the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' An edge {i, j} is selected with probability wij, independently of previous update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Such a selection process describes random daily encounters in social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If both i and j are regular, then Xi(t + 1) = Xj(t + 1) = (Xi(t) + Xj(t))/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If one of the agents is stubborn, say j, then i updates as Xi(t + 1) = (Xi(t) + z(s) j )/2, and j does not update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Other agents do not update at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The update rule can be written as X(t + 1) = Q(t)X(t) + R(t)z(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) with {[Q(t) R(t)]} a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' random matrices such that with probability wij [Q(t), R(t)] = � [Inr − 1 2(e(nr) i − e(nr) j )(e(nr) i − e(nr) j )T, 0nr,ns], if i, j ∈ Vr, [Inr − 1 2e(nr) i (e(nr) i )T, 1 2e(nr) i (e(ns) j )T], if i ∈ Vr, j ∈ Vs, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) where we use e(ns) j to represent e(ns) j−r0n for j ∈ Vs, again for notation simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the expected interaction matrices by ¯Q := EG{Q(t)} and ¯R := EG{R(t)}, where the subscript G highlights that the averaging depends on W, and hence on the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The following well-known results [2] (for rigorous analysis in the discrete- time version, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', [45, 55]) indicate that the expected final opinions depend on the expected interaction matrices and the stubborn-agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 (Stability and limit theorems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that G is connected and has at least one stubborn agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The following results hold for the gossip model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON (i) The model has a unique stationary distribution π with mean x, and X(t) converges in distribution to π, as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) The expectation of the regular-agent state vector converges to x, namely, x = lim t→∞ EG{X(t)} = (I − ¯Q)−1 ¯Rz(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) (iii) Denote S(t) := 1 t �t−1 i=0 X(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then limt→∞ S(t) = x a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The results show that, although agent states may not reach a consensus or converge to a fixed value (instead, they may fluctuate a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' [2]), they converge in distribution to a stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Also, the state time average S(t) converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This vector x characterizes the average final positions of regular agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Random Graphs with Stubborn Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To study the interplay be- tween network structure and stubborn agents, we introduce random graphs with stub- born agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We modify Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 (Random graph with stubborn agents, RG-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let nr ∈ N+ be the number of regular agents, ns ∈ N+ the number of stubborn agents, and n = nr + ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let the symmetric matrix Ψ(r) = [ψ(r) ij ] ∈ [0, 1]nr×nr be the link probability matrix between the regular, and Ψ(s) = [ψ(s) ij ] ∈ [0, 1]nr×ns be the link probability matrix between the regular and the stubborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The random graph with stubborn agents RG-S(nr, ns, Ψ(r), Ψ(s)) generates a graph G = (V, E, A) with |V| = n according to the following rule: (i) A graph on the regular agents is generated from RG(nr, Ψ(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) For i ∈ Vr and j ∈ Vs, {i, j} ∈ E with probability ψ(s) i,j−nr, independent of other edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The RG-S includes stubborn agents in the network, and the link probability matrix Ψ(s) captures the influence of the stubborn on regular agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the portion of regular agents by r0 := nr/n ∈ (0, 1), and that of the stubborn by s0 := ns/n ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' These portions can be functions of n, instead of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Gossip Dynamics over Random Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The previous subsections de- fined the random graph models and the random gossip dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To analyze opinion dynamics evolving over random graphs, we bring these two models together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assume that a graph G is generated from an RG-S and then fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Over this graph, the gossip takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' See Figure 1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From now on, by the gossip model, we mean the gossip dynamics evolving over a graph G generated from an RG-S: XG,n(t + 1) = Q(t)XG,n(t) + R(t)z(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4) where XG,n(t) is the state vector and the superscripts G and n highlight the depen- dence of the process on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Here [Q(t) R(t)] has the expression given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For the gossip model, we know from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 that the expected final opinion vector exists and is unique, if the graph G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote this vector by xG,n := lim t→∞ EG{XG,n(t)} = (I − ¯Q)−1 ¯Rz(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) Here we use the superscripts G and n to indicate that x depends on the sampled graph G and the network size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that ¯Q and ¯R are now conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To study behavior of the gossip model, we use a reference system without network randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' By averaging all possible graphs G = (V, E, A) generated from the RG-S, we obtain a graph ¯G = (V, ¯E, E{A}), where E{A} is the weighted adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Define a gossip model over this averaged graph ¯G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Illustration of the gossip model over an RG-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' On the left side of the figure, a network is generated and then fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Circles and squares represent regular and stubborn agents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' On the right, the gossip model evolves over the generated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' An edge is selected at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 (Gossip model over averaged graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Consider a random graph RG-S(nr, ns, Ψ(r), Ψ(s)) and the averaged graph ¯G = (V, ¯E, E{A}) obtained from averaging all graphs generated from the RG-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The gossip model over the averaged graph is the following model that evolves over ¯G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' X∗,n(t + 1) = Q(t)X∗,n(t) + R(t)z(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6) where X∗,n(t) is the state vector and [Q(t) R(t)], which depends on ¯G, is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assume that the averaged graph is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This gossip process also has expected final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For the gossip model over the averaged graph ¯G, its interaction prob- ability matrix is W = E{A}/E{α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let ¯Q := E ¯G{Q(t)} and ¯R := E ¯G{R(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The expected final opinions of the gossip model over the averaged graph is x∗,n := lim t→∞ E ¯G{X∗,n(t)} = (I − ¯Q)−1 ¯Rz(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Problem Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This section formulates the considered problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The first problem that we consider is when the expected final opinions xG,n con- centrate around the expected final opinions of the averaged graph x∗,n: Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Given an RG-S and the gossip model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4), provide high probability bounds for ∥xG,n − x∗,n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is well-known that random graph models have concentration properties [18, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, under certain degree conditions, the adjacency matrix A of the randomly gen- erated graph G is close to its averaged version E{A} with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Problem 1 arises naturally from this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is solved by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The second problem is to provide conditions resulting in the polarization or con- sensus of xG,n: Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Given an RG-S and the gossip model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4), provide conditions for (i) the entries of xG,n are close to the stubborn agents’ states, (ii) the entries of xG,n are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This problem concerns how network topology and stubborn agents shape the profile of the expected final opinions xG,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that E{A} has a simpler form than A, so it is easier to characterize x∗,n (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then using the solution to Problem 1, we are able to answer Problem 2 in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Finally, we derive bounds for the difference between the state time average SG,n(t) := (�t−1 i=0 XG,n(i))/t and the expected final opinions of the averaged graph x∗,n: Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Given an RG-S and the gossip model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4), provide high probability bounds for ∥SG,n(t) − x∗,n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 0 0 0 口 C RG-S 0 口 0 口 0 口 t=2 t=3 t=4 X9,n(t+ 1) = Q(t)X9,n(t) + R(t)≥(s) G=(V,,A)8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON This problem is important because only agent states can be observed in practice, rather than the expected states considered in Problems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 we know that it is possible to use the state time average to estimate the expected opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence a natural question would be when the time average becomes close to the expected final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Studying this problem can help us understand how network topology and stubborn agents affect transient behavior of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The result is given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this section, we first study the expected final opinions of the gossip model, by comparing it with those of the averaged graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We then investigate how the state time average behaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' After that we apply the results to gossip over an SBM-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A discussion on further extensions concludes the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Concentration of Expected Final Opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this section, we study properties of the expected final opinions xG,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 shows that the differ- ence ∥xG,n − x∗,n∥ can be bounded by a vanishing term with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Next, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 examines how the profile of x∗,n depends on network topology and stub- born agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Finally, we characterize the profile of xG,n in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9 by combining Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' First we introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Given a graph G = (V, E, A), for i ∈ V, by d(r) i denote the number of edges connecting i and the regular, and by d(s) i the number of edges between i and the stubborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' So di = d(r) i + d(s) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that the weighted adjacency matrix E{A} contains crucial information of a random graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let ∆r := maxi∈Vr{E{di}} be the maximum expected degree of regular agents, and ∆s = ∆sr := maxi∈Vs{E{di}} be the maximum expected degree of the stubborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In addition, denote ∆rr := maxi∈Vr{E{d(r) i }} and ∆rs := maxi∈Vr{E{d(s) i }}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Similarly, let δr := mini∈Vr{E{di}}, δs = δsr := mini∈Vs{E{di}}, δrr := mini∈Vr{E{d(r) i }}, and δrs := mini∈Vr{E{d(s) i }} be the corresponding minimum expected degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) and W = A/α, where W is the interaction probability matrix and α is the number of edges in a graph G, it holds that ¯Q = EG{Q(t)} = Ir0n − ¯ M/(2α), ¯R = EG{R(t)} = ¯U/(2α), where ¯ M := � ����� d1 −a12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −a1,r0n −a21 d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −ar0n−1,r0n −ar0n,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −ar0n,r0n−1 dr0n � ����� , ¯U := � �� a1,r0n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' a1n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' ar0n,r0n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' ar0n,n � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the expected interaction matrices of the gossip model over the averaged graph by ¯Q = Ir0n − E{ ¯ M}/(2E{α}) and ¯R = E{ ¯U}/(2E{α}) = Ψ(s)/(2E{α}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The following assumptions are used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assume that the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) δrs > 8 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) λ1(E{ ¯ M}) > 4√∆r log n, ∆r ≥ log n, and ∆rs ∨ ∆sr ≥ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) Both the gossip model over the random graph and the gossip model over the averaged graph have the same stubborn-agent states z(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) and (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Only one of them is needed for main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) is more general because δrs can be zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', some regular agents are not connected to any stubborn agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that the last two conditions in (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) cannot imply the first one: consider E{d(r) i } = (log n)2 and CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 9 E{d(s) i } = 9 log n, i ∈ Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then λ1(E{ ¯ M}) = 9 log n < √∆r log n for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' How- ever, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The lower bounds for δrs, ∆r, and ∆rs ∨ ∆sr can be relaxed to Ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This bound is necessary for graph connectivity with high prob- ability [1, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Although (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) assumes that there exists positive probability that a regular agent is connected to some stubborn agents, this regular agent need not be connected to any stubborn agents in a sampled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' □ Next we state the first main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It concerns the concentration of xG,n and provides a high-probability bound for the difference between xG,n and x∗,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 (Concentration of expected final opinions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For xG,n and x∗,n given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7), respectively, the following results hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) and (ii), it holds that P{xG,n exists, and ∥xG,n − x∗,n∥ ≤ εx,n} ≥ 1 − ηx,n = 1 − o(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) where εx,n = 4 �� (∆rs ∨ ∆sr) log n δrs + 2√∆r log n∥Ψ(s)∥ δ2rs � ∥z(s)∥, ηx,n = r0n1− δrs 8 log n + 2(1 + r0)n− 1 5 + 2n− 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) and (ii), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) holds with εx,n = 2 � � (∆rs ∨ ∆sr) log n λ1(E{ ¯ M}) − 4√∆r log n + 2√∆r log n∥Ψ(s)∥ λ1(E{ ¯ M})(λ1(E{ ¯ M}) − 4√∆r log n) � ∥z(s)∥, ηx,n = 2(1 + r0)n− 1 5 + 2n− 1 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The theorem indicates that the difference ∥xG,n − x∗,n∥ can be bounded by the norm of stubborn-agent states z(s) multiplied by a quantity depend- ing on the agent degrees, with probability that depends on the network size and the portion of the regular (and also on the smallest expected number of edges linking a regular agent and the stubborn (δrs) in (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A lower bound of λ1(E{ ¯ M}) can be found in [36], which is related to “bottleneck” of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7) give a trivial bound ∥xG,n − x∗,n∥ = O(∥z(s)∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The bound is nontrivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', o(∥z(s)∥)), if δrs or λ1(E{ ¯ M}) is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A sufficient condition in the δrs > 0 case is δrs = ω( � (∆r log n)1/2(∆rs ∨ ∆sr)), from ∥Ψ(s)∥ ≤ ∥Ψ(s)∥1 ∨ ∥Ψ(s)∥∞ ≤ ∆rs ∨ ∆sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, the influence of the stubborn on any regular agent has to be large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A direct consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) is that the opinion mean 1T nrxG,n/nr is close to its ex- pected version 1T nrx∗,n/nr with difference o(1), if |z(s) j | ≤ cx for a positive constant cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In classic concentration results for adjacency matrices [18, 34], the upper bounds only contain maximum expected degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Our results include the minimum expected degree δrs (or λ1(E{ ¯ M})), for the presence of stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The logarithm term in the upper bounds may be removed under more careful analysis [34], which is left to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 indicates that xG,n is close to x∗,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' A consequence is that xG,n i is close to x∗,n i for most i ∈ V, as given by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For ε∗ > 0 and x#, x+ ∈ Rm, denote Vε∗ := {i ∈ V : |x# i − x+ i | > ε∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If P{∥x# − x+∥ ≤ ε+} ≥ 1 − η+,n, then it holds that P � |Vε∗| ≤ ε2 + ε2∗ � ≥ 1 − η+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Set x# = xG,n, x+ = x∗,n, ε+ = εx,n, and η+ = ηx,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then |Vε∗|/nr ≤ ε2 x,n/(nrε2 ∗) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Under conditions for ε2 x,n = o(∥z(s)∥), for example δrs = ω( � (∆r log n)1/2(∆rs ∨ ∆sr)) in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4, if |z(s) j | ≤ cx for a constant cx > 0, then |Vε∗|/nr = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, xG,n i is very close to its expected version x∗,n i for all agents except a small part o(nr), with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this way it is possible to approximate the profile of xG,n by using x∗,n point-wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' □ Another advantage of relating xG,n to a reference version x∗,n is that we can obtain more properties of xG,n, which hold for almost all G, by studying x∗,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To show this, we proceed to study the profile of x∗,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let ¯L := � ��� d(r) 1 −a12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −a1,r0n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −ar0n,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' −ar0n,r0n−1 d(r) r0n � ��� be the Laplacian of the graph on regular agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The following proposition concerns the profile of x∗,n under different cases of stubborn-agent influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (Profile of x∗,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that the averaged graph ¯G is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The following results hold for x∗,n given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) (Bound for large stubborn-agent influence) If δrs > 0, then ∥x∗,n − x†,n∥ ≤ ε†,n, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) with x†,n = (diag(Ψ(s)1s0n))−1Ψ(s)z(s) and ε†,n = 2∆rr∥Ψ(s)∥∥z(s)∥/δ2 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If δrs = 0 but λ1(E{ ¯ M}) > 0 (hence (E{ ¯ M})−1 exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assume that E{d(s) i } > 0 for 1 ≤ i ≤ n1 < r0n and E{d(s) i } = 0 for n1 + 1 ≤ i ≤ n1 + n2 = r0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote E{ ¯ M} =: � ˆ M (11) ˆ M (12) ˆ M (21) ˆ M (22) � , (E{ ¯ M})−1 =: � ˜ M (11) ˜ M (12) ˜ M (21) ˜ M (22) � , Ψ(s) =: � Ψ(s) + 0n2,s0n � , where ˆ M (11), ˜ M (11) ∈ Rn1×n1, ˆ M (21), ˜ M (21) ∈ Rn2×n1, and Ψ(s) + ∈ Rn1×s0n, and δ+ rs = min1≤i≤n1{E{d(s) i }}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If δ+ rs > ∥ ˆ M (21)∥2/λ1( ˆ M (22)), then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) holds with x†,n = � (diag(Ψ(s) + 1s0n))−1 ˜ M (21) � Ψ(s) + z(s), ε†,n = 1 δ+ rs � ∥ ˆ M (21)∥2/λ1( ˆ M (22)) δ+ rs − ∥ ˆ M (21)∥2/λ1( ˆ M (22)) + 2 max1≤i≤n1{E{d(r) i }} δ+ rs � ∥Ψ(s)∥∥z(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) (Bound for small stubborn-agent influence) If λ1(E{ ¯ M}) > 0 and λ2(E{¯L}) > 2∆rs > 0, then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) holds with x†,n = 1T r0nΨ(s)z(s) r0nλ1(E{ ¯ M})1r0n, ε†,n = 2 λ2(E{¯L}) − 2∆rs � ∆rs λ1(E{ ¯ M}) + 1 � ∥Ψ(s)∥∥z(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The first part of (i) indicates that, if the minimum influence of the stubborn (δrs) is much larger than the link strength between regular agents (∆rr), then x∗,n i is almost a weighted average of stubborn-agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The weights depend only on the link probability between i and the stubborn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, polar- ization may emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The second part of (i) shows that agents not connected to the stubborn have expected final opinions that are also weighted averages of stub- born states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' But the weights, given by ˜ M (21), depend highly on network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If δ(21) rr := min1+n1≤i≤r0n{E{� 1≤j≤n1 aij}} > 0, then ∥ ˆ M (12)∥2/λ1( ˆ M (22)) has an up- per bound (∥ ˆ M (12)∥1∨∥ ˆ M (12)∥∞)2/δ(21) rr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The bound depends on how agents with and without stubborn neighbors are connected, similar to ∆rs∨∆sr and δrs in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In contrast, (ii) shows that, if the influence of stubborn agents (∆rs) is smaller than the link strength between regular agents (λ2(E{¯L})), most entries of x∗,n are close to a weighted average of stubborn states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In other words, the expected opin- ion vector is almost a consensus vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that this conclusion does not contra- dict that the consensus state is an average of regular initial states when there is no stubborn agent [12, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This is because we also need the concentration condition δrs ∨ λ1(E{ ¯ M}) > 0 to obtain the profile of xG,n (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' □ Consequently, by Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 and the triangle inequality, we are able to get the following result, characterizing the profile of xG,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9 (Profile of xG,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Under the conditions of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7, the following holds for xG,n given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) P{∥xG,n − x†,n∥ ≤ εx,n + ε†,n} ≥ 1 − ηx,n, where εx,n and ηx,n are given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, and x†,n and ε†,n in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The theorem provides high-probability bounds for the difference between the expected final opinions and a specific vector (the polarization or consensus vector given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The bound consists of two parts: the deviation of the expected final opinions from the averaged version, and the difference between the latter and the specific vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This result quantitatively characterizes how the expected final opinions are far away from typical large-scale behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Polarization is given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (i), generalizing [19] which studies a weighted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (ii) covers the consensus case, investigated also in [2], showing that the expected final opinions achieve a consensus when the influence of stubborn agents is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Moreover, we can also characterize the case in the middle by using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In that case, neither polarization nor consensus emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Instead, the expected final opinions exhibit much diversity [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is possible to obtain concentration of opinion variances as in [2], by solving the stationary covariance matrix and analyzing its concentration (one way to analyze the concentration is similar to [54] but we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=') □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Concentration of State Time Average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this subsection we study the concentration of state time average SG,n(t) = (�t−1 i=0 XG,n(i))/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We show when the time average is close to the average version of expected final opinions x∗,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The pre- vious subsection studies how to bound the difference ∥xG,n − x∗,n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 indicates that the time average SG,n(t) should be close to xG,n when t is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' By bounding ∥SG,n(t)−xG,n∥ and the triangle inequality, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11 (Concentration of state time average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that maxi∈Vr{|Xi(0)|} ∨ maxj∈Vs{|z(s) j |} ≤ cx for some cx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then the following hold for x∗,n and SG,n(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) and (ii), for εS,n > 0, t > 2¯s∗/εS,n, it holds that P{∥SG,n(t) − x∗,n∥ ≤ √r0nεS,n + εx,n} ≥ 1 − ηS,n,t − ηS,n, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) where εx,n is given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 (i), and ¯s∗ = 12√r0ncxE{α} δrs , ηS,n,t = 2r0n exp � − (tεS,n − 2¯s∗)2 2t(¯s∗)2 � , ηS,n = r0n1− δrs 8 log n + 2(1 + r0)n− 1 5 + 2n− 2 3 = on(1), (ii) Under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) and (ii), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) holds for εS,n > 0, t > 2¯s∗/εS,n with εx,n given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 (ii) and ¯s∗ = 6√r0ncxE{α} λ1(E{ ¯ M}) − 4√∆r log n, ηS,n,t = 2r0n exp � − (tεS,n − 2¯s∗)2 2t(¯s∗)2 � , ηS,n = 2(1 + r0)n− 1 5 + 2n− 1 8 = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The theorem provides high-probability bounds for the different be- tween the state time average and the averaged version of the expected final opin- ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The results capture the dependence of the bounds and the probability on agent degrees, stubborn-agent states, and time, and they thus establish concentration for almost all graphs and all large enough time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The error εS,n controls the con- centration of SG,n(t) around xG,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that εx,n has a term ∥z(s)∥ = O(√r0n), so εS,n can be set to be εx,n/∥z(s)∥ so that √r0nεS,n = o(√r0n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The probability ηS,n depends on the network size (and also on δrs in (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The terms εx,n and ηS,n do not vanish for fixed n, even if t approaches infinity, which captures the effect of the net- work size on the concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' When n is large, the concentration probability depends mostly on ηS,n,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the δrs > 0 case, assume that δrs = ω(√∆r log n), and then set εS,n = εx,n/∥z(s)∥ = O(√∆r log n/δrs) = on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' When t = Ω(n3∆r), ηS,n,t = O(n−c) with some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' That is, the difference ∥SG,n(t) − x∗,n∥ = o(√r0n) with high proba- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 ensures entry-wise concentration of SG,n(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This example indicates that the time for concentration also depends on the network size, although the time lower bound may not be tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The larger the network is, the longer we need to wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since E{X(t)} is hard to obtain in practice, computing the time average SG,n(t) helps us know the expected final positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Concentration in Gossip over an SBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this subsection, we present an example to show applications of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We assume that the network on the regular is an SBM with three equal-sized communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Conditions for general SBMs can be obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote the k-th community of the regular by Vrk with |Vrk| = nr1, 1 ≤ k ≤ 3, so Vr = ∪3 k=1Vrk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Sort the agents as follows: Vrk = {1 + (k − 1)nr1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , knr1}, 1 ≤ k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Assume that there are 2ns1 stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' So n = 3nr1 + 2ns1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The SBM(3, n, Π) with n = nr113 has a link probability matrix with πkk = p1 and πkl = p2, 1 ≤ k ̸= l ≤ 3, where p1 := (log n)β1/n, p2 := (log n)β2/n and β1 ≥ β2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let the link CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 13 probability matrix between the regular and the stubborn be Ψ(s) = � � p31nr1,ns1 0nr1,ns1 c(s) 21 p31nr1,ns1 c(s) 22 p31nr1,ns1 0nr1,ns1 p31nr1,ns1 � � , where p3 = (log n)γ/n with γ ≥ 1 and c(s) 21 , c(s) 22 ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In a graph generated by this RG-S, there are three regular communities and two stubborn communities Vsm = {3nr + (m − 1)ns1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , 3nr + mns1}, m = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Regular agents have the same intra- and inter-community link probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Also, agents in Vr1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Vr3) have the same probability linking to Vs1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Vs2) and no edges to Vs2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Vs1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Whether Vr2 is directly influenced by stubborn agents depends on c(s) 21 and c(s) 22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Such a network model describes a commonly-studied social network that consists of two leader subgroups with opposite opinions, their follower subgroups, and another fol- lower subgroup probably with fewer edges connecting to the leader groups [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The condition β1, β2, γ ≥ 1 implies that the expected degree of an agent is Ω(log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence the SBM is connected with high probability [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that nr1, ns1 = Θ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then ∆r = Θ((log n)β1∨γ) and ∆rs, ∆sr, ∥Ψ(s)∥ = Θ((log n)γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If c(s) 21 = c(s) 22 = 1, then δrs = Θ((log n)γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If c(s) 21 = c(s) 22 = 0, then λ1(E{ ¯ M}) = Ω((log n)β2∧γ) from Theorem 1 of [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In what follows we study only the δrs > 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The conditions in the δrs = 0 case can be obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, ∥xG,n − x∗,n∥ ≤ O(∥z(s)∥(log n)−γ+[(β1∨γ)+1]/2) = o(∥z(s)∥) if 2γ − (β1 ∨ γ) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It can be shown that there exist χk ∈ R, 1 ≤ k ≤ 3, such that x∗,n i = χk for i ∈ Vrk [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' When γ > β1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (i) ensures that χ1 is close to the average opinion of Vs1 (denoted by ς1), χ3 close to the average opinion of Vs2 (denoted by ς3), and χ2 is not far away from the average opinion of all stubborn agents (denoted by ς2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In contrast, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (ii) states that χk, 1 ≤ k ≤ 3, are close to the average of stubborn-agent opinions (denoted by ¯x), if λ2(E{¯L}) = Θ((log n)β2) > (log n)γ, ensured by β2 > γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From the above discussion, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5, and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9, we have the following conclusions: (i) When γ > β1 > 1 (the influence of the stubborn is large), for a small ε∗ > 0, the number of agents in Vrk such that |xG,n i − ςk| ≤ ε∗ is nr1 − O(nr((log n)1−γ ∨ (log n)2(β1−γ))) = nr1(1 − o(1)) with high probability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The expected final opinions thus polarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) When β2 > γ (the influence of stubborn agents is small) and 2γ − β1 > 1, the number of agents such that |xG,n i − ¯x| ≤ ε∗ is 3nr1 − O(nr((log n)1+β1−2γ ∨ (log n)2(γ−β2))) = 3nr1(1 − o(1)) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The expected final opinions are close to a consensus state, even if the network has a community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (iii) When β1 ≥ γ ≥ β2 (the influence of stubborn agents is moderate) and 2γ − β1 > 1, the number of agents in Vrk such that |xG,n i − χk| ≤ ε∗ is nr1 − O(nr(log n)1+β1−2γ) = nr1(1 − o(1)) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In other words, the expected final opinions have three clusters (χk need not be ςk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Now we turn to the state time average (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that E{α} = Θ(n(log n)β1∨γ), so ¯s∗ = Θ(n3/2(log n)(β1∨γ)−γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Set εS,n = εx,n/∥z(s)∥ = Θ((log n)−γ+[(β1∨γ)+1]/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then when 2γ−(β1∨γ) > 1, for time t = Ω(n3(log n)β1∨γ), the state time average SG,n(t) is close to x∗,n with error O(√nr(log n)−γ+[(β1∨γ)+1]/2) = o(√nr) with high probability for almost all G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this subsection, we discussion several extensions of the considered problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' First, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9 has a strong connection to community detection for dynamical processes, which studies how to partition agents based on only state observations [46, 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In [55] we demonstrate that, by using Polyak averaging and clustering techniques, it is possible to recover the community structure based on agent states, for the gossip model over a weighted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9 and the example given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 guarantee that such methods can still perform well for the model over SBMs, and with high probability the communities can be recovered (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', exact recovery [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Secondly, the average weight of the model is assumed to be 1/2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1), but similar concentration results with minor modifications can be derived if the weight is q ∈ (0, 1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', Xi(t + 1) = (1 − q)Xi(t) + qXj(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Furthermore, the analysis can include the case where agents are affected by random noise, for example, Xi(t + 1) = (1 − q1 − q2)Xi(t) + q1Xj(t) + q2Yi(t) with q1, q2 ∈ (0, 1) and stationary Yi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is possible to obtain parallel results by considering E{Yi(t)} as stubborn-agent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lastly, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11 shows that the state time average is close to the expected final opinions of the averaged graph after time t = Ω(n3∆r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the SBM exam- ple (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3), such concentration holds from t = Ω(n3(log n)β1∨γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For a gossip model over a weighted graph, [57] shows that agent states in the same community concentrate around the initial average opinion of that community in the time inter- val (Θ(n log n), Θ(n(log n)β1−β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' It is possible to extend this result to the SBM case (the details are omitted due to space limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' These two results together describe how the model evolves during finite periods, indicating that we can characterize not only asymptotic behavior of the model but also transient behavior in much more detail, by using random graph modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this section we present numerical simulations to illustrate the obtained theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We use the SBM studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 to generate the network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' There are three regular communities and two stubborn communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let the size of regular communities be nr1 = 25, 250, 2500, and the size of stubborn communities be ns1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2nr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The link probability matrix of the random graph is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, and we set β1 = 3 and β2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We consider three cases γ = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4, corresponding to the cases with large, moderate, and small influence of stubborn agents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We first study the case where c(s) 21 = c(s) 22 = 1 (that is, the community Vr2 has edges connected to both stubborn agent communities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For each nr1 and γ, a network is generated and then fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The stubborn agents in the first stubborn community have states generated independently and uniformly from (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9, 1) and those in the second stubborn community have states from (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The expected final opinions are computed according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Figure 2 shows that, as the network size increases, the large influence of stubborn agents boosts polarization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', agents in different communities move away from each other), whereas the expected opinions become closer when the influence of stubborn agents is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the moderate influence case, the expected opinions concentrate around their averaged counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Now we examine how the edges between regular and stubborn agents influence the profile of the expected final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Consider three cases: (i) c(s) 21 = c(s) 22 = 1, (ii) c(s) 21 = 1 and c(s) 22 = 0, (iii) c(s) 21 = c(s) 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the first case, agents in Vr2 are connected to both stubborn communities with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the second case, they are only connected to Vs1 with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the final case, they are not connected to any stubborn agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We set nr1 = 250 and γ = 4 and generate xG,n the same as earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Figure 3 shows that, in the first case, because Vr2 is influenced by both stubborn communities, the agents in this community ends in a neutral place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' However, in the second case, agents in Vr2 have states close to Vs1 because only links connecting Vr2 and Vs1 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In the third case, the expected final opinions of Vr2 is CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 15 nr1 = 25 nr1 = 250 nr1 = 2500 (a) Large influence (γ = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' nr1 = 25 nr1 = 250 nr1 = 2500 (b) Moderate influence (γ = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' nr1 = 25 nr1 = 250 nr1 = 2500 (c) Small influence (γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The profile of the expected final opinions under different stubborn influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The dashed lines represent the three distinct values of x∗,n corresponding to the communities (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (a) c(s) 21 = c(s) 22 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (b) c(s) 21 = 1 and c(s) 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (c) c(s) 21 = c(s) 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The profile of the expected final opinions xG,n under different values of c(s) 21 and c(s) 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' similar to the first case, but this similarity results from that Vr2 have the same number of edges linking to Vr1 and Vr3, rather than to the stubborn communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To illustrate the concentration of the state time average SG,n(t), we run the gossip model with nr1 = 250 (thus n = 850), γ = 4, 3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5, respectively, and c(s) 21 = c(s) 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The states of regular agents are generated uniformly from (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Figure 4 presents the histogram of SG,n(t) in the three γ cases with t = 2 × 104, 5 × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We can see that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3 I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='25 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='25 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 I 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='25 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON t = 2 × 104 t = 5 × 104 (a) γ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' t = 2 × 104 t = 5 × 104 (b) γ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' t = 2 × 104 t = 5 × 104 (c) γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The profile of the state time average SG,n(t) with t = 2 × 104 and 5 × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' the profile of SG,n(t) is similar to the profile of the expected final opinions shown in Figure 2 and concentration actually appears much earlier than the predicted bound in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that these time steps are not too large because agents interact less than 60 times on average, and interact only a few times with each neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this paper, we studied concentration in gossip opinion dy- namics over random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' High-probability bounds were derived for the concentra- tion of the expected final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We further showed how such concentration can help study the effect of network topology on the expected final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' With the help of concentration inequalities for Markov chains, we obtained concentration bounds for the state time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' We then applied the results to the gossip dynamics over an SBM, and obtained quantitative characterization of the opinion profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Future work includes investigating sharp concentration thresholds for the gossip and other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Auxiliary Concentration Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this section, we present auxiliary concentration lemmas from which the main results given in the paper are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' These lemmas are consequences of the follow- ing standard conclusions in high-dimensional probability theory and matrix analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (The Chernoff inequality, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 of [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , Xn are independent Bernoulli random variables such that P{Xi = 1} = pi = 1 − P{Xi = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let X := �n i=1 Xi and µ := E{X} = �n i=1 pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then for 0 < δ < 1, P{X ≥ (1 + δ)µ} ≤ e−µδ2/3, P{X ≤ (1 − δ)µ} ≤ e−µδ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 (The matrix Bernstein inequality, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 and Exercise 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 of [52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , YN ∈ Rn×n are independent zero-mean random matri- ces, and are such that ∥Yi∥ ≤ K a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=', 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then for a ≥ 0, it holds that P ����� N � i=1 Yi ���� ≥ a � ≤ 2n exp � −a2/2 σ2 + Ka/3 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 States10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 Statesdhil 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='15 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='05 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8 Statesdhil 1CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 17 where σ2 = ∥ �N i=1 E{Y 2 i }∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , YN ∈ Rm×n are independent, mean zero, and such that ∥Yi∥ ≤ K a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then for all a ≥ 0, it holds that P ����� N � i=1 Yi ���� ≥ a � ≤ 2(m + n) exp � −a2/2 σ2 + Ka/3 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) where σ2 = max{∥ �N i=1 E{Y T i Yi}∥, ∥ �N i=1 E{YiY T i }∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For A, B ∈ Rn×n, if A and B are symmetric, then the Weyl in- equality holds (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) of [32]): max 1≤i≤n |λi(A) − λi(B)| ≤ ∥A − B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4) If A and B are invertible, then ((5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) of [32]) ∥A−1 − B−1∥ ≤ ∥A−1∥∥B−1∥∥A − B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) First, we derive a concentration bound for the matrix ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4 (Concentration of ¯ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that ∆r ≥ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then P{∥ ¯ M − E{ ¯ M}∥ ≤ εM,n} ≥ 1−ηM,n = 1−o(1), where εM,n = 4√∆r log n and ηM,n = 2r0n− 1 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Decompose ¯ M − E{ ¯ M} = �r0n i=1 �n j=i+1 Yij, where Yij = (aij − E{aij}) (Eii + Ejj − Eij − Eji), 1 ≤ i < j ≤ r0n, and Yij = (aij − E{aij})Eii, 1 ≤ i ≤ r0n < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Here Eij = eieT j , 1 ≤ i, j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence E{Yij} = 0, Var(Yij) = 2(pij − p2 ij)(Eii + Ejj − Eij − Eji) for 1 ≤ i < j ≤ r0n, and Var(Yij) = (pij − p2 ij)Eii for 1 ≤ i ≤ r0n < j ≤ n, where pij := E{aij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote ¯Y := �r0n i=1 �n j=i+1 Var(Yij), so v2 = ∥ ¯Y ∥ ≤ 4 max1≤i≤r0n{�n j=1 pij} = 4 max1≤i≤r0n{E{di}} = 4∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2), for a > 0, since ∥Yij∥ ≤ 2, P{∥ ¯ M − E{ ¯ M}∥ > a} ≤ 2r0n exp � −a2 4(2∆r + a/3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Set a = 4√∆r log n, and from the assumption ∆r ≥ log n we have that P{∥ ¯ M − E{ ¯ M}∥ > 4 � ∆r log n} ≤ 2r0n exp � −4∆r log n 2∆r + 4√∆r log n/3 � ≤ 2r0n exp �−4 log n 2 + 4/3 � = 2r0n− 1 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' As a consequence of the preceding lemma, we can estimate the deviation between the inverse of ¯ M and E{ ¯ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 (Concentration of ¯ M −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) If δrs > 8 log n, then it holds that P{∥ ¯ M −1 − (E{ ¯ M})−1∥ ≤ ε′ M,n} ≥ 1 − η′ M,n = 1 − o(1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6) where ε′ M,n = 2εM,n/δ2 rs, η′ M,n = r0n1−δrs/(8 log n)+ηM,n, and εM,n and ηM,n are given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) If λ1(E{ ¯ M}) > εM,n and ∆r ≥ log n, then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6) holds with ε′ M,n = εM,n/ [λ1(E{ ¯ M})(λ1(E{ ¯ M}) − εM,n)] and η′ M,n = ηM,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 18 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that E{[ ¯ M]ii} = E{di}, and E{[ ¯ Mij]} = −E{aij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' So by the Ger- shgorin circle theorem, λmin(E{ ¯ M}) ≥ min1≤i≤r0n{E{di} − E{d(r) i }} = min1≤i≤r0n {E{d(s) i }} = δrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Thus, for symmetric E{ ¯ M}, (E{ ¯ M})−1 exists when δrs > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence, ∥(E{ ¯ M})−1∥ = 1 λmin(E{ ¯ M}) ≤ 1 δrs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7) Similarly, from the Gershgorin circle theorem, it follows that λmin( ¯ M) ≥ min1≤i≤r0n {di − d(r) i } = min1≤i≤r0n{d(s) i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) with δ = 1/2, we obtain that P � min 1≤i≤r0n{d(s) i } > 1 2δrs � ≥ 1 − P � r0n � i=1 � d(s) i ≤ 1 2E{d(s) i } �� ≥ 1 − r0ne−δrs/8 = 1 − r0n1−δrs/(8 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' As a result, with probability at least 1 − r0n1−δrs/(8 log n), ∥ ¯ M −1∥ = 1 λmin( ¯ M) ≤ 2 δrs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8) Therefore, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5), with probability at least 1 − r0n1−δrs/(8 log n) − ηM,n, ∥ ¯ M −1 − (E{ ¯ M})−1∥ ≤ ∥ ¯ M −1∥∥(E{ ¯ M})−1∥∥ ¯ M − E{ ¯ M}∥ ≤ 2εM,n δ2rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To show (ii), note from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4) that with probability at least 1 − ηM,n |λmin( ¯ M) − λmin(E{ ¯ M})| ≤ εM,n, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9) so λmin( ¯ M) ≥ λmin(E{ ¯ M}) − εM,n > 0 when λmin(E{ ¯ M}) > εM,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Again from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5), ∥ ¯ M −1 − (E{ ¯ M})−1∥ ≤ ∥ ¯ M −1∥∥(E{ ¯ M})−1∥∥ ¯ M − E{ ¯ M}∥ = 1 λmin( ¯ M) 1 λmin(E{ ¯ M})∥ ¯ M − E{ ¯ M}∥ ≤ εM,n λmin(E{ ¯ M})(λmin(E{ ¯ M}) − εM,n), with probability at least 1 − ηM,n, when ∆r ≥ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Similar to ¯ M, we can obtain concentration of ¯U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 (Concentration of ¯U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Suppose that ∆rs ∨∆sr ≥ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then P{∥ ¯U − Ψ(s)∥ ≤ εU,n} ≥ 1 − ηU,n = 1 − o(1), where εU,n = 2 � (∆rs ∨ ∆sr) log n and ηU,n = 2n−1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Decompose ¯U−E{ ¯U} = �r0n i=1 �n j=r0n+1 Y ′ ij, where Y ′ ij = (aij−E{aij})e(r) i (e(s) j )T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Here e(r) i := e(r0n) i and e(s) j := e(s0n) j−r0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence, ∥Y ′ ij∥ = |aij−E{aij}|∥e(r) i (e(s) j )T∥ ≤ ∥e(r) i (e(s) j )T∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Now note that ¯Y ′ := r0n � i=1 n � j=r0n+1 E{(Y ′ ij)TY ′ ij} = r0n � i=1 n � j=r0n+1 E{(aij − E{aij})2}e(s) j (e(r) i )Te(r) i (e(s) j )T CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 19 = diag � r0n � i=1 (pi,r0n+1 − p2 i,r0n+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , r0n � i=1 (pi,n − p2 i,n) � , where pij = E{aij}, so ∥ ¯Y ′∥ ≤ max r0n+1≤j≤n � r0n � i=1 (pij − p2 ij) � ≤ max r0n+1≤j≤n � r0n � i=1 pij � = ∆sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Similarly, let ¯Y ′′ := �r0n i=1 �n j=r0n+1 E{Y ′ ij(Y ′ ij)T}, and then we have that ∥ ¯Y ′′∥ ≤ ���� diag � n � j=r0n+1 (p1j − p2 1j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , n � j=r0n+1 (pr0n,j − p2 r0n,j) ����� ≤ max 1≤i≤r0n � n � j=r0n+1 (pij − p2 ij) � ≤ ∆rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let σ2 = ∆rs ∨ ∆sr and K = 1, and set a = 2 � (∆rs ∨ ∆sr) log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3), P{∥ ¯U − E{ ¯U}∥ > 2 � (∆rs ∨ ∆sr) log n} ≤ 2(s0n + r0n) exp � −2(∆rs ∨ ∆sr) log n (∆rs ∨ ∆sr) + 2 � (∆rs ∨ ∆sr) log n/3 � = 2n exp � −2 log n 1 + 2 � (log n)/(∆rs ∨ ∆sr)/3 � ≤ 2n− 1 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The preceding concentration bounds are useful in analyzing the difference ∥xG,n− x∗,n∥ = ∥(I − ¯Q)−1 ¯Rz(s) − (I − ¯Q)−1 ¯Rz(s)∥ = ∥[ ¯ M −1 ¯U − (E{ ¯ M})−1Ψ(s)]z(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' But to make sure that xG,n is well-defined, we study ¯Q and α in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (Bound of α and ρ( ¯Q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (i) Suppose that δrs > 8 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then it holds that P{[ρ( ¯Q) ≤ εQ,n < 1] ∩ [α ≥ E{α}/2 > 0]} ≥ 1 − ηQ,n = 1 − o(1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='10) where εQ,n = 1 − δrs/(6E{α}), and ηQ,n = r0n1−δrs/(8 log n) + 2n−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (ii) Suppose that λ1(E{ ¯ M}) > εM,n and ∆r ≥ log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='10) holds with εQ,n = 1 − (λ1(E{ ¯ M}) − εM,n)/(3E{α}) and ηQ,n = ηM,n + 2n−1/8, where εM,n and ηM,n are given in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) with δ = 1/2 yields that P � α − E{α} ≤ −1 2E{α} � ≤ e− E{α} 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11) When E{α} ≥ δrs > 8 log n > 0, e−E{α}/8 ≤ e− log n = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If E{α} ≥ ∆r ≥ log n > 0, e−E{α}/8 ≤ n−1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence, α ≥ E{α}/2 > 0 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that I − ¯Q = E{ ¯ M}/(2E{α}), so (I − ¯Q)−1 exists under conditions of either (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since ¯Q = I − ¯ M/(2α) is symmetric and positive semi-definite, to show ρ( ¯Q) = λmax( ¯Q) < 1, it suffices to provide a lower bound for λ1( ¯ M/(2α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' First we derive a bound under δrs > 8 log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8), we know that P{λ1( ¯ M) > δrs/2} ≥ 1 − r0n1−δrs/(8 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In addition, applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) with δ = 1/2 yields that P � 1 2α ≤ 1 3E{α} � ≤ P � α − E{α} ≥ 1 2E{α} � ≤ e− E{α} 12 ≤ n− 2 3 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='12) 20 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON so ρ( ¯Q) ≤ 1−δrs/(6E{α}) holds with probability at least 1−r0n1−δrs/(8 log n) −n−2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Thus (i) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='12) yields (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7, (I − ¯Q)−1 exists with high probability and (I − ¯Q)−1 exists under either Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) or (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In either case, xG,n and x∗,n are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If I − ¯Q is singular, define xG,n := ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For finite xG,n, it holds that ∥xG,n − x∗,n∥ = ∥(I − ¯Q)−1 ¯Rz(s) − (I − ¯Q)−1 ¯Rz(s)∥ = ���� �� ¯ M 2α �−1 ¯U 2α − � E{ ¯ M} 2E{α} �−1 Ψ(s) 2E{α} � z(s) ���� = ∥[ ¯ M −1 ¯U − (E{ ¯ M})−1Ψ(s)]z(s)∥ = ∥{ ¯ M −1( ¯U − Ψ(s)) + [ ¯ M −1 − (E{ ¯ M})−1]Ψ(s)}z(s)∥ ≤ (∥ ¯ M −1∥∥ ¯U − Ψ(s)∥ + ∥ ¯ M −1 − (E{ ¯ M})−1∥∥Ψ(s)∥)∥z(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='8), Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 (i), and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (i), it holds that ∥xG,n − x∗,n∥ ≤ � 2 δrs εU,n + ε′ M,n∥Ψ(s)∥ � ∥z(s)∥ ≤ 4 �� (∆rs ∨ ∆rs) log n δrs + 2√∆r log n∥Ψ(s)∥ δ2rs � ∥z(s)∥, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) P{xG,n exists, and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) holds} ≥ 1 − ηU,n − η′ M,n − 2n− 2 3 = 1 − r0n1− δrs 8 log n − 2(1 + r0)n− 1 5 − 2n− 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' In this way, we prove (i) of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The second part follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='9), Lem- ma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 (ii), and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote ¯S := diag(d(s) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , d(s) rn ) = ¯ M − ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since ¯G is connected, (I − ¯Q)−1 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To prove the first part of (i), note that (E{ ¯S})−1 exists when δrs > 0, so ∥x∗,n − (E{ ¯S})−1Ψ(s)z(s)∥ ≤ ∥(E{ ¯ M})−1 − (E{ ¯S})−1∥∥Ψ(s)∥∥z(s)∥ ≤ ∥(E{ ¯ M})−1∥∥(E{ ¯S})−1∥∥E{ ¯ M} − E{ ¯S}∥∥Ψ(s)∥∥z(s)∥ (From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5)) ≤ 1 δrs 1 δrs ∥E{¯L}∥∥Ψ(s)∥∥z(s)∥ ≤ 2∆rr δ2rs ∥Ψ(s)∥∥z(s)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7)) For the second part of (i), note that x∗,n − x†,n = � [ ˜ M (11) − (diag(Ψ(s) + 1s0n))−1]Ψ(s) + z(s) 0 � , so it suffices to bound ∥[ ˜ M (11) −(diag(Ψ(s) + 1s0n))−1]Ψ(s) + z(s)∥ = ∥x∗,n −x†,n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since G is connected, ˆ M (21) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Thus there exist rows of ˆ M (22) with strictly dominant diagonals, which implies that ˆ M (22) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' From the inverse formula of block matrices [31], it follows that ˜ M (11) = [ ˆ M (11) − ˆ M (12)( ˆ M (22))−1 ˆ M (21)]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence, ∥ ˜ M (11) − ( ˆ M (11))−1∥ CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 21 = ∥[ ˆ M (11) − ˆ M (12)( ˆ M (22))−1 ˆ M (21)]−1 − ( ˆ M (11))−1∥ ≤ ∥[ ˆ M (11) − ˆ M (12)( ˆ M (22))−1 ˆ M (21)]−1∥∥( ˆ M (11))−1∥∥ ˆ M (12)( ˆ M (22))−1 ˆ M (21)∥ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) = ∥ ˆ M (12)( ˆ M (22))−1 ˆ M (21)∥ λ1( ˆ M (11))λ1( ˆ M (11) − ˆ M (12)( ˆ M (22))−1 ˆ M (21)) ≤ ∥ ˆ M (12)( ˆ M (22))−1 ˆ M (21)∥ λ1( ˆ M (11))(λ1( ˆ M (11)) − ∥ ˆ M (12)( ˆ M (22))−1 ˆ M (21)∥) (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) ≤ ∥ ˆ M (21)∥2/λ1( ˆ M (22)) λ1( ˆ M (11))(λ1( ˆ M (11)) − ∥ ˆ M (21)∥2/λ1( ˆ M (22))) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) where (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='2) from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Similarly we obtain that ∥( ˆ M (11))−1 − (diag(Ψ(s) + 1s0n))−1∥ ≤ ∥ ˆ M (11) − diag(Ψ(s) + 1s0n)∥ λ1( ˆ M (11))δ+ rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4) The Gershgorin theorem yields that λ1( ˆ M (11)) ≥ δ+ rs and ∥ ˆ M (11) − diag(Ψ(s) + 1s0n)∥ ≤ 2 max1≤i≤n1{E{d(r) i }}, so the conclusion follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Now we show (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The assumption λ2(E{¯L}) > 2∆rs > 0 ensures that the eigen- value λ1(E{ ¯ M}) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' So by ξ we denote a unit eigenvector corresponding to the eigenvalue λ1(E{ ¯ M}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since E{ ¯ M} is symmetric, it has orthogonal unit eigenvalues w(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , w(r0n) corresponding to its eigenvalues λ2(E{ ¯ M}) ≤ · · · ≤ λr0n(E{ ¯ M}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Also ξ, w(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' , w(r0n) form a basis of Rr0n, and ξξT + �r0n j=2 w(j)(w(j))T = Ir0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' So ����(E{ ¯ M})−1Ψ(s)z(s) − 1 r0nλ1(E{ ¯ M})1r0n1T r0nΨ(s)z(s) ���� = ����(E{ ¯ M})−1 � ξξT + r0n � j=2 w(j)(w(j))T � Ψ(s)z(s) − 1 r0nλ1(E{ ¯ M})1r0n1T r0nΨ(s)z(s) ���� ≤ ����(E{ ¯ M})−1ξξTΨ(s)z(s) − 1 r0nλ1(E{ ¯ M})1r0n1T r0nΨ(s)z(s) ���� + ����(E{ ¯ M})−1 � r0n � j=2 w(j)(w(j))T � Ψ(s)z(s) ���� =: (I) + (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that (E{ ¯ M})−1ξ = ξ/λ1(E{ ¯ M}), so (I) = ���� 1 λ1(E{ ¯ M})ξξTΨ(s)z(s) − 1 r0nλ1(E{ ¯ M})1r0n1T r0nΨ(s)z(s) ���� ≤ 1 λ1(E{ ¯ M}) ����ξξT − 1 r0n1r0n1T r0n ����∥Ψ(s)z(s)∥ ≤ 2∥Ψ(s)∥∥z(s)∥∆rs λ1(E{ ¯ M})(λ2(E{¯L}) − 2∆rs), where the last inequality is obtained from the following lemma with A = E{ ¯ M}, B = E{¯L}, and ζ = (λ2(E{¯L}) − 2∆rs)/2, which is a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='5 in Chapter I and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='6 in Chapter V of [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let A, B ∈ Rn×n be symmetric, and µ with corresponding unit eigenvector u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' ν with unit eigenvector v) be a simple eigenvalue of A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' 22 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' XING AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' JOHANSSON B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denote r = Av − νv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If there exists ζ > 0 such that the eigenvalues of A except µ lie outside the interval [ν − ζ, ν + ζ], then ∥uuT − vvT∥ ≤ ∥r∥ ζ = ∥Av − Bv∥ ζ ≤ ∥A − B∥ ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For (II), it holds that (II) = ���� r0n � j=2 1 λj(E{ ¯ M})w(j)(w(j))TΨ(s)z(s) ���� = � � � � r0n � j=2 [(w(j))TΨ(s)z(s)]2 λ2 j(E{ ¯ M}) ≤ 1 λ2(E{ ¯ M}) � � � � r0n � j=2 [(w(j))TΨ(s)z(s)]2 = 1 λ2(E{ ¯ M}) ���� r0n � j=2 w(j)(w(j))TΨ(s)z(s) ���� = ∥(I − ξξT)Ψ(s)z(s)∥ λ2(E{ ¯ M}) ≤ 2∥Ψ(s)∥∥z(s)∥ λ2(E{ ¯ M}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' The conclusion follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='4) and then combining (I) and (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Since we have derived a bound for ∥xG,n − x∗,n∥ in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3, it suffices to bound the term ∥SG,n(t)−xG,n∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' To this end, we introduce the following concentration inequality (Lemma 1 of [55]) for the state time average of a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 (Concentration of state time average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Consider a discrete-time Markov chain {X(t)} taking values on a compact state space X and having a unique stationary distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' For a function f : X → R and ι := � X f(x)π(dx), denote g(x) := �∞ t=0 E{f(X(t))−ι|X(0) = x} and ∥g∥s := sup{|g(x)| : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' If ∥g∥s < ∞, then it holds for Sf(t) := 1 t �t−1 i=0 f(X(i)), ε > 0, and t > 2∥g∥s/ε that P{|Sf(t) − ι| ≥ ε} ≤ 2 exp � − (tε − 2∥g∥s)2 2t∥g∥2s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Conditioned on a graph G, ρ( ¯Q) < 1 ensures that the gossip model has a well-defined unique stationary distribution with mean xG,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' This result follows from a standard argument for gossip models (see [2, 45, 55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 ensures that ρ( ¯Q) < 1 holds with probability at least 1 − ηQ,n, where the probability is over the randomness of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Now we derive a bound for ∥SG,n(t) − xG,n∥ given a graph G such that ρ( ¯Q) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Let fi(x) = xi, ∀x ∈ Rr0n, 1 ≤ i ≤ r0n, and Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1 ensures that PG{|SG,n i (t) − xG,n i | ≥ ε} ≤ 2 exp � − (tε − 2∥gG,n i ∥s)2 2t∥gG,n i ∥2s � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) holds for all ε > 0 and t > 2∥gG,n i ∥s/ε, where gG,n i is the i-th component of GG,n(x) = �∞ t=0 EG{X(t) − xG,n|X(0) = x}, x ∈ Rr0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Note that ∥gG,n i ∥s < ∞ because for all CONCENTRATION IN GOSSIP OPINION DYNAMICS OVER RANDOM GRAPHS 23 x ∈ X = [−cx, cx]r0n ∥GG,n(x)∥ ≤ ∞ � t=0 ���� ¯Qtx − ∞ � i=t ¯Qi ¯Rz(s) ���� = ∞ � t=0 ���� ¯Qt � x − ∞ � i=0 ¯Qi ¯Rz(s) ����� ≤ ∞ � t=0 ∥ ¯Q∥t∥x − xG,n∥ = ∥x − xG,n∥ 1 − ρ(Q) ≤ 2√r0ncx 1 − ρ( ¯Q) =: sG,n ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Hence ∥gG,n i ∥s = supx∈X {|GG,n i (x)|} ≤ supx∈X {∥GG,n(x)∥} ≤ sG,n ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Therefore, from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='1) it follows that for all ε > 0 and t > 2sG,n ∗ /ε PG{∥SG,n(t) − xG,n∥ ≥ √r0nε} ≤ PG{∃i ∈ Vr, |SG,n i (t) − xG,n i | ≥ ε} ≤ 2r0n exp � − (tε − 2sG,n ∗ )2 2t(sG,n ∗ )2 � , Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='7 implies that P � exp � − (tε − 2sG,n ∗ )2 2t(sG,n ∗ )2 � ≤ exp � − (tε − 2¯s∗)2 2t(¯s∗)2 �� ≥ 1 − ηQ,n, where if δrs > 8 log n then ¯s∗ = 12√r0ncxE{α}/δrs and ηQ,n = r0n1−δrs/(8 log n) + 2n−2/3, and if λ1(E{ ¯ M}) > εM,n and ∆r ≥ log n then ¯s∗ = 6√r0ncxE{α}/(λ1(E{ ¯ M}) −εM,n) and ηQ,n = ηM,n + 2n−1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Denoting S1 = {ρ( ¯Q) ≤ εQ,n}, by the law of total probability we have that P{∥SG,n(t) − xG,n∥ ≥ √r0nε} = P{∥SG,n(t) − xG,n∥ ≥ √r0nε|S1}P{S1} + P{∥SG,n(t) − xG,n∥ ≥ √r0nε|Sc 1}P{Sc 1} ≤ 2r0n exp � − (tε − 2¯s∗)2 2t(¯s∗)2 � P{S1} + P{Sc 1} ≤ 2r0n exp � − (tε − 2¯s∗)2 2t(¯s∗)2 � + ηQ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' Therefore, the conclusion follows from the above bound and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E4T4oBgHgl3EQf9A55/content/2301.05352v1.pdf'} +page_content=' REFERENCES [1] E.' metadata={'source': 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Specifically, the +student’s objective function is +� +ξ ∗ ℓ(teacher’s predictions, student’s predictions) + (1 − ξ) ∗ +ℓ(given labels, student’s predictions) +� +, where ℓ is some loss function and ξ is some parameter +∈ [0, 1]. Empirically, SD has been observed to provide performance gains in several settings. +In this paper, we theoretically characterize the effect of SD in two supervised learning +problems with noisy labels. We first analyze SD for regularized linear regression and show +that in the high label noise regime, the optimal value of ξ that minimizes the expected +error in estimating the ground truth parameter is surprisingly greater than 1. Empirically, +we show that ξ > 1 works better than ξ ≤ 1 even with the cross-entropy loss for several +classification datasets when 50% or 30% of the labels are corrupted. Further, we quantify +when optimal SD is better than optimal regularization. Next, we analyze SD in the case of +logistic regression for binary classification with random label corruption and quantify the +range of label corruption in which the student outperforms the teacher in terms of accuracy. +To our knowledge, this is the first result of its kind for the cross-entropy loss. +1 +Introduction +The core idea of knowledge distillation (KD), introduced in [Hinton et al., 2015], is to train a +student model with a teacher model’s predicted soft labels (i.e., the output probability distribution +over the classes for classification problems) in addition to the original hard labels (one-hot vectors +for classification problems) on which the teacher is trained. The original rationale was to use a +teacher with large statistical capacity to better model the underlying label distribution compared +to the provided hard labels, and have the student with smaller capacity learn some mixture +of the teacher’s predicted label distribution (a.k.a. “dark knowledge”) and the provided label +distribution. Specifically, the student’s per-sample objective function in the KD framework is: +ξ ∗ ℓ +� +yT , yS(θ) +� ++ (1 − ξ) ∗ ℓ +� +y, yS(θ) +� +, +(1) +where ℓ is some loss function (usually, regularized cross-entropy loss for classification problems), +yT is the teacher’s predicted label, y is the given label on which the teacher is trained, yS(θ) is +the prediction of the student model parameterized by θ, and ξ ∈ [0, 1] is known as the imitation +parameter [Lopez-Paz et al., 2015]1. KD and its variants have been shown to be beneficial for +model compression (i.e., distilling a bigger teacher model’s knowledge into a smaller student +model), semi-supervised learning, making models robust and improving performance in general [Li +et al., 2017,Furlanello et al., 2018,Sun et al., 2019,Ahn et al., 2019,Chen et al., 2020,Xie et al., +2020,Sarfraz et al., 2021,Li et al., 2021,Pham et al., 2021,Beyer et al., 2022,Baykal et al., 2022]; +see [Gou et al., 2021] for a survey on KD. +The focus of this work is on the special case of the student and teacher having the same +architecture, which is known as self-distillation (following [Mobahi et al., 2020]); we abbreviate +1In this work, we set the temperature parameter suggested in [Hinton et al., 2015] equal to 1. +1 +arXiv:2301.13304v1 [cs.LG] 30 Jan 2023 + +it as SD henceforth. Since the teacher and student have the same capacity, one would expect +the utility of the teacher’s dark knowledge to be very limited, if any at all. However, surprisingly, +[Furlanello et al., 2018] show that SD (with ensembling) yields performance gains in both vision +and language tasks with extensive experiments. Further, [Li et al., 2017] empirically demonstrate +that SD can ameliorate learning in the presence of noisy labels. There are also a few works that +theoretically investigate SD, such as [Mobahi et al., 2020,Dong et al., 2019]; we discuss these +in detail in Section 2. The results of these papers are only with the squared loss and not the +cross-entropy loss which is the de facto loss function for classification problems. +In this work, we theoretically analyze SD in the presence of label corruption (in the supervised +setting) for the cross-entropy loss as well as the squared loss, characterizing its utility and +unveiling some new insights including a recommendation for use in practice. We summarize our +contributions next and survey the landscape of pertinent theoretical works on KD and SD in +Section 2. +Contributions: +(a) First, we consider linear regression with ℓ2-regularized squared loss in Section 3. Here, the +observed label y for a sample x is: y = ⟨θ∗, x⟩ + η, where θ∗ is the underlying parameter and η +is zero-mean random label noise. +• We show that self-distillation (SD) is associated with a bias-variance tradeoff in that +increasing ξ in eq. (1) reduces the variance but increases the bias in estimating θ∗ with +respect to the randomness in label noise; see Theorem 1 and Remark 1. +• A surprising algorithmic insight from our analysis is that the value of ξ that optimally +balances this bias-variance tradeoff can be > 1, especially in the high label noise regime (i.e., +when E[η2] is large); see Corollary 1.1 and Remark 2. This can be interpreted as actively +anti-learning (or going against) the observed (possibly noisy) labels. But as discussed +after eq. (1), ξ is tuned in [0, 1] in practice. In Section 5.1, we empirically corroborate our +insight for multi-class classification with linear probing2 using the cross-entropy loss by +showing that ξ > 1 works better than ξ ≤ 1 for several datasets with 50% or 30% of the +training set’s labels being corrupted in different ways. +• In Remark 3, we show that as the degree of label noise increases, the utility of the teacher’s +predictions in training the student increases. Intuitively, this happens because the noise +component in the teacher’s predictions is smaller compared to the original labels. We also +empirically verify this insight for the cross-entropy loss in Section 5.2. +• In Theorem 2, we provide a condition when optimal SD is better than optimal ℓ2 regular- +ization (optimal means with the best parameters); this is the first such result. +(b) Next, we look at logistic regression with ℓ2-regularized cross-entropy loss in Section 4. We +consider a balanced binary classification problem where some fraction, say p < 0.5, of the training +set’s labels are randomly flipped. Under some assumptions on the data geometry and the kernel +function, we quantify the range of p in which the student outperforms the teacher in terms of +accuracy; see Theorem 5. To our knowledge, this is the first result that provably establishes +the utility of SD in the presence of label noise for the cross-entropy loss. The main technical +challenge in the analysis is dealing with non-linear equations involving the sigmoid function. We +tackle this by employing the first-order Maclaurin series expansion of the sigmoid function and by +bounding the corresponding approximation errors; see Step 3 in the proof outline of Theorem 5. +Moreover, in Corollary 5.1, we show that the student’s predictions have smaller variability than +the teacher’s predictions which is akin to SD reducing variance in linear regression. +2i.e., learning a softmax layer on top of a pre-trained network +2 + +2 +Related Work +There is a growing body of works trying to theoretically explain KD/SD and its benefits. [Mobahi +et al., 2020] look at regression with the squared loss in Hilbert space, showing that SD essentially +amplifies regularization. However, unlike us, they do not explicitly consider the case of noisy +labels/observations or discuss the bias-variance tradeoff associated with SD in the presence of +label noise. Moreover, they restrict their analysis to ξ = 1; so unlike us, they do not have any +results on when optimal SD is better than optimal ℓ2 regularization. [Dong et al., 2019] claim +that KD is effective in transferring dark knowledge by mimicking early stopping. Further, they +propose their own SD algorithm that uses dynamically updated soft labels, and show that in the +presence of noisy labels, their algorithm is able to learn the correct labels. In this work, we focus +on the standard SD algorithm with fixed soft labels, and moreover, we quantify the range of label +corruption in which SD improves accuracy. Unlike our work, [Dong et al., 2019] do not quantify +when their proposed algorithm improves upon the standard approach of using just hard labels. +An important difference between our work and [Dong et al., 2019] as well as [Mobahi et al., 2020] +is that the results of these two papers are with the squared loss, whereas we provide results +with the cross-entropy loss in addition to squared loss. The cross-entropy loss is the customary +choice for classification problems in practice and is also more challenging to analyze. On the +note of cross-entropy loss, [Phuong and Lampert, 2019] analyze the convergence of linear student +networks trained with the cross-entropy loss, and also bound the expected difference between the +predictions of the student and teacher. [Ji and Zhu, 2020] also bound the expected difference +between the predictions of the student and teacher for wide neural networks that evolve as linear +networks under the NTK assumption. However, [Phuong and Lampert, 2019] and [Ji and Zhu, +2020] do not consider how the student might have better generalization than the teacher in +the presence of noisy labels. [Menon et al., 2021] statistically characterize “good” teachers for +distilling knowledge to a student. [Kaplun et al., 2022] show that an ensemble of teachers trained +with noisy labels can be used to label a new unlabeled dataset, which can be then employed to +train a student with good performance. We focus on the (common) case of only one teacher +and the student being trained on the same dataset as the teacher. There are also some works +such as [Cheng et al., 2020,Stanton et al., 2021,Pham et al., 2022] that empirically provide some +insights on KD. +3 +Linear Regression +Setting: The observed label y ∈ R is linearly related to the data x ∈ X ⊆ Rd as: +y = ⟨θ∗, x⟩ + η, +(2) +where θ∗ ∈ Rd and η ∈ R is label noise. Here, ⟨θ∗, x⟩ is the actual label of x. +The training set consists of n pairs of data points (drawn from X) and noisy labels {(xi, yi)}n +i=1. +Let X := [x1, . . . , xn] ∈ Rd×n be the data matrix and Y := [y1, . . . , yn]T ∈ Rn be the label +vector. Then, as per the above linear model (eq. (2)): +Y = XT θ∗ + η, +(3) +for some noise vector η ∈ Rn. We make some standard assumptions on the noise vector η. +Assumption 1. η is independent of X. Further, each coordinate of η has mean 0 and variance +γ2, and is independent of the other coordinates. +Teacher Model: The teacher tries to learn the underlying model, parameterized by θ ∈ Rd, +from (X, Y ) by applying the squared loss with ℓ2 regularization. Specifically, the teacher’s +objective function is: +fT (θ) = 1 +2∥Y − XT θ∥2 + λ +2 ∥θ∥2, +(4) +3 + +where λ > 0 is the ℓ2-regularization parameter. Now, the model learned by the teacher is3: +ˆθT := arg minθ∈Rd fT (θ) = (XXT + λId)−1XY , +(5) +where Id is the identity matrix of size d × d. Plugging in Y from eq. (3) in eq. (5), we get: +ˆθT = (XXT + λId)−1X(XT θ∗ + η). +(6) +Student Model Trained with Self-Distillation: Following eq. (1), here the student is trained +with a weighted sum of (i) the ℓ2-regularized squared loss between the student’s predictions +and the teacher’s predictions, and (ii) the ℓ2-regularized squared loss between the student’s +predictions and the original labels on which the teacher was trained. For the ith sample, the +teacher’s prediction is ˆyi = ⟨ˆθT , xi⟩. Define ˆY := [ˆy1, . . . , ˆyn]T ∈ Rn; note that ˆY = XT ˆθT . +The student’s objective function is: +fS(θ; ξ) = ξ +�1 +2∥ ˆY − XT θ∥2 + λ +2 ∥θ∥2� ++ (1 − ξ) +�1 +2∥Y − XT θ∥2 + λ +2 ∥θ∥2� += ξ +�1 +2∥ ˆY − XT θ∥2� ++ (1 − ξ) +�1 +2∥Y − XT θ∥2� ++ λ +2 ∥θ∥2, +(7) +where ξ ∈ R is known as the imitation parameter [Lopez-Paz et al., 2015] and λ > 0 is the same +regularization parameter that was used by the teacher. Even though it is standard practice to +restrict ξ ∈ [0, 1], we do not impose this condition. Now, the model learned by the student is: +ˆθS(ξ) := arg minθ∈Rd fS(θ; ξ) = (XXT + λId)−1X(ξ ˆY + (1 − ξ)Y ) += ξ(XXT + λId)−1XXT ˆθT + (1 − ξ)ˆθT , +(8) +where eq. (8) is obtained by using ˆY = XT ˆθT and eq. (5). Note that ξ = 0 corresponds to the +teacher, i.e. ˆθS(0) = ˆθT . +Finally, plugging in ˆθT from eq. (6) in eq. (8), we get: +ˆθS(ξ) = +� +ξ(XXT + λId)−1XXT + (1 − ξ)Id +� +(XXT + λId)−1X(XT θ∗ + η). +(9) +3.1 +Estimation Error Comparison: Bias-Variance Tradeoff +Let us denote the student’s error in estimating the ground truth parameter θ∗ with imitation +parameter ξ as ϵS(ξ) := ˆθS(ξ) − θ∗. Note that ϵS(0) := ˆθS(0) − θ∗ = ˆθT − θ∗ is the teacher’s +estimation error. We shall analyze the expected squared norm of the estimation error w.r.t. the +random label noise η, i.e. Eη[∥ϵS(ξ)∥2], as a function of ξ4. +It will be illustrative to analyze Eη[∥ϵS(ξ)∥2] in terms of the SVD of X. Let rank(X) = r (note +that r ≤ min(d, n)) and the SVD decomposition of X be �r +j=1 σjujvT +j , where σ1 ≥ . . . ≥ σr > 0, +and each uj ∈ Rd and vj ∈ Rn. Also, let {u1, . . . , ud} be the full set of left singular vectors of X +(i.e., even those corresponding to the zero singular values); note that this forms an orthonormal +basis for Rd. +Following standard bias-variance decomposition, we have: +Eη +���ϵS(ξ) +��2� += +��Eη[ϵS(ξ)] +��2 +� +�� +� +squared bias ++ Eη +���ϵS(ξ) − Eη[ϵS(ξ)] +��2� +� +�� +� +variance +. +(10) +Now we shall quantify the squared bias and variance in eq. (10) as a function of ξ. +3Throughout this work, we shall assume that we can converge to the exact optimum of the objective function. +All the objective functions in this work are convex, and hence (stochastic) gradient descent will converge to the +optimum in all the cases. +4We do not analyze the expected squared prediction error, i.e. Eη,x +�� +⟨ˆθS(ξ), x⟩ − ⟨θ∗, x⟩ +�2�, because that +would force us to make assumptions on the distribution of x (the data) as well. However, it is worth noting that +with the standard assumption of x ∼ N(⃗0d, Id), the expected squared prediction error is the same as the expected +squared norm of the error in estimating θ∗. +4 + +Theorem 1 (Bias2 and Variance). Suppose Assumption 1 holds. Then, +(i) the squared bias is: +��Eη[ϵS(ξ)] +��2 = +r +� +j=1 +� +⟨θ∗, uj⟩ +�2 +� +λ/σ2 +j +1 + λ/σ2 +j +�2� +1 + +ξ +1 + λ/σ2 +j +�2 ++ +d +� +j=r+1 +� +⟨θ∗, uj⟩ +�2.5 +(11) +(ii) the variance is: +Eη +���ϵS(ξ) − Eη[ϵS(ξ)] +��2� += γ2 +λ +� +r +� +j=1 +λ/σ2 +j +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +��2� +, +(12) +where γ2 is the per-coordinate label noise variance (as per Assumption 1). +The proof of Theorem 1 is in Appendix A. +Remark 1 (Bias-Variance Tradeoff as a Function of ξ). Let us restrict our attention to +ξ ∈ [0, 1] which is the range of ξ used in practice [Lopez-Paz et al., 2015, Li et al., 2017, Sun +et al., 2019]. From eq. (11), note that +��Eη[ϵS(ξ)] +��2 is an increasing function of ξ, i.e. the +bias increases as the student tries to imitate the teacher more. However, from eq. (12), we +see that Eη +���ϵS(ξ) − Eη[ϵS(ξ)] +��2� +is a decreasing function of ξ, i.e., the variance (due to label +noise) reduces as the student tries to imitate the teacher more. Thus, SD is associated with a +bias-variance tradeoff – a higher value of the imitation parameter ξ mitigates the impact of +label noise variance at the cost of increasing the estimation bias (and vice versa). +Plugging in eq. (11) and eq. (12) in eq. (10), we obtain Eη[∥ϵS(ξ)∥2]; note that it is a quadratic +function of ξ. Corollary 1.1 provides the optimal value of ξ, say ξ∗, that minimizes Eη[∥ϵS(ξ)∥2] +(obtained by simple differentiation). +Corollary 1.1. Let cj := λ/σ2 +j and θ∗ +j := +� +⟨θ∗, uj⟩ +�2. Then: +ξ∗ = arg minξ∈REη[∥ϵS(ξ)∥2] = +�r +j=1 +� γ2 +λ − θ∗ +j +� +c2 +j +(1+cj)3 +�r +j=1 +� γ2 +λ cj + θ∗ +j +� +c2 +j +(1+cj)4 +. +(13) +Thus, setting ξ = ξ∗ yields the optimal balance between the squared bias and variance. +Remark 2 (Anti-Learning Observed Labels in Noisy Settings). There are scenarios when +ξ∗ obtained in Corollary 1.1 is more than 16, especially when γ is large, i.e., there is a lot of label +noise. For e.g., note that limγ→∞ ξ∗ = +�r +j=1 c2 +j/(1+cj)3 +�r +j=1 c3 +j/(1+cj)4 > 17. However, the imitation parameter ξ +is restricted to and tuned in [0, 1] [Lopez-Paz et al., 2015,Li et al., 2017,Sun et al., 2019]. Based +on our analysis, we advocate not restricting ξ ∈ [0, 1] and also trying ξ > 1 in the high noise +regime. Setting ξ > 1 can be interpreted as “anti-learning” (or going against) the observed labels. +In Section 5.1, we provide empirical evidence showing that ξ > 1 works better than ξ ≤ 1 even +with the cross-entropy loss for several noisy datasets; see Table 1. +5Note the �d +j=r+1 +� +⟨θ∗, uj⟩ +�2 term. If r < d, then this quantity is equal to the squared norm of the component +of θ∗ along the non-empty null-space of XT ; this component is not recoverable by any algorithm. +6ξ∗ can be negative too, but we shall not focus on this case in this work. +7This is because �r +j=1 +c3 +j +(1+cj)4 = �r +j=1 +cj +(1 + cj) +� +�� +� +<1 +� +c2 +j +(1+cj)3 +� +< �r +j=1 +c2 +j +(1+cj)3 . +5 + +Remark 3 (Utility of Teacher’s Predicted Labels). In Proposition 1 (Appendix B), we +show that ξ∗ is an increasing function of the label noise variance γ2, i.e., we should assign more +weight to the teacher’s predicted labels as γ2 increases. So in linear regression, the benefit of using +the teacher’s predictions (which is the core idea of SD) increases with the degree of label noise. +We make a similar observation in our experiments on multi-class classification in Section 5.2, +where SD with ξ = 1 – which corresponds to only using the teacher’s predictions (and completely +ignoring the original labels) – does not yield any gains (over the teacher) with zero label corruption +but it consistently yields higher gains as the amount of label corruption increases. +Is Optimal Self-Distillation Better than Optimal ℓ2 Regularization? Let e(λ, ξ) := +Eη +� +∥ϵS(ξ)∥2� +(recall ϵS(ξ) is a function of the ℓ2-regularization parameter λ too). Since ξ = 0 +corresponds to using plain ℓ2 regularization, we define ereg(λ) := e(λ, 0) as the estimation error +obtained using only ℓ2 regularization (and no SD) with parameter λ. Next, let us define esd(λ) +as the error obtained using SD with ℓ2-regularization parameter = λ and the optimal value of +ξ = ξ∗ from Corollary 1.1 (which is itself a function of λ), i.e., esd(λ) := e(λ, ξ∗). By definition, +esd(λ) ≤ ereg(λ) ∀ λ; we wish to know when and if minλ esd(λ) < minλ ereg(λ) (note the strict +inequality), i.e., when and if optimal SD is better than optimal ℓ2-regularization by tuning +over λ. +Theorem 2. Let λ∗ +reg := arg minλereg(λ). It holds that esd(λ∗ +reg) = ereg(λ∗ +reg) and desd(λ) +dλ +�� +λ=λ∗reg = +0, i.e., λ∗ +reg is a stationary point of esd(λ) also. It is a local maximum point of esd(λ) when: +r +� +k=1 +k−1 +� +j=1 +σ2 +j σ2 +k(σ2 +j − σ2 +k)(θ∗ +k − θ∗ +j) +(λ∗reg + σ2 +j )4(λ∗reg + σ2 +k)4 < 0, +(14) +with θ∗ +j := (⟨θ∗, uj⟩)2. When the above holds, optimal self-distillation is better than optimal +ℓ2-regularization. +The detailed version and proof of Theorem 2 appear in Appendix C. +One case when eq. (14) holds is θ∗ +1 > . . . > θ∗ +r (since σ1 ≥ . . . ≥ σr). In general, when the +squared projections of θ∗ along the most significant left singular vectors of X (i.e., the ones with +“large” singular values) follow the same ordering as the corresponding singular values and the +noise variance is large enough, λ∗ +reg will be a local maximum point of esd(λ). We formalize this +next. +Theorem 3. Without loss of generality, let ∥θ∗∥ = 1 and σ1 = 1. Further, suppose σj ≤ δ for +j ∈ {q + 1, . . . , r} and θ∗ +1 > . . . > θ∗ +q. Then, λ = λ∗ +reg is a local maximum point of esd(λ) when +δ ≤ O( 1 +r) and γ2 ≥ +maxj∈{1,...,r} θ∗ +j +r−1 +. +The detailed statement and proof of Theorem 3 appear in Appendix D. In practice, X is usually +low rank and only a few of its singular values are large. So, the assumption of Theorem 3 is +realistic and that too with q ≪ r. +To the best of our knowledge, there are no results comparable to Theorems 2 and 3 +quantifying when optimal SD is better than optimal ℓ2 regularization. Now we consider a +synthetic example to verify the previous discussion. Suppose θ∗ = +1 +√ +2 +� +u1 +u2 +� +, n > d = 100 and +σj = 1 +j for j ∈ {1, . . . , d} (so only few singular values are large). Note that eq. (14) is satisfied. +We consider 3 values of γ = {0.125, 0.25, 0.5} & 10 values of λ = {2i−3γ2} with i ∈ {1, . . . , 10}. +In Figure 1, we plot ereg(λ) and esd(λ) for these values of γ and λ; see the figure caption for +discussion. +If esd(λ) does not have a local maximum at λ∗ +reg, it is difficult to say whether λ∗ +reg is a +sub-optimal local minimum point or the global minimum point of esd(λ); also see Appendix C. If +6 + +2 +2 +2 +1 +20 +21 +22 +23 +24 +25 +26 +27 +( / +2) +2 +2 +20 +22 +24 +Estimation Error += 0.125 +ereg( ) +esd( ) +(a) γ = 0.125 +2 +2 +2 +1 +20 +21 +22 +23 +24 +25 +26 +27 +( / +2) +2 +2 +2 +1 +20 +21 +22 +23 +24 +Estimation Error += 0.25 +ereg( ) +esd( ) +(b) γ = 0.25 +2 +2 +2 +1 +20 +21 +22 +23 +24 +25 +26 +27 +( / +2) +2 +1 +20 +21 +22 +23 +Estimation Error += 0.5 +ereg( ) +esd( ) +(c) γ = 0.5 +Figure 1: Estimation errors of vanilla ℓ2 regularization ereg(λ) and SD esd(λ) vs. λ for the +synthetic example at the end of Section 3. As per Theorem 2, note that the global minimum of +ereg(λ) is a local maximum of esd(λ). Observe that minλ esd(λ) < minλ ereg(λ). So, optimal SD +does better than optimal ℓ2-regularization here. +λ∗ +reg is the global minimum point of esd(λ), then optimal SD is not better than (i.e., does not yield +any improvement over) optimal regularization because esd(λ∗ +reg) = ereg(λ∗ +reg). To complement +this, we present the following result (proved in Appendix E). +Theorem 4. There exists θ∗ and X s.t. for any noise variance γ2, λ∗ +reg is the global minimum +point of esd(λ). +So there are cases when optimal SD does not yield any improvement over optimal regularization. +4 +Logistic Regression +We now move onto logistic regression with the cross-entropy loss. Note that linear probing [Alain +and Bengio, 2016,Kumar et al., 2022] is the same as logistic regression with features obtained +from a pre-trained model. It is also worth mentioning here that our analysis for logistic regression +is significantly different from and harder than linear regression. +Setting: We consider a binary classification problem where each sample x ∈ X has a dis- +crete label y(x) ∈ {0, 1}. Let the marginal distribution of the sample space (with support X) be +denoted by P. We assume that there is a feature map φ : X −→ � +X and we have access to a sample +in terms of its features. We are given 2n pairs of data points in terms of features and corrupted +labels {(φ(xi), ˆyi)}2n +i=1, where each ˆyi ∈ {0, 1} and xi ∼ +iid P. Let the corresponding actual +labels be {yi}2n +i=1; we assume that the dataset is balanced, i.e., |i : yi = 1| = |i : yi = 0| = n. +Specifically, without loss of generality (w.l.o.g.), let yi = 1 for i ∈ S1 := {1, . . . , n} and yi = 0 +7 + +for i ∈ S0 := {n + 1, . . . , 2n}; our training algorithms are not privy to this. We consider the +following corruption model: ˆn < n/2 samples of each class, chosen randomly, are provided to us +with flipped labels (again, our training algorithms are not privy to this). Specifically, w.l.o.g., let: +ˆyi = +� +� +� +� +� +� +� +� +� +� +� +1 − yi for i ∈ {1, . . . , ˆn} +� +�� +� +:=S1,bad +∪ {n + 1, . . . , n + ˆn} +� +�� +� +:=S0,bad +, +yi for i ∈ {ˆn + 1, . . . , n} +� +�� +� +:=S1,good +∪ {n + ˆn + 1, . . . , 2n} +� +�� +� +:=S0,good +. +Define p := ˆn +n as the label corruption fraction; note that p < 1 +2. +Our goal is to learn a separator for the data w.r.t. the actual labels by training a logistic +regression model on {(φ(xi), ˆyi)}2n +i=1. Specifically, for a sample x with feature φ(x) ∈ � +X, the +prediction for the label y(x) is modeled as: +P(y(x) = 1) = σ(⟨θ, φ(x)⟩), 8 +(15) +where θ ∈ � +X is the parameter that we wish to learn, and σ(z) = +1 +1+e−z for z ∈ R is the +sigmoid function. We use the binary cross-entropy loss for training; we denote this by BCE : +[0, 1] × (0, 1) −→ R≥0 and it is defined as: +BCE(q, ˆq) = − +� +q log(ˆq) + (1 − q) log(1 − ˆq) +� +. +(16) +Next, we state our assumptions on the feature map φ(.). +Assumption 2 (Orthonormality). The features have unit norm, i.e., ∥φ(x)∥2 = 1 ∀ x ∈ +X. +Further, the space of samples in feature space with labels 0 and 1 are orthogonal, i.e., +⟨φ(x), φ(x′)⟩ = 0 ∀ x ∈ X, x′ ∈ X with different labels. +Assumption 2 ensures that the data is separable and indeed there exists a separator. +Assumption 3 (Feature Correlation in the Training Set). ⟨φ(xi), φ(xi′)⟩ = c ∈ (0, 1) ∀ +i ̸= i′ such that yi = yi′. +It is true that at face value, Assumption 3 seems strong. Instead, an assumption in expectation +like Ex,x′ +� +⟨φ(x), φ(x′)⟩ +���x and x′ have the same label +� += c is more realistic; let us call this As- +sumption 3′ for the sake of discussion. For n → ∞ and when the labels are corrupted randomly, +we hypothesize that the average9 prediction (i.e., soft score ∈ (0, 1) assigned to a particular +class) of a model under Assumption 3′ is the same as that under Assumption 3. We provide +empirical evidence to support this hypothesis in Appendix F. Thus, for large n, we argue that +Assumption 3 is reasonable and an important case to analyze. +Teacher Model: To learn the logistic regression parameter, the teacher minimizes the ℓ2- +regularized binary cross-entropy loss with the provided labels as its targets, i.e., the teacher’s +objective is: +fT(θ) = 1 +2n +2n +� +i=1 +BCE +� +ˆyi, σ +� +⟨θ, φ(xi)⟩ +�� ++ λ∥θ∥2 +2 +. +(17) +In eq. (17), λ > 0 is the ℓ2-regularization parameter. The teacher’s estimated parameter is θ∗ +T := +arg minθfT(θ). The teacher’s predicted soft label for the ith sample is y(T) +i +:= σ(⟨θ∗ +T, φ(xi)⟩); +these are used to train the student. +8The bias term can be absorbed within the feature vector φ(.) itself. +9This is taken over the training set. +8 + +Student Model Trained Only with Teacher’s Soft Labels: Here we set the imitation +parameter ξ = 1 in eq. (1). Thus, the student minimizes the ℓ2-regularized binary cross-entropy +loss with the teacher’s predicted soft labels as its targets, i.e., the student’s objective is: +fS(θ) = 1 +2n +2n +� +i=1 +BCE +� +y(T) +i +, σ +� +⟨θ, φ(xi)⟩ +�� ++ λ∥θ∥2 +2 +. +(18) +In eq. (18), λ is the same ℓ2-regularization parameter that is used by the teacher. The student’s +estimated parameter is θ∗ +S := arg minθfS(θ). +4.1 +Comparison of Student and Teacher +We shall now characterize the conditions under which the student outperforms the teacher w.r.t. +classification accuracy; to our knowledge, this is the first result of its kind. For the sake of +avoiding any ambiguity, the teacher’s population accuracy is defined as 100 ∗ Ex∼P +� +1 +� +y(x) = +1 +� +σ(⟨θ∗ +T, φ(x)⟩) > 1 +2 +��� +%10. The student’s accuracy is defined similarly with θ∗ +S replacing θ∗ +T. +Theorem 5 (When is Student’s Accuracy > Teacher’s Accuracy?). Suppose we have +access to the population, i.e., n → ∞. Further, let Assumptions 2 and 3 hold with c = Θ(1) in +Assumption 3 (recall that c < 1). Define ˆλ := 2nλ and r := (1−c) +4ˆλ . Suppose λ is chosen so that +ˆλ ∈ +� +1−c +2.16, 1−c +0.40 +� +, which corresponds to r ∈ [0.10, 0.54]. If the label corruption fraction +p ∈ +� +max +�1.08 − r +2.08 +, 1 + r +3.7 +� +, 1 − 0.51(1 + r)2 +1 + 2r +� +, +then the student achieves 100% population accuracy (w.r.t. the true labels), while the teacher +only achieves a population accuracy of 100(1-p)% (again, w.r.t. the true labels). +Discussion: In our setup, there exists 0 < plow < phigh < 0.5 such that (i) when p ≤ plow, +the teacher attains 100% accuracy and so there is no need for SD, (ii) when p ∈ (plow, phigh), +the student attains 100% accuracy while the teacher attains 100(1 − p)% accuracy, and (iii) +when p ≥ phigh, both the teacher and student attain 100(1 − p)% accuracy. The range of p in +Theorem 5 ⊆ (plow, phigh); our range is more conservative than the actual range because we had +to impose some more restrictions on p in order to control certain error terms in our analysis. In +Figure 2, we plot the teacher’s and student’s accuracies as a function of p for r = {0.2, 0.3, 0.4} +obtained by exactly solving for θ∗ +T and θ∗ +S (through a computer). In all the cases, it can be seen +that the range of p where the student outperforms the teacher as per Theorem 5 falls within the +actual range of p where the student outperforms the teacher. +The detailed proof of Theorem 5 can be found in Appendix G; we now outline the key steps in +the proof. +Step 1 (Details in Appendix G.1). It can be shown that the teacher’s learned parameter +θ∗ +T = arg minθfT(θ) = �2n +i=1 αiφ(xi) for some real numbers {αi}2n +i=1 which are known as the +teacher’s dual-space coordinates. In Lemma 2, we obtain expressions for {αi}2n +i=1 which then +enables us to obtain the teacher’s predicted soft labels +� +y(T) +i +�2n +i=1. Specifically, we get: +y(T) +i += +� +� +� +� +� +� +� +� +� +� +� +ˆλˆα for i ∈ S1,bad, +1 − ˆλα for i ∈ S1,good, +1 − ˆλˆα for i ∈ S0,bad, +ˆλα for i ∈ S0,good, +(19) +101(.) is the indicator function. Specifically, 1(z) = 1 if z is true and 0 if z is false. +9 + +(a) r = 0.2. Derived bound in Theorem 5: +p ∈ (0.423, 0.475). +(b) r = 0.3. Derived bound in Theorem 5: +p ∈ (0.375, 0.461). +(c) r = 0.4. Derived bound in Theorem 5: +p ∈ (0.378, 0.444). +Figure 2: Comparison of student’s and teacher’s accuracies for different values of label corruption +fraction p obtained by exactly solving eq. (20) and eq. (21) for the teacher and eq. (23) and +eq. (24) for the student. We set c = 0.1 and n = 5000 here. In all the cases, note that our +predicted range of p where the student outperforms the teacher as per Theorem 5 falls within +the actual range of p where the student outperforms the teacher. +where α ≥ 0 and ˆα ≥ 0 are obtained by jointly solving: +σ +� +cn +� +α − (α + ˆα)p) − (1 − c)ˆα +� += ˆλˆα, +(20) +and +σ +� +cn +� +α − (α + ˆα)p) + (1 − c)α +� += 1 − ˆλα. +(21) +We focus on the interesting case of: +(a) p being large enough so that the teacher misclassifies the incorrectly labeled points (S1,bad ∪ +S0,bad) because otherwise, there is no need for SD, and +(b) ˆλ being chosen sensibly so that the teacher at least correctly classifies the correctly labeled +points (S1,good ∪ S0,good) because otherwise, SD is hopeless. +Later in Step 3, we impose conditions on p (a lower bound) and ˆλ such that (a) and (b) +hold by requiring ˆλˆα < 1 +2 and ˆλα < 1 +2. +Step 2 (Details in Appendix G.3). Similar to the teacher in Step 1, in Lemma 3, we +10 + +r = 0.2 +100 +90 +Accuracy +80 +70 +60 +Teacher +Student +50 +0.36 +0.40 +0.43 +0.46 +0.48 0.49 +Label Corruption Fraction pr = 0.3 +100 +90 +Accuracy +80 +70 +60 +Teacher +Student +50 +0.33 +0.36 +0.39 +0.41 +0.43 +0.45 +0.47 +0.49 +Label Corruption Fraction pr = 0.4 +100 +Teacher +Student +90 +Accuracy +80 +70 +60 +50 +0.30 +0.33 +0.36 +0.39 +0.41 +0.43 +0.45 +0.47 +0.49 +Label Corruption Fraction pshow that the student’s predicted soft label for the ith sample, y(S) +i +, turns out to be: +y(S) +i += +� +� +� +� +� +� +� +� +� +� +� +ˆλˆα + ˆλˆβ for i ∈ S1,bad, +1 − ˆλα − ˆλβ for i ∈ S1,good, +1 − ˆλˆα − ˆλˆβ for i ∈ S0,bad, +ˆλα + ˆλβ for i ∈ S0,good, +(22) +where β ≥ 0 and ˆβ ≥ 0 (assuming ˆλˆα < 1 +2 and ˆλα < 1 +2) are obtained by jointly solving: +σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� += ˆλˆα + ˆλˆβ, +(23) +and +σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� += 1 − ˆλα − ˆλβ. +(24) +Now note that if ˆλˆα + ˆλˆβ > 1 +2 and ˆλα + ˆλβ < 1 +2, then the student has managed to correctly +classify all the points in the training set; we ensure this in Step 3 by upper bounding p. The +tradeoff here is that the (1-0) accuracy of the student increases at the cost of decreased confidence +in classifying the correctly labeled points compared to the teacher. +Step 3 (Details in Appendix G.5). Now we come to the challenging part of the proof. +To obtain a range for p, we need to analytically solve eq. (20) and eq. (21) for the teacher +and then eq. (23) and eq. (24) for the student, which is particularly challenging due to the +non-linearity of the sigmoid function present in these equations. Our novel proof technique +involves employing the first-order Maclaurin series expansion of the sigmoid function which +enables us to bound α, ˆα, β and ˆβ as a function of p, ˆλ and c in a small range (while imposing +some conditions on p and ˆλ to ensure the range is small). Using this, we can bound the teacher’s +and student’s predictions, and then impose conditions on p and ˆλ such that the teacher only +correctly classifies the correctly labeled points and errs on all the incorrectly labeled points +(i.e., ˆλα < 1 +2 and ˆλˆα < 1 +2; see Step 1) but the student correctly classifies all the points (i.e., +ˆλα + ˆλβ < 1 +2 and ˆλˆα + ˆλˆβ > 1 +2; see Step 2). Finally, since n → ∞, population accuracy → +training accuracy (we formalize this at the end in Appendix G.5). +4.2 +Variability of Predictions of Student and Teacher +Corollary 5.1 (Variability of predictions of points within the same class). Define +∆T := maxi̸=i′,yi=yi′ |y(T) +i +− y(T) +i′ | as the teacher’s variability, i.e., the maximum difference +between the teacher’s predictions on two points having the same ground truth label. Similarly, +∆S := maxi̸=i′,yi=yi′ |y(S) +i +− y(S) +i′ | is defined as the student’s variability. Under the conditions of +Theorem 5, ∆S < ∆T. +In other words, the student’s predictions are more homogeneous than the teacher’s predictions as +per Corollary 5.1. This is analogous to SD mitigating the variance term due to label noise in +linear regression (Remark 1) leading to smaller variability. +We prove Corollary 5.1 in Appendix H and corroborate it with empirical evidence in Section 5.3. +5 +Empirical Results +For our experiments, we consider multi-class classification with the cross-entropy loss on sev- +eral vision datasets available in PyTorch’s torchvision, namely, CIFAR-100 with 100 classes, +Caltech-256 [Griffin et al., 2007] with 257 classes, Food-101 [Bossard et al., 2014] with 101 classes, +11 + +StanfordCars [Krause et al., 2013] with 196 classes and Flowers-102 [Nilsback and Zisserman, +2008] with 102 classes. Since Caltech-256 does not have any train/test split provided by default, +we pick 25k random images from the full dataset to form the training set, while the remaining +images form the test set. For all the datasets, we train a softmax layer on top of a pre-trained +ResNet-34/VGG-16 model on ImageNet which is kept fixed, i.e., we do linear probing on ResNet- +34/VGG-16. No data augmentation is involved. Next, we describe the different types of label +corruption that we experiment on. +Label Corruption Type 1 (Random Corruption): Suppose the set of labels is [C] := +{1, . . . , C}. Consider a sample whose true label is c ∈ [C]. A corruption level of 100p % means +we observe this sample’s label as c with a probability of (1 − p) or some random i ∈ [C] \ c with +a probability of p/(C − 1) for each such i ̸= c. We call this random corruption11. +Label Corruption Type 2 (Hierarchical Corruption [Hendrycks et al., 2018]): Here, +the label corruption only occurs between semantically similar classes. This is a more realistic +type of corruption compared to random corruption. By default, CIFAR-100 comes with 20 +super-classes each containing 5 semantically similar classes; for e.g., the super-class “fish” consists +of aquarium fish, flatfish, ray, shark and trout, while the super-class “small mammals” consists of +hamster, mouse, rabbit, shrew and squirrel. Unfortunately, the other datasets do not have any +semantically similar classes provided by default. +Now, we describe the exact corruption scheme. Consider a sample whose true class is c and +super-class is S = {c1, . . . , c|S|}. A corruption level of 100p % means we observe this sample’s +label as c with a probability of (1 − p) or some random c′ ∈ S \ c with a probability of p/(|S| − 1) +(for each such c′ ̸= c). Following [Hendrycks et al., 2018], we call this hierarchical corruption. +Label Corruption Type 3 (Adversarial Corruption): Instead of semantically similar +classes, we determine “hard” classes for each class by looking at the output of the teacher in +the noiseless case (i.e., when there is no corruption) and induce label corruption only among +these hard classes. +Specifically, in the noiseless case, for a sample x, let pT(x, c) be the +teacher’s predicted probability of x belonging to class c ∈ {1, . . . , C}. Also, let Xc be the +set of samples in the training set belonging to class c. Now, for each class c, we compute +νc = +� +1 +|Xc| +� +x∈Xc pT(x, 1), . . . , +1 +|Xc| +� +x∈Xc pT(x, C) +� +∈ RC, and define the k hardest classes for +class c to be the indices in {1, . . . , C} \ c corresponding to the k largest values in νc. For our +experiments, we take k = 5. +Now, we describe the corruption scheme. Consider a sample whose true class is c and the set +of hardest 5 classes for c is S. A corruption level of 100p % means we observe this sample’s label +as c with a probability of (1 − p) or some random c′ ∈ S with a probability of p/5. We call this +adversarial corruption. +5.1 +Verifying Remark 2 +In Remark 2, we advocated trying ξ > 1 in the high noise regime. We shall now test our +recommendation on several noisy datasets. The teacher is trained with the ℓ2-regularized cross- +entropy loss and the student’s per-sample loss is given by eq. (1) where ℓ is the ℓ2-regularized +cross-entropy loss. Following our theory setting, the teacher and student are both trained with the +same ℓ2-regularization parameter; the common weight decay value (PyTorch’s ℓ2-regularization +parameter) is set to 5 × 10−4. Note that this weight decay value was the first one that we +tried (i.e., it was not cherry-picked); in fact, we show results with other weight decay values in +Appendix I.2. We defer the remaining experimental details to Appendix J. In Table 1, we list the +11This has been also called symmetric noise in prior work; see for e.g., [Chen et al., 2019] +12 + +student’s improvement over the teacher (i.e., student’s test accuracy - teacher’s test accuracy)12 +averaged across 3 different runs for different values of ξ with ResNet-34 and VGG-16 in the case +of 50% random, hierarchical and adversarial corruption. In all these experiments, note that the +value of ξ yielding the biggest improvement is > 1. Table 5 (in Appendix I.1) shows results with +30% corruption in Stanford Cars and Flowers-102; even there, ξ > 1 does better than ξ ≤ 1. +5.2 +Verifying Remark 3 +In Remark 3, we claimed that the utility of the teacher’s predictions increases with the amount +of label noise. To demonstrate this, we train the student with ξ = 1 which corresponds to setting +the teacher’s predicted soft labels as the student’s targets (just as we did in Section 4) and +completely ignoring the provided labels. All other experimental details (including weight decay) +are the same as in Section 5.1. In Table 2, we show the student’s improvement over the teacher +averaged across 3 different runs for varying degrees and types of label corruption with ResNet-34; +see the table caption for discussion. +5.3 +Verifying Corollary 5.1 +We now provide empirical evidence for our claim of the student’s predictions being more +homogeneous than the teacher’s predictions in Corollary 5.1. Since our experiments are for the +multi-class (and not binary) case, we look at a slightly different metric to quantify variability +which we introduce next. For a sample x belonging to class c(x), let ˆpT(x) and ˆpS(x) be the +teacher’s and student’s predicted probability of x belonging to c(x), respectively. Also, let X ′ +c be +the set of samples in the test set belonging to class c. To quantify the variability of the teacher and +student for class c, we look at maxx1,x2∈X ′c |ˆpT(x1) − ˆpT(x2)| and maxx1,x2∈X ′c |ˆpS(x1) − ˆpS(x2)|, +i.e., the range of ˆpT(x) and ˆpS(x) w.r.t. x ∈ X ′ +c, respectively. In Figure 3, we plot the per-class +variability as defined here for three of the cases of Table 2 covering all three types of label +corruption; please see the caption for discussion. +12The individual accuracies of the teacher and student can be found in Appendix J; we omit them in the main +text for brevity. +13 + +ξ +Improvement of +student over teacher +0.2 +2.22 ± 0.12 % +0.5 +5.18 ± 0.03 % +0.7 +6.84 ± 0.06 % +1.0 +8.54 ± 0.29 % +1.2 +9.66 ± 0.23 % +1.5 +10.04 ± 0.51 % +1.7 +9.81 ± 0.55 % +2.0 +8.56 ± 0.73 % +(a) 50% Random Corruption in Caltech-256 +with ResNet-34 +ξ +Improvement of +student over teacher +0.5 +0.89 ± 0.10 % +1.0 +2.01 ± 0.14 % +1.5 +3.13 ± 0.11 % +2.0 +4.22 ± 0.20 % +2.5 +5.28 ± 0.13 % +3.0 +5.78 ± 0.12 % +3.5 +5.86 ± 0.18 % +4.0 +5.32 ± 0.33 % +(b) 50% Random Corruption in Caltech-256 +with VGG-16 +ξ +Improvement of +student over teacher +0.2 +0.98 ± 0.12 % +0.5 +2.46 ± 0.11 % +0.7 +3.38 ± 0.02 % +1.0 +4.19 ± 0.09 % +1.2 +4.46 ± 0.19 % +1.5 +4.46 ± 0.17 % +1.7 +4.32 ± 0.18 % +2.0 +3.52 ± 0.23 % +(c) 50% Hierarchical Corruption in +CIFAR-100 with ResNet-34 +ξ +Improvement of +student over teacher +0.2 +1.10 ± 0.09 % +0.5 +2.69 ± 0.02 % +0.7 +3.72 ± 0.05 % +1.0 +5.29 ± 0.11 % +1.2 +6.26 ± 0.09 % +1.5 +7.20 ± 0.14 % +1.7 +7.23 ± 0.17 % +2.0 +6.42 ± 0.26 % +(d) 50% Hierarchical Corruption in +CIFAR-100 with VGG-16 +ξ +Improvement of +student over teacher +0.2 +0.13 ± 0.08 % +0.5 +0.97 ± 0.04 % +0.7 +1.45 ± 0.01 % +1.0 +1.85 ± 0.09 % +1.2 +1.87 ± 0.06 % +1.5 +1.86 ± 0.08 % +1.7 +1.80 ± 0.05 % +2.0 +1.53 ± 0.02 % +(e) 50% Adversarial Corruption in Food-101 +with ResNet-34 +ξ +Improvement of +student over teacher +0.2 +0.79 ± 0.23 % +0.5 +2.14 ± 0.09 % +0.7 +2.96 ± 0.04 % +1.0 +3.85 ± 0.05 % +1.2 +4.22 ± 0.15 % +1.5 +4.39 ± 0.29 % +1.7 +4.20 ± 0.34 % +2.0 +3.53 ± 0.49 % +(f) 50% Adversarial Corruption in Food-101 +with VGG-16 +Table 1: Average (± 1 std.) improvement of student over teacher (i.e., student’s test set accuracy +- teacher’s test set accuracy) with different values of the imitation parameter ξ; recall that +ξ = 0 corresponds to the teacher. Observe that in all cases, the value of ξ yielding the biggest +improvement is more than 1 (although in Food-101 with ResNet-34, ξ = 1 does just as well as +ξ > 1). This is consistent with our message in Remark 2, where we advocate trying ξ > 1 in the +high noise regime. +14 + +Corruption level +Random corruption: +Improvement of student +Adversarial corruption: +Improvement of student +0% +−0.04 ± 0.02 % +−0.04 ± 0.02 % +10% +2.51 ± 0.11 % +2.32 ± 0.10 % +30% +6.14 ± 0.16 % +5.08 ± 0.25 % +50% +8.54 ± 0.29 % +5.77 ± 0.19 % +(a) Caltech-256 (Random and Adversarial Corruption) +Corruption level +Random corruption: +Improvement of student +Hierarchical corruption: +Improvement of student +0% +−0.23 ± 0.06 % +−0.23 ± 0.06 % +10% +0.63 ± 0.11 % +1.19 ± 0.08 % +30% +1.34 ± 0.13 % +2.80 ± 0.06 % +50% +2.11 ± 0.15 % +4.19 ± 0.09 % +(b) CIFAR-100 (Random and Hierarchical Corruption) +Corruption level +Random corruption: +Improvement of student +Adversarial corruption: +Improvement of student +0% +−0.37 ± 0.10 % +−0.37 ± 0.10 % +10% +0.10 ± 0.04 % +0.25 ± 0.05 % +30% +0.47 ± 0.04 % +0.77 ± 0.06 % +50% +1.12 ± 0.08 % +1.85 ± 0.09 % +(c) Food-101 (Random and Adversarial Corruption) +Table 2: ResNet-34 with ξ = 1: Average (± 1 std.) improvement of student over teacher (i.e., +student’s test set accuracy - teacher’s test set accuracy) with different kinds and varying levels of +label corruption. Observe that as the corruption level increases, so does the improvement of the +student over the teacher for all types of corruption. This shows that the utility of the teacher’s +predictions (which is the core idea of SD) increases with the amount of label noise corroborating +our claim in Remark 3. +0 +8 +16 +24 +32 +40 +48 +56 +64 +72 +80 +88 +96 +104 +112 +120 +128 +136 +144 +152 +160 +168 +176 +184 +192 +200 +208 +216 +224 +232 +240 +248 +Class Number +Teacher +Student +Caltech-256 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +(a) Caltech-256 with 50% +random corruption +0 +3 +6 +9 +12 +15 +18 +21 +24 +27 +30 +33 +36 +39 +42 +45 +48 +51 +54 +57 +60 +63 +66 +69 +72 +75 +78 +81 +84 +87 +90 +93 +96 +99 +Class Number +Teacher +Student +CIFAR-100 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +(b) CIFAR-100 with 50% +hierarchical corruption +0 +3 +6 +9 +12 +15 +18 +21 +24 +27 +30 +33 +36 +39 +42 +45 +48 +51 +54 +57 +60 +63 +66 +69 +72 +75 +78 +81 +84 +87 +90 +93 +96 +99 +Class Number +Teacher +Student +Food-101 +0.5 +0.6 +0.7 +0.8 +0.9 +(c) Food-101 with 50% +adversarial corruption +Figure 3: ResNet-34 with ξ = 1: Comparison of the per-class variability of the teacher and +student (i.e., range of the teacher’s and student’s predictions of belonging to the correct class, as +defined in Section 5.3) for three of the cases of Table 2 as a heat map. Note that a darker shade +corresponds to a lower value; in all the cases, the student’s heat map has a darker shade than +the teacher’s heat map which means that the student has a smaller variability than the teacher. +This is consistent with the claim in Corollary 5.1. +15 + +6 +Conclusion +In this work, we analyzed the utility of self-distillation (SD) in supervised learning with noisy +labels. Our main algorithmic contribution was introducing the idea of trying ξ > 1 in the high +label noise regime. On the theoretical side, for a binary classification problem where some fraction +of the sample’s labels are flipped, we quantified the range of label corruption fraction in which +the student outperforms the teacher under some assumptions on the data. We also characterized +when optimal SD is better than optimal regularization in linear regression. +There are some limitations of our work which pave the way for interesting directions of future +work. Our results in Section 4 for logistic regression are under Assumption 3; it would be nice to +derive similar results under a weaker assumption such as in expectation (see Assumption 3′ in +the discussion after Assumption 3) or by assuming that the feature inner products are bounded +in some range. Also, our results for logistic regression are with ξ = 1; one could try to obtain +results with a general ξ to shed some light on how to better tune ξ for noisy datasets, like we did +for linear regression. Further, our empirical results are with linear probing; experiments with full +network fine-tuning are left for future work. +7 +Acknowledgement +This work was supported by NSF TRIPODS grant 1934932. +References +[Ahn et al., 2019] Ahn, S., Hu, S. X., Damianou, A., Lawrence, N. D., and Dai, Z. (2019). +Variational information distillation for knowledge transfer. 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In Proceedings of the IEEE/CVF conference on +computer vision and pattern recognition, pages 10687–10698. +18 + +Appendix +Contents +• Appendix A: Proof of Theorem 1 +• Appendix B: Behavior of ξ∗ w.r.t. γ2 +• Appendix C: Detailed Version and Proof of Theorem 2 +• Appendix D: Detailed Version and Proof of Theorem 3 +• Appendix E: Proof of Theorem 4 +• Appendix F: Empirical Motivation for Assumption 3 +• Appendix G: Proof of Theorem 5 +• Appendix H: Proof of Corollary 5.1 +• Appendix I: More Empirical Results +• Appendix J: Detailed Empirical Results +19 + +A +Proof of Theorem 1 +With the SVD notation of X, we can rewrite ˆθS(ξ) (from eq. (9)) as: +ˆθS(ξ) = +r +� +j=1 +⟨θ∗, uj⟩ +� +1 + λ/σ2 +j +� +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +uj + +r +� +j=1 +⟨η, vj⟩/σj +� +1 + λ/σ2 +j +� +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +uj. +(25) +Also, since {u1, . . . , ud} forms an orthonormal basis for Rd, we have: +θ∗ = +d +� +j=1 +⟨θ∗, uj⟩uj. +So, using eq. (25): +ϵS(ξ) = − +r +� +j=1 +⟨θ∗, uj⟩ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +1 + +ξ +1 + λ/σ2 +j +� +uj − +d +� +j=r+1 +⟨θ∗, uj⟩uj ++ +r +� +j=1 +⟨η, vj⟩/σj +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +uj. +(26) +Using Assumption 1, we have: +Eη[ϵS(ξ)] = − +r +� +j=1 +⟨θ∗, uj⟩ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +1 + +ξ +1 + λ/σ2 +j +� +uj − +d +� +j=r+1 +⟨θ∗, uj⟩uj. +(27) +Thus, using the orthonormality of {u1, . . . , ud}, we get: +��Eη[ϵS(ξ)] +��2 = +r +� +j=1 +� +⟨θ∗, uj⟩ +�2 +� +λ/σ2 +j +1 + λ/σ2 +j +�2� +1 + +ξ +1 + λ/σ2 +j +�2 ++ +d +� +j=r+1 +� +⟨θ∗, uj⟩ +�2. +(28) +Next: +Eη +���ϵS(ξ) − Eη[ϵS(ξ)] +��2� += Eη +������ +r +� +j=1 +⟨η, vj⟩/σj +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +�� +uj +����� +2� +(29) += +r +� +j=1 +Eη +�� +⟨η, vj⟩ +�2� +σ2 +j +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +��2 +(30) += +r +� +j=1 +vT +j Eη +� +ηηT � +vj +σ2 +j +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +��2 +(31) += γ2 +� +r +� +j=1 +1 +σ2 +j +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +��2� +. +(32) +Equation (30) follows from the orthonormality of the uj’s, eq. (31) follows because the vj’s +are independent of η from Assumption 1, and eq. (32) follows because Eη +� +ηηT � += γ2In from +Assumption 1 and because vT +j vj = 1 for all j ∈ {1, . . . , r}. Rewriting eq. (32) slightly differently, +we get: +Eη +���ϵS(ξ) − Eη[ϵS(ξ)] +��2� += γ2 +λ +� +r +� +j=1 +λ/σ2 +j +� +1 + λ/σ2 +j +�2 +� +1 − ξ +� +λ/σ2 +j +1 + λ/σ2 +j +��2� +. +(33) +20 + +B +Behavior of ξ∗ w.r.t. γ2 +Proposition 1. ξ∗ (in Corollary 1.1) is an increasing function of γ2. +Proof. Let ρ = γ2. Then from Corollary 1.1: +ξ∗ = +�r +j=1 +� ρ +λ − θ∗ +j +� +c2 +j +(1+cj)3 +�r +j=1 +� ρ +λcj + θ∗ +j +� +c2 +j +(1+cj)4 +. +(34) +Now, +∂ξ∗ +∂ρ = +� �r +j=1 +c2 +j +(1+cj)3 +�� �r +j=1 +θ∗ +j c2 +j +(1+cj)4 +� ++ +� �r +j=1 +c3 +j +(1+cj)4 +�� �r +j=1 +θ∗ +j c2 +j +(1+cj)3 +� +λ +� �r +j=1 +� ρ +λcj + θ∗ +j +� +c2 +j +(1+cj)4 +�2 +> 0. +(35) +Thus, ξ∗ is an increasing function of ρ, i.e., γ2. +■ +C +Detailed Version and Proof of Theorem 2 +Theorem 6 (Detailed Version of Theorem 2). The following hold with θ∗ +j := +� +⟨θ∗, uj⟩ +�2 +(and with ′ denoting the derivative w.r.t. λ): +esd(λ) = ereg(λ) − +� +e′ +reg(λ) +�2 +h(λ) +and e′ +sd(λ) = e′ +reg(λ) +� +1 − 2e′′ +reg(λ) +h(λ) ++ e′ +reg(λ)h′(λ) +(h(λ))2 +� +, +where ereg(λ) = +r +� +j=1 +λ2θ∗ +j +(λ + σ2 +j )2 + +d +� +j=r+1 +θ∗ +j+ +r +� +j=1 +γ2σ2 +j +(λ + σ2 +j )2 and h(λ) = 4 +r +� +j=1 +�γ2 +σ2 +j ++ θ∗ +j +� +σ4 +j +(λ + σ2 +j )4 . +(36) +Let λ∗ +reg := arg minλereg(λ). Then, esd(λ∗ +reg) = ereg(λ∗ +reg) and e′ +sd(λ∗ +reg) = 0, i.e., λ = λ∗ +reg is a +stationary point of esd(λ) also. It is a local maximum point of esd(λ) when: +r +� +k=1 +k−1 +� +j=1 +σ2 +j σ2 +k(σ2 +j − σ2 +k)(θ∗ +k − θ∗ +j) +(λ∗reg + σ2 +j )4(λ∗reg + σ2 +k)4 < 0. +(37) +When the above holds13, optimal self-distillation is better than optimal ℓ2-regularization. +Note that if λ = λ∗ +reg is not a local maximum point of esd(λ), it could be a sub-optimal local +minimum point or the global minimum point of esd(λ). The other stationary points of esd(λ) are +obtained by solving (this follows from eq. (36)): +1 − 2e′′ +reg(λ) +h(λ) ++ e′ +reg(λ)h′(λ) +(h(λ))2 += 0. +(38) +Unfortunately, it seems difficult to determine whether a root of eq. (38) or λ∗ +reg will be the +global minimum point of esd(λ). If λ∗ +reg is the global minimum point of esd(λ), then optimal +SD is not better than (i.e., does not yield any improvement over) optimal ℓ2-regularization as +esd(λ∗ +reg) = ereg(λ∗ +reg). +13Also, assume that λ∗ +reg ≥ 0 as the ℓ2-regularization parameter is supposed to be non-negative. +21 + +Proof. Using eq. (11) and eq. (12) in eq. (10) while using our notation of cj = λ/σ2 +j and +θ∗ +j = +� +⟨θ∗, uj⟩ +�2 from Corollary 1.1, we get: +e(λ, ξ) = +r +� +j=1 +θ∗ +j +� +cj +1 + cj +�2� +1 + +ξ +1 + cj +�2 ++ +d +� +j=r+1 +θ∗ +j + γ2 +λ +� +r +� +j=1 +cj +(1 + cj)2 +� +1 − ξ +� +cj +1 + cj +��2� +. +(39) +Thus, +ereg(λ) := e(λ, 0) = +r +� +j=1 +θ∗ +j +� +cj +1 + cj +�2 ++ +d +� +j=r+1 +θ∗ +j + γ2 +λ +r +� +j=1 +cj +(1 + cj)2 . +(40) +Next, we compute esd(λ) := e(λ, ξ∗). +Lemma 1. +esd(λ) = ereg(λ) − +� �r +j=1 +� +θ∗ +j − γ2 +λ +� +c2 +j +(1+cj)3 +�2 +�r +j=1 +� γ2 +λ cj + θ∗ +j +� +c2 +j +(1+cj)4 +. +(41) +Lemma 1 involves a little bit of algebra; we prove it in Appendix C.1. +Since the cj’s depend on λ, let us substitute cj in eq. (40) and eq. (41) and rewrite them. +ereg(λ) = +r +� +j=1 +λ2θ∗ +j +(λ + σ2 +j )2 + +d +� +j=r+1 +θ∗ +j + +r +� +j=1 +γ2σ2 +j +(λ + σ2 +j )2 . +(42) +esd(λ) = ereg(λ) − +�� +r +� +j=1 +� +λθ∗ +j − γ2� +σ2 +j +(λ + σ2 +j )3 +� +�� +� +:=g(λ) +�2��� r +� +j=1 +�γ2 +σ2 +j ++ θ∗ +j +� +σ4 +j +(λ + σ2 +j )4 +� +. +(43) +Interestingly, it can be checked that g(λ) = 1 +2e′ +reg(λ); here ′ indicates the derivative w.r.t. λ. +Plugging this in eq. (43), we get: +esd(λ) = ereg(λ) − +� +e′ +reg(λ) +�2 +h(λ) +, where h(λ) = 4 +r +� +j=1 +�γ2 +σ2 +j ++ θ∗ +j +� +σ4 +j +(λ + σ2 +j )4 . +(44) +Now note that: +e′ +sd(λ) = e′ +reg(λ) +� +1 − 2e′′ +reg(λ) +h(λ) ++ e′ +reg(λ)h′(λ) +(h(λ))2 +� +. +(45) +Thus, e′ +reg(λ) = 0 =⇒ e′ +sd(λ) = 0, i.e., any stationary point of ereg(λ) is also a stationary point +of esd(λ). +Next, λ∗ +reg := arg minλereg(λ) satisfies: +e′ +reg(λ∗ +reg) = 2 +r +� +j=1 +� +λ∗ +regθ∗ +j − γ2� +σ2 +j +(λ∗reg + σ2 +j )3 = 0. +(46) +From eq. (45), e′ +sd(λ∗ +reg) = 0, i.e., λ = λ∗ +reg is a stationary point of esd(λ) also. We shall now show +that λ = λ∗ +reg can be a local maximum point of esd(λ) in many cases. For that, we need to check +the sign of e′′ +sd(λ∗ +reg). Note that: +e′′ +sd(λ∗ +reg) = e′′ +reg(λ∗ +reg) +� +1 − 2e′′ +reg(λ∗ +reg) +h(λ∗reg) +� +. +(47) +22 + +The above follows by just differentiating eq. (45) and evaluating it at λ = λ∗ +reg while using the +fact that e′ +reg(λ∗ +reg) = 0. Also note that e′′ +reg(λ∗ +reg) > 0 as λ = λ∗ +reg is a minimizer of ereg(λ). Let +us now examine the sign of t = +� +1 − +2e′′ +reg(λ∗ +reg) +h(λ∗reg) +� +. After a bit of algebra: +t = 1 − +�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )4 +� +θ∗ +jσ2 +j + 3γ2 − 2λ∗ +regθ∗ +j +� +�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )4 +� +γ2 + θ∗ +jσ2 +j +� += +2�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )4 +� +λ∗ +regθ∗ +j − γ2� +�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )4 +� +γ2 + θ∗ +jσ2 +j +� +(48) +The denominator of t is positive so we only need to analyze the sign of the numerator, +�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )4 +� +λ∗ +regθ∗ +j − γ2� +; let us refer to it as t2 for brevity. From eq. (46), we have that: +λ∗ +reg = +γ2 �r +j=1 +σ2 +j +(λ∗reg+σ2 +j )3 +�r +j=1 +θ∗ +j σ2 +j +(λ∗reg+σ2 +j )3 +. +(49) +Using this, we get: +t2 = +� +γ2 +�r +j=1 +θ∗ +j σ2 +j +(λ∗reg+σ2 +j )3 +� +� +�� +� +>0 +� � +j,k +σ2 +j σ2 +k(θ∗ +j − θ∗ +k) +(λ∗reg + σ2 +j )4(λ∗reg + σ2 +k)3 +� +� +�� +� +:=t3 +(50) +Simplifying t3 a bit, we get: +t3 = +r +� +k=1 +k−1 +� +j=1 +σ2 +j σ2 +k(σ2 +j − σ2 +k)(θ∗ +k − θ∗ +j) +(λ∗reg + σ2 +j )4(λ∗reg + σ2 +k)4 . +(51) +So, t3 < 0 =⇒ t2 < 0 =⇒ t < 0 =⇒ e′′ +sd(λ∗ +reg) < 0; but this means λ = λ∗ +reg is a local +maximum point of esd(λ). +■ +C.1 +Proof of Lemma 1 +Proof. Note that e(λ, ξ) is a quadratic function of ξ; specifically, it is of the form aξ2 + bξ + c, +where: +a = +r +� +j=1 +�γ2 +λ cj + θ∗ +j +� +c2 +j +(1 + cj)4 , b = 2 +r +� +j=1 +� +θ∗ +j − γ2 +λ +� +c2 +j +(1 + cj)3 , and c = ereg(λ). +(52) +By simple differentiation, ξ∗ = arg minξ∈Re(λ, ξ) = − b +2a (which is what we obtained in Corol- +lary 1.1). A little bit of algebra gives us: +e(λ, ξ∗) = c − b2 +4a. +(53) +Plugging in the values of a, b and c from eq. (52) in yields: +esd(λ) := e(λ, ξ∗) = ereg(λ) − +� �r +j=1 +� +θ∗ +j − γ2 +λ +� +c2 +j +(1+cj)3 +�2 +�r +j=1 +� γ2 +λ cj + θ∗ +j +� +c2 +j +(1+cj)4 +. +(54) +This finishes the proof. +■ +23 + +D +Detailed Version and Proof of Theorem 3 +Theorem 7 (Detailed Version of Theorem 3). Without loss of generality, let ∥θ∗∥ = 1 and +σ1 = 1. Further, suppose σj ≤ δ for j ∈ {q + 1, . . . , r} and θ∗ +1 > . . . > θ∗ +q. Also, suppose λ∗ +reg > 0. +For any ν > 1, if δ ≤ +1 +√ +2νr +� +mink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +and γ2 ≥ +maxj∈{1,...,r} θ∗ +j +ν−1 +, then +λ = λ∗ +reg is a local maximum point of esd(λ). +Theorem 3 is obtained by using ν = r in Theorem 7. +Proof. Define vk := �k−1 +j=1 +σ2 +j σ2 +k(σ2 +j −σ2 +k)(θ∗ +k−θ∗ +j ) +(λ∗reg+σ2 +j )4(λ∗reg+σ2 +k)4 . For λ = λ∗ +reg to be a local maximum point of +esd(λ), we must have �r +k=1 vk < 0 as per Theorem 2. +Let us analyze vk for k > q first. Using σk ≤ δ for k > q, (σ2 +j − σ2 +k) ≤ σ2 +j ≤ σ2 +1 = 1 for j < k +and |θ∗ +k − θ∗ +j| ≤ ∥θ∗∥2 = 1, we get for k > q: +|vk| ≤ δ2 +k−1 +� +j=1 +σ2 +j +(λ∗reg + σ2 +j )4(λ∗reg + σ2 +k)4 . +(55) +Now since λ∗ +reg > 0, we can further simplify eq. (55): +|vk| ≤ δ2 +k−1 +� +j=1 +σ2 +j +(λ∗reg)8 = +δ2 +(λ∗reg)8 +� +q +� +j=1 +σ2 +j +���� +≤1 ++ +r +� +j=q+1 +σ2 +j +���� +≤δ2 +� +≤ δ2(q + rδ2) +(λ∗reg)8 +. +(56) +Summing up eq. (56) from k = q + 1 through to k = r, we get: +r +� +k=q+1 +vk ≤ +r +� +k=q+1 +|vk| ≤ rδ2(q + rδ2) +(λ∗reg)8 +. +(57) +Let us now look at k ≤ q. Since θ∗ +1 > . . . > θ∗ +q, we have that vk < 0 for all k ≤ q. Note that for +each k ≤ q: +vk ≤ +σ2 +k(1 − σ2 +k)(θ∗ +k − θ∗ +1) +(λ∗reg + 1)4(λ∗reg + σ2 +k)4 ≤ σ2 +k(1 − σ2 +k)(θ∗ +k − θ∗ +1) +(λ∗reg + 1)8 +, +(58) +where the last step follows using λ∗ +reg > 0. Thus, +q +� +k=1 +vk ≤ +1 +(λ∗reg + 1)8 +q +� +k=1 +σ2 +k(1 − σ2 +k)(θ∗ +k − θ∗ +1) ≤ −q mink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +(λ∗reg + 1)8 +. +(59) +Using eq. (57) and eq. (59), we get: +r +� +k=1 +vk = +q +� +k=1 +vk + +r +� +k=q+1 +vk ≤ −q mink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +(λ∗reg + 1)8 ++ rδ2(q + rδ2) +(λ∗reg)8 +. +(60) +So to ensure �r +k=1 vk < 0, ensuring: +rδ2(q + rδ2) +(λ∗reg)8 +< q mink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +(λ∗reg + 1)8 +(61) +suffices. This implies: +λ∗ +reg + 1 +λ∗reg +< +� +q mink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +rδ2(q + rδ2) +�1/8 +� +�� +� +:=z +. +(62) +24 + +For any ν > 1, note that z > ν for δ2 < +1 +2νrmink∈{1,...,q} +� +σ2 +k(1 − σ2 +k)(θ∗ +1 − θ∗ +k) +� +. In that case, we +must have λ∗ +reg > +1 +z−1, which can be ensured by having: +λ∗ +reg > +1 +ν − 1. +(63) +From eq. (46), recall that λ∗ +reg = +γ2 �r +j=1 +σ2 +j +(λ∗reg+σ2 +j )3 +�r +j=1 +θ∗ +j σ2 +j +(λ∗reg+σ2 +j )3 +. Now since λ∗ +reg > 0, we have that: +λ∗ +reg ≥ +γ2 �r +j=1 +σ2 +j +(λ∗reg+σ2 +j )3 +θ∗max +�r +j=1 +σ2 +j +(λ∗reg+σ2 +j )3 +≥ +γ2 +θ∗max +, +(64) +where θ∗ +max = maxj∈{1,...,r} θ∗ +j. Using this, if γ2 > θ∗ +max +ν−1 , then λ∗ +reg ≥ +γ2 +θ∗max > +1 +ν−1 > +1 +z−1. This +completes the proof. +■ +E +Proof of Theorem 4 +Proof. We provide a 2-dimensional example, i.e., d = 2. Suppose n > 2. Take θ∗ = +1 +√ +2(u1 + u2); +so, θ∗ +1 = θ∗ +2 = 1 +2. Also, suppose σ1 = 1 and σ2 = 1 +2. For this case, we get (by using the formulas +in Theorem 6): +ereg(λ) = λ2 +2 +� +1 +(λ + 1)2 + +16 +(4λ + 1)2 +� ++ γ2 +� +1 +(λ + 1)2 + +4 +(4λ + 1)2 +� +, +(65) +and +e′ +reg(λ) = (λ − 2γ2) +� +1 +(λ + 1)3 + +16 +(4λ + 1)3 +� +. +(66) +From eq. (66), we have that λ∗ +reg = arg minλ>0ereg(λ) = 2γ2. +From Theorem 6, we have that: +e′ +sd(λ) = e′ +reg(λ) +� +1 − 2e′′ +reg(λ) +h(λ) ++ e′ +reg(λ)h′(λ) +(h(λ))2 +� +, where h(λ) = +� +4γ2 + 2 +(λ + 1)4 + 256γ2 + 32 +(4λ + 1)4 +� +. (67) +After a lot of algebraic heavy lifting, we get: +e′ +sd(λ) = +288(λ − 2γ2)3 +(λ + 1)5(4λ + 1)5 +� +2γ2+1 +(λ+1)4 + 128γ2+16 +(4λ+1)4 +�2 +� +1 +(λ + 1)3 + +16 +(4λ + 1)3 +� +. +(68) +Using eq. (68), we can conclude that arg minλ>0esd(λ) = 2γ2 = λ∗ +reg. +■ +25 + +F +Empirical Motivation for Assumption 3 +We consider the same logistic regression setting as Section 4. Note that the Gram matrix +K ∈ R2n×2n (w.r.t. φ(.)) is of the form K = +� +K1 +0n×n +0n×n +K0 +� +, where 0n×n is the n × n matrix +of all 0’s and K1 and K0 are both PSD matrices with diagonal entries = 1. For our simulations, +the diagonal elements of K1 are set equal to 1 and the off-diagonal elements are set equal to the +corresponding off-diagonal element of 1 +nZ1ZT +1 , where each element of Z1 ∈ Rn×n is drawn i.i.d. +from (i) Unif[0, 1], and (ii) Bernoulli(0.8)14. K0 is constructed in the same way. Note that K +is PSD. In the case of (i) (resp., (ii)), the expected off-diagonal element of both K1 and K0 +is 0.25 (resp., 0.64), and so we compare against Assumption 3 with c = 0.25 (resp., c = 0.64). +Specifically, for our two Gram matrices, we compare the average predictions (average being over +the training set) of our logistic regression model against the corresponding predictions under +Assumption 3. We consider four values of n, namely, 1000, 5000, 10000 and 50000. +In Table 3, we show results for (i) when p = 0.45 (top) and p = 0.35 (bottom) with ˆλ = 1 − c +(recall that ˆλ ∈ +� 1−c +2.16, 1−c +0.40 +� +as per Theorem 5). In Table 4, we show results for (ii) when p = 0.3 +(top) and p = 0.2 (bottom) with ˆλ = 1−c +0.50 = 2(1 − c). Please see the table captions for a detailed +discussion, but in summary, we conclude that Assumption 3 is a reasonable assumption to analyze +the average behavior of a linear model on a large dataset under random label corruption. +14If X ∼ Bernoulli(p), then P(X = 1) = p and P(X = 0) = 1 − p. +26 + +Teacher +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.4413 +0.4400 +0.6372 +0.6400 +5k +0.4399 +0.6397 +10k +0.4399 +0.6399 +50k +0.4400 +0.6400 +Student +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.5287 +0.5280 +0.5645 +0.5680 +5k +0.5279 +0.5676 +10k +0.5279 +0.5679 +50k +0.5280 +0.5680 +(a) p = 0.45 +Teacher +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.5243 +0.5200 +0.7146 +0.7200 +5k +0.5198 +0.7195 +10k +0.5198 +0.7198 +50k +0.5200 +0.7200 +Student +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.6264 +0.6240 +0.6568 +0.6640 +5k +0.6235 +0.6631 +10k +0.6236 +0.6636 +50k +0.6240 +0.6640 +(b) p = 0.35 +Table 3: (i) Unif[0, 1]: Results (up to fourth decimal point) for p = 0.45 (top) and p = 0.35 +(bottom) with ˆλ = 1 − c on points with true label = 1; points with true label = 0 follow the same +trend by symmetry of the problem. In the table, “bad” (resp., “good”) points mean incorrectly +(resp., correctly) labeled points, and A3 is Assumption 3. Also, “pred.” is the predicted probability +of the label being 1 and “Avg. pred. for bad points” (resp., “Avg. pred. for good points”) is +the empirical average over all bad (resp., good) points with true label = 1; please note that +this is with the actual Gram matrix. Under Assumption 3, all bad/good points have the same +prediction (see Equations (19) and (22) or Lemmas 2 and 3) due to which the corresponding +columns do not have the word “Avg.”. Observe that as n increases, the average prediction for +both good and bad points (with the actual Gram matrix) matches the corresponding predictions +under Assumption 3 (and n → ∞). Thus, Assumption 3 is a reasonable assumption to analyze +the average behavior of a linear model on a large dataset under random label corruption. +27 + +Teacher +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.6213 +0.6222 +0.7324 +0.7333 +5k +0.6220 +0.7332 +10k +0.6221 +0.7332 +50k +0.6222 +0.7333 +Student +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.6895 +0.6913 +0.7018 +0.7037 +5k +0.6910 +0.7033 +10k +0.6911 +0.7035 +50k +0.6913 +0.7037 +(a) p = 0.3 +Teacher +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.7097 +0.7111 +0.8208 +0.8222 +5k +0.7108 +0.8219 +10k +0.7109 +0.8221 +50k +0.7111 +0.8222 +Student +n +Avg. pred. for +bad points +Pred. for +bad points under +A3 & n → ∞ +Avg. pred. for +good points +Pred. for +good points under +A3 & n → ∞ +1k +0.7872 +0.7901 +0.7995 +0.8024 +5k +0.7895 +0.8019 +10k +0.7898 +0.8021 +50k +0.7901 +0.8024 +(b) p = 0.2 +Table 4: (ii) Bernoulli(0.8): Same as Table 3 except for p = 0.3 (top) and p = 0.2 (bottom) +with ˆλ = 2(1−c). Just like in Table 3, as n increases, the average prediction for both good and bad +points (with the actual Gram matrix) matches the corresponding predictions under Assumption 3 +(and n → ∞). Thus, Assumption 3 is a reasonable assumption to analyze the average behavior +of a linear model on a large dataset under random label corruption. +28 + +G +Proof of Theorem 5 +G.1 +Step 1 in Detail +The teacher’s estimated parameter θ∗ +T := arg minθfT(θ) satisfies ∇fT(θ∗ +T) = +1 +2n +�2n +i=1 +� +σ(⟨θ∗ +T, φ(xi)⟩)− +ˆyi +� +φ(xi) + λθ∗ +T = ⃗0. From this, we get: +θ∗ +T = +2n +� +i=1 +1 +2nλ +� +ˆyi − σ(⟨θ∗ +T, φ(xi)⟩) +� +� +�� +� +:=αi +φ(xi) = +2n +� +i=1 +αiφ(xi), +(69) +for some real numbers {αi}2n +i=1 which are known as the teacher’s dual-space coordinates. Recall +that we defined ˆλ := 2nλ in the theorem statement. +Lemma 2 (Teacher’s Dual-Space Coordinates and Predictions). Suppose Assumptions +2 and 3 hold. Then: +αi = +� +� +� +� +� +� +� +� +� +� +� +−ˆα for i ∈ S1,bad, +α for i ∈ S1,good, +ˆα for i ∈ S0,bad, +−α for i ∈ S0,good, +(70) +where α ≥ 0 and ˆα ≥ 0 are obtained by jointly solving: +σ +� +cn +� +α − (α + ˆα)p) − (1 − c)ˆα +� += ˆλˆα, +(71) +and +σ +� +cn +� +α − (α + ˆα)p) + (1 − c)α +� += 1 − ˆλα. +(72) +Also, the teacher’s prediction for the ith sample, y(T) +i +, turns out to be: +y(T) +i += +� +� +� +� +� +� +� +� +� +� +� +ˆλˆα for i ∈ S1,bad, +1 − ˆλα for i ∈ S1,good, +1 − ˆλˆα for i ∈ S0,bad, +ˆλα for i ∈ S0,good. +(73) +Lemma 2 is proved next in Appendix G.2. +As mentioned in the proof sketch in the main text, we shall focus on the interesting case +of: +(a) p being large enough so that the teacher misclassifies the incorrectly labeled points because +otherwise, there is no need for SD, and +(b) ˆλ being chosen sensibly so that the teacher at least correctly classifies the correctly labeled +points because otherwise, SD is hopeless. +Later in Appendix G.5, we shall impose a lower bound on p (in terms of c and ˆλ) so that +(a) is ensured. Specifically, the teacher misclassifies the incorrectly labeled points (with indices +S1,bad = {1, . . . , ˆn} and S0,bad = {n + 1, . . . , n + ˆn}) when +ˆλˆα < 1 +2. +(74) +Moreover, in Appendix G.5, we shall also restrict ˆλ (in terms of c) so that (b) is ensured. +Specifically, the teacher correctly classifies the correctly labeled points (with indices S1,good = +{ˆn + 1, . . . , n} and S0,good = {n + ˆn + 1, . . . , 2n}) when +1 − ˆλα > 1 +2 =⇒ ˆλα < 1 +2. +(75) +29 + +G.2 +Proof of Lemma 2 +Proof. From eq. (69), we have: +2nλαi = ˆyi − σ +� 2n +� +j=1 +αj⟨φ(xj), φ(xi)⟩ +� +, +(76) +for all i ∈ {1, . . . , 2n}. For ease of notation, let us define vi := �2n +j=1 αj⟨φ(xj), φ(xi)⟩. Then, the +above equation can be rewritten as: +2nλαi = ˆyi − σ(vi). +(77) +Note here that the teacher’s predictions are: +y(T) +i +:= σ(vi) = ˆyi − 2nλαi, +(78) +for i ∈ {1, . . . , 2n}. Next, using Assumptions 2 and 3, we have: +vi = +� +αi + c � +j∈{1,...,n}\i αj = αi(1 − c) + c �n +j=1 αj for i ∈ {1, . . . , n}, +αi + c �n +j∈{n+1,...,2n}\i αj = αi(1 − c) + c �2n +j=n+1 αj for i ∈ {n + 1, . . . , 2n}. +(79) +Let us focus on i ∈ {1, . . . , n}. Let S = �n +j=1 αj. Then, we have the following equations: +2nλαi = −σ(αi(1 − c) + cS) for i ∈ {1, . . . , ˆn}, +(80) +and +2nλαi = 1 − σ(αi(1 − c) + cS) for i ∈ {ˆn + 1, . . . , n}. +(81) +Using the monotonicity of the sigmoid function, we conclude that: +αi = +� +−ˆα for i ∈ {1, . . . , ˆn} +α for i ∈ {ˆn + 1, . . . , n}, +(82) +for some α, ˆα ≥ 0. Using a similar argument, we can conclude that for i ∈ {n + 1, . . . , 2n}: +αi = +� +ˆα2 for i ∈ {n + 1, . . . , n + ˆn} +−α2 for i ∈ {n + ˆn + 1, . . . , 2n}, +(83) +for some α2, ˆα2 ≥ 0. We further claim that: +α2 = α and ˆα2 = ˆα. +(84) +Let us verify if this indeed holds up. Note that with such a solution: +n +� +j=1 +αj = − +2n +� +j=n+1 +αj = α(n − ˆn) − ˆαˆn = αn − (α + ˆα)ˆn. +(85) +Plugging this back in eq. (79) for i ∈ {1, . . . , n} and then in eq. (77), we get (after a bit of +rewriting): +σ +� +− (1 − c)ˆα + cαn − c(α + ˆα)ˆn +� += 2nλˆα. +(86) +σ +� +(1 − c)α + cαn − c(α + ˆα)ˆn +� += 1 − 2nλα. +(87) +Doing the same but for i ∈ {n + 1, . . . , 2n} with α2 = α and ˆα2 = ˆα, we get (again, after a bit +of rewriting): +σ +� +(1 − c)ˆα − cαn + c(α + ˆα)ˆn +� += 1 − 2nλˆα. +(88) +30 + +σ +� +− (1 − c)α − cαn + c(α + ˆα)ˆn +� += 2nλα. +(89) +Now note that eq. (86) and eq. (88), and eq. (87) and eq. (89) are the same – this is because +σ(−z) = 1 − σ(z) for all z ∈ R. Thus, our claim in eq. (84) is true. +Hence, we can consider only eq. (86) and eq. (87), and solve them to find the two unknown +variables α and ˆα in order to obtain θ∗ +T. Recalling ˆn = np, we can rewrite eq. (86) and eq. (87) +as follows: +σ +� +cn +� +α − (α + ˆα)p) − (1 − c)ˆα +� += 2nλˆα. +(90) +σ +� +cn +� +α − (α + ˆα)p) + (1 − c)α +� += 1 − 2nλα. +(91) +Thus, we have: +αi = +� +� +� +� +� +� +� +� +� +� +� +−ˆα for i ∈ {1, . . . , ˆn}, +α for i ∈ {ˆn + 1, . . . , n}, +ˆα for i ∈ {n + 1, . . . , n + ˆn}, +−α for i ∈ {n + ˆn + 1, . . . , 2n}, +(92) +where α and ˆα are obtained by solving eq. (90) and eq. (91). +From eq. (78), recall that the teacher’s predictions for the ith sample is: +y(T) +i +:= ˆyi − 2nλαi. +(93) +Now using eq. (92) in eq. (93), we get: +y(T) +i += +� +� +� +� +� +� +� +� +� +� +� +2nλˆα for i ∈ {1, . . . , ˆn}, +1 − 2nλα for i ∈ {ˆn + 1, . . . , n}, +1 − 2nλˆα for i ∈ {n + 1, . . . , n + ˆn}, +2nλα for i ∈ {n + ˆn + 1, . . . , 2n}. +(94) +Replacing 2nλ with ˆλ in equations (90), (91) and (94), and plugging in S1,bad = {1, . . . , ˆn}, +S1,good = {ˆn+1, . . . , n}, S0,bad = {n+1, . . . , n+ ˆn} and S0,good = {n+ ˆn+1, . . . , 2n} throughout +finishes the proof. +■ +G.3 +Step 2 in Detail +Just like eq. (69) for the teacher, it can be shown that: +θ∗ +S = +2n +� +i=1 +βiφ(xi), +(95) +for some real numbers {βi}2n +i=1 which are known as the student’s dual-space coordinates. +Lemma 3 (Student’s Dual-Space Coordinates and Predictions). Suppose Assumptions +2 and 3 hold, and the teacher correctly classifies the correctly labeled points but misclassifies the +incorrectly labeled points, i.e., ˆλα < 1 +2 and ˆλˆα < 1 +2 in Lemma 2. Then: +βi = +� +� +� +� +� +� +� +� +� +� +� +−ˆβ for i ∈ S1,bad, +β for i ∈ S1,good, +ˆβ for i ∈ S0,bad, +−β for i ∈ S0,good, +(96) +31 + +where β ≥ 0 and ˆβ ≥ 0 are obtained by jointly solving: +σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� += ˆλˆα + ˆλˆβ, +(97) +and +σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� += 1 − ˆλα − ˆλβ. +(98) +Also, the student’s prediction for the ith sample, y(S) +i +, turns out to be: +y(S) +i += +� +� +� +� +� +� +� +� +� +� +� +ˆλˆα + ˆλˆβ for i ∈ S1,bad, +1 − ˆλα − ˆλβ for i ∈ S1,good, +1 − ˆλˆα − ˆλˆβ for i ∈ S0,bad, +ˆλα + ˆλβ for i ∈ S0,good. +(99) +We prove Lemma 3 in Appendix G.4. +Now note that if ˆλˆα + ˆλˆβ > +1 +2 and ˆλα + ˆλβ < +1 +2, then the student has managed to cor- +rectly classify all the points in the training set. We ensure this in Appendix G.5 by imposing an +upper bound on p. +G.4 +Proof of Lemma 3 +Proof. The student’s estimated parameter θ∗ +S = arg minθfS(θ) satisfies ∇fS(θ∗ +S) = ⃗0, from which +we get: +θ∗ +S = +2n +� +i=1 +1 +2nλ +� +y(T) +i +− σ(⟨θ∗ +S, φ(xi)⟩) +� +� +�� +� +:=βi +φ(xi). +(100) +Thus the student’s ith dual coordinate βi (as defined in eq. (95)) satisfies: +2nλβi = y(T) +i +− σ(⟨θ∗ +S, φ(xi)⟩). +(101) +By following the same approach as the one we took in the proof of Lemma 2 for the teacher +(with hard labels replaced by soft labels), we can show that: +βi = +� +� +� +� +� +� +� +� +� +� +� +−ˆβ for i ∈ {1, . . . , ˆn}, +β for i ∈ {ˆn + 1, . . . , n}, +ˆβ for i ∈ {n + 1, . . . , n + ˆn}, +−β for i ∈ {n + ˆn + 1, . . . , 2n}, +(102) +where β ∈ R and ˆβ ∈ R are obtained by solving the following two equations: +σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� += 2nλˆα + 2nλˆβ, +(103) +and +σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� += 1 − 2nλα − 2nλβ. +(104) +We shall now show that β ≥ 0 and ˆβ ≥ 0. We shall prove this by contradiction – specifically, by +showing that the other cases lead to a contradiction. +Case 1: β ≤ 0 and ˆβ ≤ 0. In this case: +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ ≥ cn +� +β − (β + ˆβ)p) + (1 − c)β, +(105) +32 + +which implies (by the increasing nature of the sigmoid function): +σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� +� +�� +� +=2nλˆα+2nλˆβ from eq. (103) +≥ σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� +� +�� +� +=1−2nλα−2nλβ from eq. (104) +. +(106) +Now using eq. (103) and eq. (104), we get: +2nλˆα + 2nλˆβ ≥ 1 − 2nλα − 2nλβ =⇒ 2nλˆα ≥ 1 − 2nλα −2nλ(β + ˆβ) +� +�� +� +≥0 +=⇒ 2nλˆα ≥ 1 − 2nλα. +(107) +But this is a contradiction because as per eq. (74) and eq. (75), we had: +2nλˆα < 1 +2 and 1 − 2nλα > 1 +2 =⇒ 2nλˆα < 1 − 2nλα. +(108) +Hence, β ≤ 0 and ˆβ ≤ 0 is not possible. +Case 2: β ≥ 0 and ˆβ ≤ 0. In this case: +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ ≥ 0 =⇒ σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� +� +�� +� +=2nλˆα+2nλˆβ from eq. (103) +≥ 1 +2. +(109) +Using the above and eq. (103), we get that: +2nλˆα + 2nλˆβ +� �� � +≤0 +≥ 1 +2 =⇒ 2nλˆα ≥ 1 +2. +(110) +But this is again a contradiction as 2nλˆα < 1 +2 as per eq. (74). Hence, β ≥ 0 and ˆβ ≤ 0 is also +ruled out. +Case 3: β ≤ 0 and ˆβ ≥ 0. In this case: +cn +� +β − (β + ˆβ)p) + (1 − c)β ≤ 0 =⇒ σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� +� +�� +� +=1−2nλα−2nλβ from eq. (104) +≤ 1 +2. +(111) +Using the above and eq. (104), we get that: +1 − 2nλα − 2nλβ +� �� � +≤0 +≤ 1 +2 =⇒ 1 − 2nλα ≤ 1 +2. +(112) +But this is also a contradiction as 1 − 2nλα > 1 +2 as per eq. (75). Hence, β ≤ 0 and ˆβ ≥ 0 is also +ruled out. +So, only β ≥ 0 and ˆβ ≥ 0 is possible. Recall that β and ˆβ are solutions to: +σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� += 2nλˆα + 2nλˆβ, +(113) +and +σ +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +� += 1 − 2nλα − 2nλβ. +(114) +33 + +Just like we obtained the teacher’s predictions +� +y(T) +i +�2n +i=1, the student’s predictions are: +y(S) +i += +� +� +� +� +� +� +� +� +� +� +� +2nλˆα + 2nλˆβ for i ∈ {1, . . . , ˆn}, +1 − 2nλα − 2nλβ for i ∈ {ˆn + 1, . . . , n}, +1 − 2nλˆα − 2nλˆβ for i ∈ {n + 1, . . . , n + ˆn}, +2nλα + 2nλβ for i ∈ {n + ˆn + 1, . . . , 2n}. +(115) +Finally, replacing 2nλ with ˆλ in equations (113), (114) and (115), and plugging in S1,bad = +{1, . . . , ˆn}, S1,good = {ˆn+1, . . . , n}, S0,bad = {n+1, . . . , n+ ˆn} and S0,good = {n+ ˆn+1, . . . , 2n} +throughout gives us the desired result. +■ +G.5 +Step 3 in Detail +Proof. Here, we shall obtain analytical expressions for the teacher’s and student’s predictions by +solving eq. (71) and eq. (72) (in Lemma 2) for the teacher and then eq. (97) and eq. (98) (in +Lemma 3) for the student. Our approach will involve employing the first-order Maclaurin series +expansion of the sigmoid function; specifically, we will use: +σ(z) = 1 +2 + z +4 + ε(z), +(116) +where ε(z) is the residual error function. Note that: +ε(z) +� +� +� +� +� +< 0 for z > 0 or equivalently when σ(z) > 1 +2 += 0 for z = 0 or equivalently when σ(z) = 1 +2 +> 0 for z < 0 or equivalently when σ(z) < 1 +2. +(117) +It also holds that ε(z) is a decreasing function. So, +sup +z∈[−1,0] +ε(z) = ε(−1) < 0.02 or equivalently +sup +z:σ(z)∈[σ(−1),0.5] +ε(z) < 0.02, +(118) +and +inf +z∈[0,1] ε(z) = ε(1) > −0.02 or equivalently +inf +z:σ(z)∈[0.5,σ(1)] ε(z) > −0.02. +(119) +Let us start with the teacher. Rewriting eq. (71) and eq. (72) while using the Maclaurin series +expansion of the sigmoid function (from eq. (116)) and the fact that σ(−z) = 1 − σ(z) ∀ z ∈ R, +we have: +ˆλˆα = σ +� +cn +� +α − (α + ˆα)p) − (1 − c)ˆα +� += 1 +2 + +� +cn +� +α − (α + ˆα)p) − (1 − c)ˆα +4 +� ++ ε1, +(120) +and +ˆλα = σ +� +− cn +� +α − (α + ˆα)p) − (1 − c)α +� += 1 +2 − +� +cn +� +α − (α + ˆα)p) + (1 − c)α +4 +� ++ ε2, (121) +for some real numbers ε1, ε2. Solving the above two equations in the limit of n → ∞, when +c = Θ(1) and ˆλ < O(n) (this will be ensured subsequently), gives us: +lim +n→∞ α = p(1 + ε1 + ε2) +ˆλ + 1−c +4 +and +lim +n→∞ ˆα = (1 − p)(1 + ε1 + ε2) +ˆλ + 1−c +4 +. +(122) +Henceforth, we shall drop the limn→∞ notation, and it is implied directly. +34 + +Let us now bound ε1 + ε2 by imposing some more constraints. +First, recall from eq. (74) +and eq. (75) that we want ˆλˆα < 1 +2 (i.e., the teacher does not correctly classify the incorrectly +labeled points) and ˆλα < 1 +2 (i.e., the teacher correctly classifies the correctly labeled points). Now +since we are solving eq. (120) and eq. (121), we must have ˆλˆα = σ +� +cn +� +α−(α+ ˆα)p)−(1−c)ˆα +� +< 1 +2 +and ˆλα = σ +� +− cn +� +α − (α + ˆα)p) − (1 − c)α +� +< 1 +2; in this case, we must have that ε1 > 0 and +ε2 > 0 from eq. (117). Next, we shall obtain upper bounds for ε1 and ε2. Using eq. (121), if +σ +� +− cn +� +α − (α + ˆα)p) − (1 − c)α +� += ˆλα > σ(−1), then ε2 < 0.02 from eq. (118). Note that +since ε1 + ε2 > 0 and p < 1 +2, ˆα > α. So if ˆλα > σ(−1) holds, then so does ˆλˆα > σ(−1), in which +case ε1 < 0.02. But using the fact that ε1 + ε2 > 0, having +ˆλp +ˆλ + 1−c +4 +> σ(−1), +(123) +ensures ˆλα > σ(−1) (as well as, ˆλˆα > σ(−1)). Recalling that r = (1−c)/4 +ˆλ +and using the fact that +σ(−1) = +1 +1+e, we get: +p > 1 + r +1 + e. +(124) +But we must also have p < 1 +2 due to which we should have 1+r +1+e < 1 +2; this holds when: +r = (1 − c)/4 +ˆλ +< e − 1 +2 +=⇒ ˆλ > +1 − c +2(e − 1). +(125) +The above two conditions can be evaluated and simplified a bit more to get: +p > 1 + r +3.7 +and r < 0.85 or ˆλ > 1 − c +3.4 , +(126) +and under these conditions, ε1 < 0.02 and ε2 < 0.02. Combining all this, eq. (122) can be +rewritten as (while also dropping the limn→∞ notation): +α = p(1 + ζ) +ˆλ + 1−c +4 +and ˆα = (1 − p)(1 + ζ) +ˆλ + 1−c +4 +, +(127) +where ζ ∈ (0, 0.04). Next, recall that we want ˆλα < 1 +2 and ˆλˆα < 1 +2. Since, ˆα > α, both these +conditions can be satisfied by just ensuring ˆλˆα < 1 +2 which itself can be ensured by imposing: +1.04ˆλ(1 − p) +ˆλ + 1−c +4 += 1.04(1 − p) +1 + r +< 1 +2. +(128) +The above is obtained by making use of eq. (127) and the fact that ζ < 0.04. This gives us: +p > 1 − +�1 + r +2.08 +� +. +(129) +But again, we must have p < 1 +2 due to which we should also have 1 − +� +1+r +2.08 +� +< 1 +2; this holds +when: +r = (1 − c)/4 +ˆλ +> 0.04 =⇒ ˆλ < 1 − c +0.16 . +(130) +So to recap, for the teacher, we have: +α = p(1 + ζ) +ˆλ + 1−c +4 +and ˆα = (1 − p)(1 + ζ) +ˆλ + 1−c +4 +, +(131) +35 + +where ζ ∈ (0, 0.04), with ˆλα < ˆλˆα < 1 +2 for p > max +� +1 − +� +1+r +2.08 +� +, 1+r +3.7 +� +. All this is valid when +r ∈ +� +0.04, 0.85 +� +or equivalently when ˆλ ∈ +� +1−c +3.4 , 1−c +0.16 +� +. +Let us do a sanity check to verify that the above range of p ensures ˆλα < ˆλˆα < 1 +2. First, we +shall show that ζ = ε1 +ε2 ≥ 0 by contradiction; so suppose ζ < 0. Then using eq. (127), we have +ˆλˆα = (1+ζ)(1−p) +1+r +< 1.04(1−p) +1+r +< 1 +2, where the last step follows because p > 1− +� 1+r +2.08 +� +. But if ˆλˆα < 1 +2, +we must have ε1 > 0 (using eq. (117)) as we are solving ˆλˆα = σ +� +cn +� +α − (α + ˆα)p +� +− (1 − c)ˆα +� +. +Similarly, we must also have ε2 > 0 as ˆλα is also < 1 +2 (which is easy to see because 0 < α < ˆα +since p < 1 +2). But then ζ = ε1+ε2 > 0, which is a contradiction to our earlier supposition of ζ < 0. +Hence, we must have ζ ≥ 0. But then using eq. (127), we have ˆλα = (1+ζ)p +1+r +> +p +1+r > σ(−1), +where the last step follows because p > 1+r +3.7 . But if ˆλα > σ(−1), we must have ε2 < 0.02 +(using eq. (118)) as we are solving ˆλα = σ +� +− cn +� +α − (α + ˆα)p +� +− (1 − c)α +� +. Similarly, we must +also have ε1 < 0.02 as ˆλˆα is also > σ(−1) (again, because α < ˆα). Combining all this, we get +ζ = ε1 +ε2 < 0.04. So, ˆλα < ˆλˆα = (1+ζ)(1−p) +1+r +< 1.04(1−p) +1+r +< 1 +2, where the last step follows because +p > 1 − +� 1+r +2.08 +� +. So our prescribed range of p indeed ensures ˆλα < ˆλˆα < 1 +2. +Let us now move onto the student. Rewriting eq. (97) and eq. (98) while using the Maclaurin +series expansion of the sigmoid function (from eq. (116)) and the fact that σ(−z) = 1 − σ(z) ∀ +z ∈ R, we get: +ˆλˆα + ˆλˆβ = σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� += 1 +2 + +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +4 +� ++ ε3, (132) +and +ˆλα+ˆλβ = σ +� +−cn +� +β −(β + ˆβ)p)−(1−c)β +� += 1 +2 − +� +cn +� +β − (β + ˆβ)p) + (1 − c)β +4 +� ++ε4, (133) +for some real numbers ε3 and ε4. Solving the above two equations in the limit of n → ∞ (when +c = Θ(1) and ˆλ < O(n)) while using the values of α and ˆα from eq. (131), we get: +lim +n→∞ β = +p +ˆλ + 1−c +4 +� +− +ˆλ(1 + ζ) +ˆλ + 1−c +4 ++(1+ζ′) +� +and +lim +n→∞ +ˆβ = +1 − p +ˆλ + 1−c +4 +� +− +ˆλ(1 + ζ) +ˆλ + 1−c +4 ++(1+ζ′) +� +, (134) +with ζ′ := ε3 + ε4. Again, we shall drop the limn→∞ notation subsequently, and it is implied +directly. +Next, we get: +α + β = +p +ˆλ + 1−c +4 +� +( 1−c +4 )(1 + ζ) +ˆλ + 1−c +4 ++ (1 + ζ′) +� +, +(135) +and +ˆα + ˆβ = +1 − p +ˆλ + 1−c +4 +� +( 1−c +4 )(1 + ζ) +ˆλ + 1−c +4 ++ (1 + ζ′) +� +. +(136) +Now, recall that if ˆλ(ˆα + ˆβ) > 1 +2 and ˆλ(α + β) < 1 +2, then the student has managed to correctly +classify all the points in the training set. Let us first impose ˆλ(ˆα + ˆβ) ∈ +� 1 +2, σ(1) +� +. Then, since +we are solving eq. (132), σ +� +cn +� +β − (β + ˆβ)p) − (1 − c)ˆβ +� +∈ +� 1 +2, σ(1) +� +, and so ε3 ∈ (−0.02, 0) +using eq. (119). Now, we shall be imposing ˆλ(α + β) < 1 +2. Additionally, we ensured earlier that +ˆλα > σ(−1) and showed in Lemma 3 that β ≥ 0. Therefore, we will have ˆλ(α + β) ∈ +� +σ(−1), 1 +2 +� +. +36 + +Since we are solving eq. (133), σ +� +− cn +� +β − (β + ˆβ)p) − (1 − c)β +� +∈ +� +σ(−1), 1 +2 +� +, due to which +ε4 ∈ (0, 0.02) using eq. (118). Thus, ζ′ = ε3 + ε4 ∈ (−0.02, 0.02). +Now, using eq. (135) and eq. (136), and plugging in r = (1−c)/4 +ˆλ +, we get: +ˆλ(α + β) = +p +1 + r +�r(1 + ζ) +1 + r ++ (1 + ζ′) +� +, +(137) +and +ˆλ(ˆα + ˆβ) = 1 − p +1 + r +�r(1 + ζ) +1 + r ++ (1 + ζ′) +� +, +(138) +with ζ ∈ (0, 0.04) and ζ′ ∈ (−0.02, 0.02). Let us first ensure ˆλ(ˆα + ˆβ) ∈ +� 1 +2, σ(1) +� +. Using the +bounds on ζ and ζ′, this can be ensured by having: +1 − p +1 + r +�1.04r +1 + r + 1.02 +� +< σ(1) = +e +1 + e, +(139) +and +1 − p +1 + r +� +r +1 + r + 0.98 +� +> 1 +2. +(140) +Solving and simplifying the above two equations gives us: +p ∈ +� +1 − 0.7(1 + r)2 +1 + 2r +, 1 − 0.51(1 + r)2 +1 + 2r +� +. +(141) +Note that: +1 − 0.7(1 + r)2 +1 + 2r +< 1 − 0.51(1 + r)2 +1 + 2r +< 1 +2 +(142) +for all r > 0, and so we are good here. But recall that from the teacher’s analysis (see the +discussion after eq. (131)), we had p > max +� +1− +� +1+r +2.08 +� +, 1+r +3.7 +� +. Combining everything, our current +bound on p is: +p ∈ +� +max +� +1 − +�1 + r +2.08 +� +, 1 + r +3.7 , 1 − 0.7(1 + r)2 +1 + 2r +� +, 1 − 0.51(1 + r)2 +1 + 2r +� +. +(143) +But the above is only meaningful when the lower bound on p is smaller than the upper bound on +it. So we must find the range of r for which: +1 − +�1 + r +2.08 +� +< 1 − 0.51(1 + r)2 +1 + 2r +and 1 + r +3.7 +< 1 − 0.51(1 + r)2 +1 + 2r +. +1− 0.7(1+r)2 +1+2r +is trivially smaller than 1− 0.51(1+r)2 +1+2r +so we do not need to worry about that. Combining +the range of r obtained from the above equation with the previous range of r ∈ (0.04, 0.85) (that +we obtained from the teacher), we get: +r ∈ [0.07, 0.54] =⇒ ˆλ ∈ +�1 − c +2.16 , 1 − c +0.28 +� +. +(144) +Finally, we need to ensure ˆλ(α + β) < 1 +2. Using eq. (137) and the bounds on ζ and ζ′, this can +be ensured by imposing: +p +1 + r +�1.04r +1 + r + 1.02 +� +< 1 +2. +(145) +This can be simplified to: +p < 0.485(1 + r)2 +1 + 2r +. +37 + +But recall that we already have an upper bound on p of 1 − 0.51(1+r)2 +1+2r +. It can be checked that +1 − 0.51(1+r)2 +1+2r +< 0.485(1+r)2 +1+2r +for r ≥ 0.08. Thus, for r ∈ [0.08, 0.54] or ˆλ ∈ +� +1−c +2.16, 1−c +0.32 +� +, our bound +on p remains the same as eq. (143), i.e., +p ∈ +� +max +� +1 − +�1 + r +2.08 +� +, 1 + r +3.7 , 1 − 0.7(1 + r)2 +1 + 2r +� +, 1 − 0.51(1 + r)2 +1 + 2r +� +. +(146) +Finally, to simplify our bound on p a bit, we consider r ∈ [0.10, 0.54], where: +max +� +1 − +�1 + r +2.08 +� +, 1 + r +3.7 , 1 − 0.7(1 + r)2 +1 + 2r +� += max +� +1 − +�1 + r +2.08 +� +, 1 + r +3.7 +� +. +(147) +Thus, our final bound on p is: +p ∈ +� +max +� +1 − +�1 + r +2.08 +� +, 1 + r +3.7 +� +, 1 − 0.51(1 + r)2 +1 + 2r +� +, +(148) +for +r ∈ [0.10, 0.54] or ˆλ ∈ +�1 − c +2.16 , 1 − c +0.40 +� +. +(149) +Finally, note that the prescribed range of ˆλ is < O(n) (as required in eq. (122) and eq. (134)) +since c = Θ(1). So we are good here. +Also, since n → ∞, the generalization gap (i.e., population accuracy - training accuracy) +→ 0; see for e.g., the margin bounds (with ℓ2-regularization) in [Kakade et al., 2008] where it is +shown that the generalization gap goes down as O(1/√n). Therefore, the population accuracy of +the student (resp., teacher) is the same as the training accuracy of the student (resp., teacher). +This finishes the proof. +■ +H +Proof of Corollary 5.1 +Proof. From eq. (73), we have: +∆T = 1 − ˆλ(α + ˆα), +(150) +where ˆλ = 2nλ. Similarly, using eq. (99), we have: +∆S = 1 − ˆλ(α + β + ˆα + ˆβ). +(151) +Next, using eq. (127) in eq. (150), we get: +∆T = 1 − +�1 + ζ +1 + r +� +, +(152) +where ζ ∈ (0, 0.04) and r = (1−c) +4ˆλ . Similarly, using eq. (137) and eq. (138), we get: +∆S = 1 − +1 +1 + r +�r(1 + ζ) +1 + r ++ (1 + ζ′) +� +, +(153) +where ζ′ ∈ (−0.02, 0.02). Rewriting eq. (153) slightly, we get: +∆S = 1 − +�1 + ζ +1 + r +�� +r +1 + r + 1 + ζ′ +1 + ζ +� +(154) +≤ 1 − +�1 + ζ +1 + r +��0.1 +1.1 + 0.98 +1.04 +� +(155) +< 1 − +�1 + ζ +1 + r +� +(156) += ∆T. +(157) +In eq. (155), we have used the fact that r ≥ 0.1 (from the condition of Theorem 5), ζ′ ≥ −0.02 +and ζ ≤ 0.04. +■ +38 + +I +More Empirical Results +I.1 +Verifying Remark 2 (Continued) +In Section 5.1, we compared the performance of different values of ξ with 50% corruption. In +Table 5, we show results with 30% corruption in Stanford Cars and Flowers-10215 with the same +weight decay value as in Section 5 (viz., 5 × 10−4); even here, the improvement with ξ > 1 is +more than that with ξ ≤ 1. Again, the individual accuracies of the teacher and student and the +experimental details are in Appendix J. +ξ +Improvement of student +over teacher (i.e., ξ = 0) +0.2 +0.89 ± 0.15 % +0.5 +2.15 ± 0.06 % +0.7 +2.75 ± 0.10 % +1.0 +3.32 ± 0.11 % +1.2 +3.53 ± 0.16 % +1.5 +3.38 ± 0.12 % +1.7 +2.96 ± 0.24 % +2.0 +1.79 ± 0.29 % +(a) 30% Random Corruption in Stanford Cars +with ResNet-34 +ξ +Improvement of student +over teacher (i.e., ξ = 0) +0.5 +−0.12 ± 0.20 % +1.0 +0.54 ± 0.02 % +1.5 +0.86 ± 0.01 % +2.0 +1.57 ± 0.34 % +2.5 +2.05 ± 0.27 % +3.0 +2.49 ± 0.25 % +3.5 +2.62 ± 0.12 % +4.0 +2.87 ± 0.09 % +4.5 +3.01 ± 0.22 % +5.0 +3.21 ± 0.07 % +5.5 +2.94 ± 0.33 % +6.0 +3.06 ± 0.09 % +(b) 30% Adversarial Corruption in Flowers-102 +with ResNet-34 +Table 5: Average (± 1 std.) improvement of student over teacher (i.e., student’s test set accuracy +- teacher’s test set accuracy) with different values of the imitation parameter ξ. Just like in +Table 1, note that the value of ξ yielding the biggest improvement is more than 1. +I.2 +Results with Other Weight Decay Values +All our previous results were with weight decay = 5 × 10−4. Here, we verify Remarks 2 and 3 for +two other weight decay values which are 1 × 10−3 and 1 × 10−4. +(i) Verifying Remark 2: In Table 6, we list the student’s improvement over the teacher (i.e., +student’s test accuracy - teacher’s test accuracy) averaged across 3 different runs for different +values of ξ in the case of (a) Caltech-256 with 50% random corruption & weight decay = 1 × 10−4 +and (b) CIFAR-100 with 50% hierarchical corruption & weight decay = 1 × 10−3. As was the +case with weight decay = 5 × 10−4 in Tables 1 and 5, note that the value of ξ yielding the biggest +improvement here is also > 1. +(ii)Verifying Remark 3: The setup is the same as Section 5.2, i.e., the student is trained +with ξ = 1. In Table 7, we show the student’s improvement over the teacher averaged across 3 +different runs for varying degrees of label corruption in the case of (a) Caltech-256 with random +corruption & weight decay = 1 × 10−4 and (b) CIFAR-100 with hierarchical corruption & weight +decay = 1 × 10−3. As was the case with weight decay = 5 × 10−4 in Table 2, note that the +improvement of the student (trained with ξ = 1) over the teacher increases as the corruption +level increases. +The individual accuracies of the teacher and student and the experimental details appear in +Appendix J. +15For Flowers-102, we include the provided validation set in the training set. +39 + +ξ +Improvement of student +over teacher +0.2 +2.04 ± 0.16 % +0.5 +5.05 ± 0.10 % +0.7 +6.82 ± 0.16 % +1.0 +9.07 ± 0.17 % +1.2 +10.43 ± 0.15 % +1.5 +11.78 ± 0.16 % +1.7 +12.30 ± 0.19 % +2.0 +13.07 ± 0.20 % +2.2 +12.89 ± 0.43 % +2.5 +11.74 ± 0.60 % +(a) ResNet-34: 50% Random Corruption in +Caltech-256 with weight decay = 1 × 10−4 +ξ +Improvement of student +over teacher +0.2 +0.39 ± 0.09 % +0.5 +1.90 ± 0.08 % +0.7 +2.80 ± 0.09 % +1.0 +3.82 ± 0.04 % +1.2 +4.17 ± 0.05 % +1.5 +4.51 ± 0.07 % +1.7 +4.56 ± 0.02 % +2.0 +4.15 ± 0.07 % +(b) ResNet-34: 50% Hierarchical Corruption in +CIFAR-100 with weight decay = 1 × 10−3 +Table 6: Average (± 1 std.) improvement of student over teacher (i.e., student’s test set accuracy +- teacher’s test set accuracy) with different values of ξ. Just like with weight decay = 5 × 10−4 +(Tables 1 and 5), note that the value of ξ yielding the biggest improvement with both weight +decay values here is more than 1. This is consistent with our message in Remark 2. +Corruption level +Improvement of student over teacher +0% +0.25 ± 0.04 % +10% +1.39 ± 0.10 % +30% +6.31 ± 0.11 % +50% +9.07 ± 0.17 % +(a) Random Corruption in Caltech-256 with weight decay = 1 × 10−4 +Corruption level +Improvement of student over teacher +0% +−0.73 ± 0.09 % +10% +0.03 ± 0.10 % +30% +1.77 ± 0.19 % +50% +3.82 ± 0.04 % +(b) Hierarchical Corruption in CIFAR-100 with weight decay = 1 × 10−3 +Table 7: ResNet-34 with ξ = 1: Average (± 1 std.) improvement of student over teacher (i.e., +student’s test set accuracy - teacher’s test set accuracy) with varying levels of label corruption. +Just like with weight decay = 5 × 10−4 (Table 2), note that the improvement of the student +over the teacher increases as the corruption level increases. This is consistent with our claim in +Remark 3. +40 + +J +Detailed Empirical Results +We list the individual accuracies of the teacher and student (along with the student’s improve- +ment) corresponding to the results of Table 1 in Tables 8-13, Table 5 in Tables 14-15, Table 2 in +Tables 16-21, Table 6 in Tables 22-23 and Table 7 in Tables 24-25. +Experimental Details: In all the cases, we use SGD with momentum = 0.9 and batch +size = 128 for training. Since we are training only the softmax layer (i.e., doing logistic regres- +sion), we use an exponentially decaying learning rate scheme with decay parameter = 0.98 (for +every epoch) and the initial learning rate is tuned16 over {0.001, 0.005, 0.01, 0.05, 0.1, 0.5}. The +maximum number of epochs is 200. +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +57.61 ± 0.03 % +0 % +0.2 +59.83 ± 0.12 % +2.22 ± 0.12 % +0.5 +62.79 ± 0.04 % +5.18 ± 0.03 % +0.7 +64.45 ± 0.09 % +6.84 ± 0.06 % +1.0 +66.15 ± 0.27 % +8.54 ± 0.29 % +1.2 +67.27 ± 0.25 % +9.66 ± 0.23 % +1.5 +67.65 ± 0.54 % +10.04 ± 0.51 % +1.7 +67.42 ± 0.58 % +9.81 ± 0.55 % +2.0 +66.17 ± 0.77 % +8.56 ± 0.73 % +Table 8: Detailed Version of Table 1a (50% Random Corruption in Caltech-256 with ResNet-34) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +61.15 ± 0.09 % +0 % +0.5 +62.04 ± 0.02 % +0.89 ± 0.10 % +1.0 +63.16 ± 0.06 % +2.01 ± 0.14 % +1.5 +64.28 ± 0.06 % +3.13 ± 0.11 % +2.0 +65.37 ± 0.12 % +4.22 ± 0.20 % +2.5 +66.43 ± 0.05 % +5.28 ± 0.13 % +3.0 +66.93 ± 0.03 % +5.78 ± 0.12 % +3.5 +67.01 ± 0.13 % +5.86 ± 0.18 % +4.0 +66.47 ± 0.25 % +5.32 ± 0.33 % +Table 9: Detailed Version of Table 1b (50% Random Corruption in Caltech-256 with VGG-16) +16The tuning is done by picking the learning rate which yields the lowest training loss with the observed (noisy) +labels. This is consistent with our theory setup where we assume convergence to the optimum of the training loss +w.r.t. the observed labels. +41 + +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +50.80 ± 0.04 % +0 % +0.2 +51.78 ± 0.14 % +0.98 ± 0.12 % +0.5 +53.26 ± 0.14 % +2.46 ± 0.11 % +0.7 +54.18 ± 0.03 % +3.38 ± 0.02 % +1.0 +54.99 ± 0.08 % +4.19 ± 0.09 % +1.2 +55.26 ± 0.18 % +4.46 ± 0.19 % +1.5 +55.26 ± 0.15 % +4.46 ± 0.17 % +1.7 +55.12 ± 0.16 % +4.32 ± 0.18 % +2.0 +54.32 ± 0.20 % +3.52 ± 0.23 % +Table 10: Detailed Version of Table 1c (50% Hierarchical Corruption in CIFAR-100 with ResNet- +34) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +41.60 ± 0.08 % +0 % +0.2 +42.70 ± 0.03 % +1.10 ± 0.09 % +0.5 +44.29 ± 0.06 % +2.69 ± 0.02 % +0.7 +45.32 ± 0.05 % +3.72 ± 0.05 % +1.0 +46.89 ± 0.05 % +5.29 ± 0.11 % +1.2 +47.86 ± 0.06 % +6.26 ± 0.09 % +1.5 +48.80 ± 0.16 % +7.20 ± 0.14 % +1.7 +48.83 ± 0.18 % +7.23 ± 0.17 % +2.0 +48.02 ± 0.25 % +6.42 ± 0.26 % +Table 11: Detailed Version of Table 1d (50% Hierarchical Corruption in CIFAR-100 with VGG-16) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +48.93 ± 0.08 % +0 % +0.2 +49.06 ± 0.05 % +0.13 ± 0.08 % +0.5 +49.90 ± 0.04 % +0.97 ± 0.04 % +0.7 +50.38 ± 0.09 % +1.45 ± 0.01 % +1.0 +50.78 ± 0.07 % +1.85 ± 0.09 % +1.2 +50.80 ± 0.06 % +1.87 ± 0.06 % +1.5 +50.79 ± 0.03 % +1.86 ± 0.08 % +1.7 +50.73 ± 0.04 % +1.80 ± 0.05 % +2.0 +50.46 ± 0.09 % +1.53 ± 0.02 % +Table 12: Detailed Version of Table 1e (50% Adversarial Corruption in Food-101 with ResNet-34) +42 + +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +37.01 ± 0.46 % +0 % +0.2 +37.80 ± 0.23 % +0.79 ± 0.23 % +0.5 +39.15 ± 0.37 % +2.14 ± 0.09 % +0.7 +39.97 ± 0.42 % +2.96 ± 0.04 % +1.0 +40.86 ± 0.51 % +3.85 ± 0.05 % +1.2 +41.23 ± 0.60 % +4.22 ± 0.15 % +1.5 +41.40 ± 0.71 % +4.39 ± 0.29 % +1.7 +41.21 ± 0.76 % +4.20 ± 0.34 % +2.0 +40.54 ± 0.90 % +3.53 ± 0.49 % +Table 13: Detailed Version of Table 1f (50% Adversarial Corruption in Food-101 with VGG-16) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +25.01 ± 0.20 % +0 % +0.2 +25.90 ± 0.09 % +0.89 ± 0.15 % +0.5 +27.16 ± 0.16 % +2.15 ± 0.06 % +0.7 +27.76 ± 0.21 % +2.75 ± 0.10 % +1.0 +28.33 ± 0.14 % +3.32 ± 0.11 % +1.2 +28.54 ± 0.11 % +3.53 ± 0.16 % +1.5 +28.39 ± 0.10 % +3.38 ± 0.12 % +1.7 +27.97 ± 0.20 % +2.96 ± 0.24 % +2.0 +26.80 ± 0.27 % +1.79 ± 0.29 % +Table 14: Detailed Version of Table 5a (30% Random Corruption in Stanford Cars with ResNet-34) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +50.34 ± 0.23 % +0 % +0.5 +50.22 ± 0.23 % +−0.12 ± 0.20 % +1.0 +50.88 ± 0.24 % +0.54 ± 0.02 % +1.5 +51.20 ± 0.22 % +0.86 ± 0.01 % +2.0 +51.91 ± 0.41 % +1.57 ± 0.34 % +2.5 +52.39 ± 0.44 % +2.05 ± 0.27 % +3.0 +52.83 ± 0.28 % +2.49 ± 0.25 % +3.5 +52.96 ± 0.28 % +2.62 ± 0.12 % +4.0 +53.21 ± 0.31 % +2.87 ± 0.09 % +4.5 +53.35 ± 0.15 % +3.01 ± 0.22 % +5.0 +53.55 ± 0.25 % +3.21 ± 0.07 % +5.5 +53.28 ± 0.50 % +2.94 ± 0.33 % +6.0 +53.40 ± 0.28 % +3.06 ± 0.09 % +Table 15: Detailed Version of Table 5b (30% Adversarial Corruption in Flowers-102 with ResNet- +34) +43 + +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +83.97 ± 0.10 % +83.93 ± 0.12 % +−0.04 ± 0.02 % +10% +77.86 ± 0.14 % +80.37 ± 0.04 % +2.51 ± 0.11 % +30% +68.09 ± 0.21 % +74.23 ± 0.08 % +6.14 ± 0.16 % +50% +57.61 ± 0.03 % +66.15 ± 0.27 % +8.54 ± 0.29 % +Table 16: Detailed Version of Random Corruption in Caltech-256 with ResNet-34 and ξ = 1 +(Table 2a) +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +83.97 ± 0.10 % +83.93 ± 0.12 % +−0.04 ± 0.02 % +10% +77.01 ± 0.23 % +79.33 ± 0.13 % +2.32 ± 0.10 % +30% +64.21 ± 0.36 % +69.29 ± 0.12 % +5.08 ± 0.25 % +50% +48.66 ± 0.10 % +54.43 ± 0.29 % +5.77 ± 0.19 % +Table 17: Detailed Version of Adversarial Corruption in Caltech-256 with ResNet-34 and ξ = 1 +(Table 2a) +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +72.77 ± 0.07 % +72.54 ± 0.07 % +−0.23 ± 0.06 % +10% +70.57 ± 0.14 % +71.20 ± 0.02 % +0.63 ± 0.11 % +30% +66.80 ± 0.06 % +68.14 ± 0.07 % +1.34 ± 0.13 % +50% +62.47 ± 0.10 % +64.58 ± 0.10 % +2.11 ± 0.15 % +Table 18: Detailed Version of Random Corruption in CIFAR-100 with ResNet-34 and ξ = 1 +(Table 2b) +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +72.77 ± 0.07 % +72.54 ± 0.07 % +−0.23 ± 0.06 % +10% +69.39 ± 0.09 % +70.58 ± 0.08 % +1.19 ± 0.08 % +30% +62.18 ± 0.12 % +64.98 ± 0.10 % +2.80 ± 0.06 % +50% +50.80 ± 0.04 % +54.99 ± 0.08 % +4.19 ± 0.09 % +Table 19: Detailed Version of Hierarchical Corruption in CIFAR-100 with ResNet-34 and ξ = 1 +(Table 2b) +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +63.65 ± 0.08 % +63.28 ± 0.04 % +−0.37 ± 0.10 % +10% +62.44 ± 0.03 % +62.54 ± 0.03 % +0.10 ± 0.04 % +30% +59.38 ± 0.20 % +59.85 ± 0.18 % +0.47 ± 0.04 % +50% +54.76 ± 0.13 % +55.88 ± 0.06 % +1.12 ± 0.08 % +Table 20: Detailed Version of Random Corruption in Food-101 with ResNet-34 and ξ = 1 +(Table 2c) +44 + +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +63.65 ± 0.08 % +63.28 ± 0.04 % +−0.37 ± 0.10 % +10% +61.92 ± 0.13 % +62.16 ± 0.11 % +0.25 ± 0.05 % +30% +57.03 ± 0.16 % +57.80 ± 0.22 % +0.77 ± 0.06 % +50% +48.93 ± 0.08 % +50.78 ± 0.07 % +1.85 ± 0.09 % +Table 21: Detailed Version of Adversarial Corruption in Food-101 with ResNet-34 and ξ = 1 +(Table 2c) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +39.78 ± 0.18 % +0 % +0.2 +41.82 ± 0.06 % +2.04 ± 0.16 % +0.5 +44.83 ± 0.11 % +5.05 ± 0.10 % +0.7 +46.60 ± 0.09 % +6.82 ± 0.16 % +1.0 +48.85 ± 0.06 % +9.07 ± 0.17 % +1.2 +50.21 ± 0.03 % +10.43 ± 0.15 % +1.5 +51.56 ± 0.10 % +11.78 ± 0.16 % +1.7 +52.08 ± 0.03 % +12.30 ± 0.19 % +2.0 +52.85 ± 0.05 % +13.07 ± 0.20 % +2.2 +52.67 ± 0.33 % +12.89 ± 0.43 % +2.5 +51.52 ± 0.51 % +11.74 ± 0.60 % +Table 22: Detailed Version of Table 6a (50% Random Corruption in Caltech-256 w/ ResNet-34 +and wt. decay = 1 × 10−4) +ξ +Student’s test acc. +Improvement of student +over teacher +0.0 (=Teacher) +54.71 ± 0.05 % +0 % +0.2 +55.10 ± 0.04 % +0.39 ± 0.09 % +0.5 +56.61 ± 0.04 % +1.90 ± 0.08 % +0.7 +57.51 ± 0.04 % +2.80 ± 0.09 % +1.0 +58.53 ± 0.05 % +3.82 ± 0.04 % +1.2 +58.88 ± 0.03 % +4.17 ± 0.05 % +1.5 +59.22 ± 0.11 % +4.51 ± 0.07 % +1.7 +59.27 ± 0.06 % +4.56 ± 0.02 % +2.0 +58.86 ± 0.02 % +4.15 ± 0.07 % +Table 23: Detailed Version of Table 6b (50% Hierarchical Corruption in CIFAR-100 w/ ResNet-34 +and wt. decay = 1 × 10−3) +45 + +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +82.95 ± 0.02 % +83.20 ± 0.04 % +0.25 ± 0.04 % +10% +74.29 ± 0.12 % +75.68 ± 0.03 % +1.39 ± 0.10 % +30% +54.65 ± 0.13 % +60.96 ± 0.18 % +6.31 ± 0.11 % +50% +39.78 ± 0.18 % +48.85 ± 0.06 % +9.07 ± 0.17 % +Table 24: Detailed Version of Table 7a (Random Corruption in Caltech-256 w/ ResNet-34, ξ = 1 +and wt. decay = 1 × 10−4) +Corruption level +Teacher’s test acc. +Student’s test acc. +Improvement of student +over teacher +0% +72.99 ± 0.09 % +72.26 ± 0.01 % +−0.73 ± 0.09 % +10% +70.59 ± 0.04 % +70.62 ± 0.07 % +0.03 ± 0.10 % +30% +64.64 ± 0.12 % +66.41 ± 0.09 % +1.77 ± 0.19 % +50% +54.71 ± 0.05 % +58.53 ± 0.05 % +3.82 ± 0.04 % +Table 25: Detailed Version of Table 7b (Hierarchical Corruption in CIFAR-100 w/ ResNet-34, +ξ = 1 and wt. decay = 1 × 10−3) +46 + diff --git a/NdFQT4oBgHgl3EQfWTbR/content/tmp_files/load_file.txt b/NdFQT4oBgHgl3EQfWTbR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be9eace00e2c7f7fdab32d58526eb75718aacdb3 --- /dev/null +++ b/NdFQT4oBgHgl3EQfWTbR/content/tmp_files/load_file.txt @@ -0,0 +1,2963 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf,len=2962 +page_content='Understanding Self-Distillation in the Presence of Label Noise Rudrajit Das* and Sujay Sanghavi* UT Austin Abstract Self-distillation (SD) is the process of first training a “teacher” model and then using its predictions to train a “student” model with the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, the student’s objective function is � ξ ∗ ℓ(teacher’s predictions, student’s predictions) + (1 − ξ) ∗ ℓ(given labels, student’s predictions) � , where ℓ is some loss function and ξ is some parameter ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Empirically, SD has been observed to provide performance gains in several settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this paper, we theoretically characterize the effect of SD in two supervised learning problems with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We first analyze SD for regularized linear regression and show that in the high label noise regime, the optimal value of ξ that minimizes the expected error in estimating the ground truth parameter is surprisingly greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Empirically, we show that ξ > 1 works better than ξ ≤ 1 even with the cross-entropy loss for several classification datasets when 50% or 30% of the labels are corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, we quantify when optimal SD is better than optimal regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, we analyze SD in the case of logistic regression for binary classification with random label corruption and quantify the range of label corruption in which the student outperforms the teacher in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To our knowledge, this is the first result of its kind for the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 1 Introduction The core idea of knowledge distillation (KD), introduced in [Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015], is to train a student model with a teacher model’s predicted soft labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the output probability distribution over the classes for classification problems) in addition to the original hard labels (one-hot vectors for classification problems) on which the teacher is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The original rationale was to use a teacher with large statistical capacity to better model the underlying label distribution compared to the provided hard labels, and have the student with smaller capacity learn some mixture of the teacher’s predicted label distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' “dark knowledge”) and the provided label distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, the student’s per-sample objective function in the KD framework is: ξ ∗ ℓ � yT , yS(θ) � + (1 − ξ) ∗ ℓ � y, yS(θ) � , (1) where ℓ is some loss function (usually, regularized cross-entropy loss for classification problems), yT is the teacher’s predicted label, y is the given label on which the teacher is trained, yS(θ) is the prediction of the student model parameterized by θ, and ξ ∈ [0, 1] is known as the imitation parameter [Lopez-Paz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' KD and its variants have been shown to be beneficial for model compression (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', distilling a bigger teacher model’s knowledge into a smaller student model), semi-supervised learning, making models robust and improving performance in general [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2017,Furlanello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2018,Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019,Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019,Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020,Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020,Sarfraz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021,Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021,Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021,Beyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2022,Baykal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2022];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see [Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021] for a survey on KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The focus of this work is on the special case of the student and teacher having the same architecture, which is known as self-distillation (following [Mobahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we abbreviate 1In this work, we set the temperature parameter suggested in [Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015] equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13304v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='LG] 30 Jan 2023 it as SD henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since the teacher and student have the same capacity, one would expect the utility of the teacher’s dark knowledge to be very limited, if any at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, surprisingly, [Furlanello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2018] show that SD (with ensembling) yields performance gains in both vision and language tasks with extensive experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2017] empirically demonstrate that SD can ameliorate learning in the presence of noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' There are also a few works that theoretically investigate SD, such as [Mobahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020,Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we discuss these in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The results of these papers are only with the squared loss and not the cross-entropy loss which is the de facto loss function for classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this work, we theoretically analyze SD in the presence of label corruption (in the supervised setting) for the cross-entropy loss as well as the squared loss, characterizing its utility and unveiling some new insights including a recommendation for use in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We summarize our contributions next and survey the landscape of pertinent theoretical works on KD and SD in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Contributions: (a) First, we consider linear regression with ℓ2-regularized squared loss in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Here, the observed label y for a sample x is: y = ⟨θ∗, x⟩ + η, where θ∗ is the underlying parameter and η is zero-mean random label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We show that self-distillation (SD) is associated with a bias-variance tradeoff in that increasing ξ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (1) reduces the variance but increases the bias in estimating θ∗ with respect to the randomness in label noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Theorem 1 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' A surprising algorithmic insight from our analysis is that the value of ξ that optimally balances this bias-variance tradeoff can be > 1, especially in the high label noise regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', when E[η2] is large);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This can be interpreted as actively anti-learning (or going against) the observed (possibly noisy) labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But as discussed after eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (1), ξ is tuned in [0, 1] in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, we empirically corroborate our insight for multi-class classification with linear probing2 using the cross-entropy loss by showing that ξ > 1 works better than ξ ≤ 1 for several datasets with 50% or 30% of the training set’s labels being corrupted in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Remark 3, we show that as the degree of label noise increases, the utility of the teacher’s predictions in training the student increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Intuitively, this happens because the noise component in the teacher’s predictions is smaller compared to the original labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We also empirically verify this insight for the cross-entropy loss in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Theorem 2, we provide a condition when optimal SD is better than optimal ℓ2 regular- ization (optimal means with the best parameters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' this is the first such result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (b) Next, we look at logistic regression with ℓ2-regularized cross-entropy loss in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We consider a balanced binary classification problem where some fraction, say p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5, of the training set’s labels are randomly flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Under some assumptions on the data geometry and the kernel function, we quantify the range of p in which the student outperforms the teacher in terms of accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To our knowledge, this is the first result that provably establishes the utility of SD in the presence of label noise for the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The main technical challenge in the analysis is dealing with non-linear equations involving the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We tackle this by employing the first-order Maclaurin series expansion of the sigmoid function and by bounding the corresponding approximation errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Step 3 in the proof outline of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Moreover, in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, we show that the student’s predictions have smaller variability than the teacher’s predictions which is akin to SD reducing variance in linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', learning a softmax layer on top of a pre-trained network 2 2 Related Work There is a growing body of works trying to theoretically explain KD/SD and its benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Mobahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020] look at regression with the squared loss in Hilbert space, showing that SD essentially amplifies regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, unlike us, they do not explicitly consider the case of noisy labels/observations or discuss the bias-variance tradeoff associated with SD in the presence of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Moreover, they restrict their analysis to ξ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' so unlike us, they do not have any results on when optimal SD is better than optimal ℓ2 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] claim that KD is effective in transferring dark knowledge by mimicking early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, they propose their own SD algorithm that uses dynamically updated soft labels, and show that in the presence of noisy labels, their algorithm is able to learn the correct labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this work, we focus on the standard SD algorithm with fixed soft labels, and moreover, we quantify the range of label corruption in which SD improves accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Unlike our work, [Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] do not quantify when their proposed algorithm improves upon the standard approach of using just hard labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' An important difference between our work and [Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] as well as [Mobahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020] is that the results of these two papers are with the squared loss, whereas we provide results with the cross-entropy loss in addition to squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The cross-entropy loss is the customary choice for classification problems in practice and is also more challenging to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' On the note of cross-entropy loss, [Phuong and Lampert, 2019] analyze the convergence of linear student networks trained with the cross-entropy loss, and also bound the expected difference between the predictions of the student and teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Ji and Zhu, 2020] also bound the expected difference between the predictions of the student and teacher for wide neural networks that evolve as linear networks under the NTK assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, [Phuong and Lampert, 2019] and [Ji and Zhu, 2020] do not consider how the student might have better generalization than the teacher in the presence of noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021] statistically characterize “good” teachers for distilling knowledge to a student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Kaplun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2022] show that an ensemble of teachers trained with noisy labels can be used to label a new unlabeled dataset, which can be then employed to train a student with good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We focus on the (common) case of only one teacher and the student being trained on the same dataset as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' There are also some works such as [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020,Stanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021,Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2022] that empirically provide some insights on KD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 3 Linear Regression Setting: The observed label y ∈ R is linearly related to the data x ∈ X ⊆ Rd as: y = ⟨θ∗, x⟩ + η, (2) where θ∗ ∈ Rd and η ∈ R is label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Here, ⟨θ∗, x⟩ is the actual label of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The training set consists of n pairs of data points (drawn from X) and noisy labels {(xi, yi)}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let X := [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , xn] ∈ Rd×n be the data matrix and Y := [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , yn]T ∈ Rn be the label vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, as per the above linear model (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (2)): Y = XT θ∗ + η, (3) for some noise vector η ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We make some standard assumptions on the noise vector η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' η is independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, each coordinate of η has mean 0 and variance γ2, and is independent of the other coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Teacher Model: The teacher tries to learn the underlying model, parameterized by θ ∈ Rd, from (X, Y ) by applying the squared loss with ℓ2 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, the teacher’s objective function is: fT (θ) = 1 2∥Y − XT θ∥2 + λ 2 ∥θ∥2, (4) 3 where λ > 0 is the ℓ2-regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, the model learned by the teacher is3: ˆθT := arg minθ∈Rd fT (θ) = (XXT + λId)−1XY , (5) where Id is the identity matrix of size d × d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Plugging in Y from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (3) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (5), we get: ˆθT = (XXT + λId)−1X(XT θ∗ + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (6) Student Model Trained with Self-Distillation: Following eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (1), here the student is trained with a weighted sum of (i) the ℓ2-regularized squared loss between the student’s predictions and the teacher’s predictions, and (ii) the ℓ2-regularized squared loss between the student’s predictions and the original labels on which the teacher was trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For the ith sample, the teacher’s prediction is ˆyi = ⟨ˆθT , xi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Define ˆY := [ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆyn]T ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' note that ˆY = XT ˆθT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The student’s objective function is: fS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ξ) = ξ �1 2∥ ˆY − XT θ∥2 + λ 2 ∥θ∥2� + (1 − ξ) �1 2∥Y − XT θ∥2 + λ 2 ∥θ∥2� = ξ �1 2∥ ˆY − XT θ∥2� + (1 − ξ) �1 2∥Y − XT θ∥2� + λ 2 ∥θ∥2, (7) where ξ ∈ R is known as the imitation parameter [Lopez-Paz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015] and λ > 0 is the same regularization parameter that was used by the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Even though it is standard practice to restrict ξ ∈ [0, 1], we do not impose this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, the model learned by the student is: ˆθS(ξ) := arg minθ∈Rd fS(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ξ) = (XXT + λId)−1X(ξ ˆY + (1 − ξ)Y ) = ξ(XXT + λId)−1XXT ˆθT + (1 − ξ)ˆθT , (8) where eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (8) is obtained by using ˆY = XT ˆθT and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that ξ = 0 corresponds to the teacher, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ˆθS(0) = ˆθT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Finally, plugging in ˆθT from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (6) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (8), we get: ˆθS(ξ) = � ξ(XXT + λId)−1XXT + (1 − ξ)Id � (XXT + λId)−1X(XT θ∗ + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Estimation Error Comparison: Bias-Variance Tradeoff Let us denote the student’s error in estimating the ground truth parameter θ∗ with imitation parameter ξ as ϵS(ξ) := ˆθS(ξ) − θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that ϵS(0) := ˆθS(0) − θ∗ = ˆθT − θ∗ is the teacher’s estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We shall analyze the expected squared norm of the estimation error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the random label noise η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Eη[∥ϵS(ξ)∥2], as a function of ξ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It will be illustrative to analyze Eη[∥ϵS(ξ)∥2] in terms of the SVD of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let rank(X) = r (note that r ≤ min(d, n)) and the SVD decomposition of X be �r j=1 σjujvT j , where σ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ≥ σr > 0, and each uj ∈ Rd and vj ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, let {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ud} be the full set of left singular vectors of X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', even those corresponding to the zero singular values);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' note that this forms an orthonormal basis for Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Following standard bias-variance decomposition, we have: Eη ���ϵS(ξ) ��2� = ��Eη[ϵS(ξ)] ��2 � �� � squared bias + Eη ���ϵS(ξ) − Eη[ϵS(ξ)] ��2� � �� � variance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (10) Now we shall quantify the squared bias and variance in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (10) as a function of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 3Throughout this work, we shall assume that we can converge to the exact optimum of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' All the objective functions in this work are convex, and hence (stochastic) gradient descent will converge to the optimum in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 4We do not analyze the expected squared prediction error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Eη,x �� ⟨ˆθS(ξ), x⟩ − ⟨θ∗, x⟩ �2�, because that would force us to make assumptions on the distribution of x (the data) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, it is worth noting that with the standard assumption of x ∼ N(⃗0d, Id), the expected squared prediction error is the same as the expected squared norm of the error in estimating θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 4 Theorem 1 (Bias2 and Variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, (i) the squared bias is: ��Eη[ϵS(ξ)] ��2 = r � j=1 � ⟨θ∗, uj⟩ �2 � λ/σ2 j 1 + λ/σ2 j �2� 1 + ξ 1 + λ/σ2 j �2 + d � j=r+1 � ⟨θ∗, uj⟩ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 (11) (ii) the variance is: Eη ���ϵS(ξ) − Eη[ϵS(ξ)] ��2� = γ2 λ � r � j=1 λ/σ2 j � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j ��2� , (12) where γ2 is the per-coordinate label noise variance (as per Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The proof of Theorem 1 is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Remark 1 (Bias-Variance Tradeoff as a Function of ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us restrict our attention to ξ ∈ [0, 1] which is the range of ξ used in practice [Lopez-Paz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2017, Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (11), note that ��Eη[ϵS(ξ)] ��2 is an increasing function of ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the bias increases as the student tries to imitate the teacher more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (12), we see that Eη ���ϵS(ξ) − Eη[ϵS(ξ)] ��2� is a decreasing function of ξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the variance (due to label noise) reduces as the student tries to imitate the teacher more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, SD is associated with a bias-variance tradeoff – a higher value of the imitation parameter ξ mitigates the impact of label noise variance at the cost of increasing the estimation bias (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Plugging in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (11) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (12) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (10), we obtain Eη[∥ϵS(ξ)∥2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' note that it is a quadratic function of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 provides the optimal value of ξ, say ξ∗, that minimizes Eη[∥ϵS(ξ)∥2] (obtained by simple differentiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let cj := λ/σ2 j and θ∗ j := � ⟨θ∗, uj⟩ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then: ξ∗ = arg minξ∈REη[∥ϵS(ξ)∥2] = �r j=1 � γ2 λ − θ∗ j � c2 j (1+cj)3 �r j=1 � γ2 λ cj + θ∗ j � c2 j (1+cj)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (13) Thus, setting ξ = ξ∗ yields the optimal balance between the squared bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Remark 2 (Anti-Learning Observed Labels in Noisy Settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' There are scenarios when ξ∗ obtained in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 is more than 16, especially when γ is large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', there is a lot of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', note that limγ→∞ ξ∗ = �r j=1 c2 j/(1+cj)3 �r j=1 c3 j/(1+cj)4 > 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' However, the imitation parameter ξ is restricted to and tuned in [0, 1] [Lopez-Paz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2015,Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2017,Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Based on our analysis, we advocate not restricting ξ ∈ [0, 1] and also trying ξ > 1 in the high noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Setting ξ > 1 can be interpreted as “anti-learning” (or going against) the observed labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, we provide empirical evidence showing that ξ > 1 works better than ξ ≤ 1 even with the cross-entropy loss for several noisy datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5Note the �d j=r+1 � ⟨θ∗, uj⟩ �2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' If r < d, then this quantity is equal to the squared norm of the component of θ∗ along the non-empty null-space of XT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' this component is not recoverable by any algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 6ξ∗ can be negative too, but we shall not focus on this case in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 7This is because �r j=1 c3 j (1+cj)4 = �r j=1 cj (1 + cj) � �� � <1 � c2 j (1+cj)3 � < �r j=1 c2 j (1+cj)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5 Remark 3 (Utility of Teacher’s Predicted Labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Proposition 1 (Appendix B), we show that ξ∗ is an increasing function of the label noise variance γ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', we should assign more weight to the teacher’s predicted labels as γ2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So in linear regression, the benefit of using the teacher’s predictions (which is the core idea of SD) increases with the degree of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We make a similar observation in our experiments on multi-class classification in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2, where SD with ξ = 1 – which corresponds to only using the teacher’s predictions (and completely ignoring the original labels) – does not yield any gains (over the teacher) with zero label corruption but it consistently yields higher gains as the amount of label corruption increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Is Optimal Self-Distillation Better than Optimal ℓ2 Regularization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let e(λ, ξ) := Eη � ∥ϵS(ξ)∥2� (recall ϵS(ξ) is a function of the ℓ2-regularization parameter λ too).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since ξ = 0 corresponds to using plain ℓ2 regularization, we define ereg(λ) := e(λ, 0) as the estimation error obtained using only ℓ2 regularization (and no SD) with parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, let us define esd(λ) as the error obtained using SD with ℓ2-regularization parameter = λ and the optimal value of ξ = ξ∗ from Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 (which is itself a function of λ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', esd(λ) := e(λ, ξ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' By definition, esd(λ) ≤ ereg(λ) ∀ λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we wish to know when and if minλ esd(λ) < minλ ereg(λ) (note the strict inequality), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', when and if optimal SD is better than optimal ℓ2-regularization by tuning over λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let λ∗ reg := arg minλereg(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It holds that esd(λ∗ reg) = ereg(λ∗ reg) and desd(λ) dλ �� λ=λ∗reg = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', λ∗ reg is a stationary point of esd(λ) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It is a local maximum point of esd(λ) when: r � k=1 k−1 � j=1 σ2 j σ2 k(σ2 j − σ2 k)(θ∗ k − θ∗ j) (λ∗reg + σ2 j )4(λ∗reg + σ2 k)4 < 0, (14) with θ∗ j := (⟨θ∗, uj⟩)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' When the above holds, optimal self-distillation is better than optimal ℓ2-regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The detailed version and proof of Theorem 2 appear in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' One case when eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (14) holds is θ∗ 1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' > θ∗ r (since σ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ≥ σr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In general, when the squared projections of θ∗ along the most significant left singular vectors of X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the ones with “large” singular values) follow the same ordering as the corresponding singular values and the noise variance is large enough, λ∗ reg will be a local maximum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We formalize this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Without loss of generality, let ∥θ∗∥ = 1 and σ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, suppose σj ≤ δ for j ∈ {q + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , r} and θ∗ 1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' > θ∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, λ = λ∗ reg is a local maximum point of esd(λ) when δ ≤ O( 1 r) and γ2 ≥ maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',r} θ∗ j r−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The detailed statement and proof of Theorem 3 appear in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In practice, X is usually low rank and only a few of its singular values are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, the assumption of Theorem 3 is realistic and that too with q ≪ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To the best of our knowledge, there are no results comparable to Theorems 2 and 3 quantifying when optimal SD is better than optimal ℓ2 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now we consider a synthetic example to verify the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose θ∗ = 1 √ 2 � u1 +u2 � , n > d = 100 and σj = 1 j for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , d} (so only few singular values are large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (14) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We consider 3 values of γ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='125, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5} & 10 values of λ = {2i−3γ2} with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Figure 1, we plot ereg(λ) and esd(λ) for these values of γ and λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see the figure caption for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' If esd(λ) does not have a local maximum at λ∗ reg, it is difficult to say whether λ∗ reg is a sub-optimal local minimum point or the global minimum point of esd(λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' also see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' If 6 2 2 2 1 20 21 22 23 24 25 26 27 ( / 2) 2 2 20 22 24 Estimation Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='125 ereg( ) esd( ) (a) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='125 2 2 2 1 20 21 22 23 24 25 26 27 ( / 2) 2 2 2 1 20 21 22 23 24 Estimation Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 ereg( ) esd( ) (b) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 2 2 2 1 20 21 22 23 24 25 26 27 ( / 2) 2 1 20 21 22 23 Estimation Error = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 ereg( ) esd( ) (c) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 Figure 1: Estimation errors of vanilla ℓ2 regularization ereg(λ) and SD esd(λ) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' λ for the synthetic example at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' As per Theorem 2, note that the global minimum of ereg(λ) is a local maximum of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Observe that minλ esd(λ) < minλ ereg(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, optimal SD does better than optimal ℓ2-regularization here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' λ∗ reg is the global minimum point of esd(λ), then optimal SD is not better than (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', does not yield any improvement over) optimal regularization because esd(λ∗ reg) = ereg(λ∗ reg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To complement this, we present the following result (proved in Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' There exists θ∗ and X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for any noise variance γ2, λ∗ reg is the global minimum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So there are cases when optimal SD does not yield any improvement over optimal regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 4 Logistic Regression We now move onto logistic regression with the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that linear probing [Alain and Bengio, 2016,Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2022] is the same as logistic regression with features obtained from a pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It is also worth mentioning here that our analysis for logistic regression is significantly different from and harder than linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Setting: We consider a binary classification problem where each sample x ∈ X has a dis- crete label y(x) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let the marginal distribution of the sample space (with support X) be denoted by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We assume that there is a feature map φ : X −→ � X and we have access to a sample in terms of its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We are given 2n pairs of data points in terms of features and corrupted labels {(φ(xi), ˆyi)}2n i=1, where each ˆyi ∈ {0, 1} and xi ∼ iid P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let the corresponding actual labels be {yi}2n i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we assume that the dataset is balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', |i : yi = 1| = |i : yi = 0| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, without loss of generality (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ), let yi = 1 for i ∈ S1 := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n} and yi = 0 7 for i ∈ S0 := {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' our training algorithms are not privy to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We consider the following corruption model: ˆn < n/2 samples of each class, chosen randomly, are provided to us with flipped labels (again, our training algorithms are not privy to this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', let: ˆyi = � � � � � � � � � � � 1 − yi for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn} � �� � :=S1,bad ∪ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn} � �� � :=S0,bad , yi for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n} � �� � :=S1,good ∪ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n} � �� � :=S0,good .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Define p := ˆn n as the label corruption fraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' note that p < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Our goal is to learn a separator for the data w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the actual labels by training a logistic regression model on {(φ(xi), ˆyi)}2n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, for a sample x with feature φ(x) ∈ � X, the prediction for the label y(x) is modeled as: P(y(x) = 1) = σ(⟨θ, φ(x)⟩), 8 (15) where θ ∈ � X is the parameter that we wish to learn, and σ(z) = 1 1+e−z for z ∈ R is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We use the binary cross-entropy loss for training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we denote this by BCE : [0, 1] × (0, 1) −→ R≥0 and it is defined as: BCE(q, ˆq) = − � q log(ˆq) + (1 − q) log(1 − ˆq) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (16) Next, we state our assumptions on the feature map φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Assumption 2 (Orthonormality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The features have unit norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ∥φ(x)∥2 = 1 ∀ x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, the space of samples in feature space with labels 0 and 1 are orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ⟨φ(x), φ(x′)⟩ = 0 ∀ x ∈ X, x′ ∈ X with different labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Assumption 2 ensures that the data is separable and indeed there exists a separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Assumption 3 (Feature Correlation in the Training Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ⟨φ(xi), φ(xi′)⟩ = c ∈ (0, 1) ∀ i ̸= i′ such that yi = yi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It is true that at face value, Assumption 3 seems strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Instead, an assumption in expectation like Ex,x′ � ⟨φ(x), φ(x′)⟩ ���x and x′ have the same label � = c is more realistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' let us call this As- sumption 3′ for the sake of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For n → ∞ and when the labels are corrupted randomly, we hypothesize that the average9 prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', soft score ∈ (0, 1) assigned to a particular class) of a model under Assumption 3′ is the same as that under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We provide empirical evidence to support this hypothesis in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, for large n, we argue that Assumption 3 is reasonable and an important case to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Teacher Model: To learn the logistic regression parameter, the teacher minimizes the ℓ2- regularized binary cross-entropy loss with the provided labels as its targets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the teacher’s objective is: fT(θ) = 1 2n 2n � i=1 BCE � ˆyi, σ � ⟨θ, φ(xi)⟩ �� + λ∥θ∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (17) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (17), λ > 0 is the ℓ2-regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The teacher’s estimated parameter is θ∗ T := arg minθfT(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The teacher’s predicted soft label for the ith sample is y(T) i := σ(⟨θ∗ T, φ(xi)⟩);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' these are used to train the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 8The bias term can be absorbed within the feature vector φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 9This is taken over the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 8 Student Model Trained Only with Teacher’s Soft Labels: Here we set the imitation parameter ξ = 1 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, the student minimizes the ℓ2-regularized binary cross-entropy loss with the teacher’s predicted soft labels as its targets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the student’s objective is: fS(θ) = 1 2n 2n � i=1 BCE � y(T) i , σ � ⟨θ, φ(xi)⟩ �� + λ∥θ∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (18) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (18), λ is the same ℓ2-regularization parameter that is used by the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The student’s estimated parameter is θ∗ S := arg minθfS(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Comparison of Student and Teacher We shall now characterize the conditions under which the student outperforms the teacher w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' classification accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' to our knowledge, this is the first result of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For the sake of avoiding any ambiguity, the teacher’s population accuracy is defined as 100 ∗ Ex∼P � 1 � y(x) = 1 � σ(⟨θ∗ T, φ(x)⟩) > 1 2 ��� %10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The student’s accuracy is defined similarly with θ∗ S replacing θ∗ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Theorem 5 (When is Student’s Accuracy > Teacher’s Accuracy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose we have access to the population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, let Assumptions 2 and 3 hold with c = Θ(1) in Assumption 3 (recall that c < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Define ˆλ := 2nλ and r := (1−c) 4ˆλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose λ is chosen so that ˆλ ∈ � 1−c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16, 1−c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 � , which corresponds to r ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' If the label corruption fraction p ∈ � max �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 − r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 � , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r � , then the student achieves 100% population accuracy (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the true labels), while the teacher only achieves a population accuracy of 100(1-p)% (again, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the true labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Discussion: In our setup, there exists 0 < plow < phigh < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 such that (i) when p ≤ plow, the teacher attains 100% accuracy and so there is no need for SD, (ii) when p ∈ (plow, phigh), the student attains 100% accuracy while the teacher attains 100(1 − p)% accuracy, and (iii) when p ≥ phigh, both the teacher and student attain 100(1 − p)% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The range of p in Theorem 5 ⊆ (plow, phigh);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' our range is more conservative than the actual range because we had to impose some more restrictions on p in order to control certain error terms in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Figure 2, we plot the teacher’s and student’s accuracies as a function of p for r = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4} obtained by exactly solving for θ∗ T and θ∗ S (through a computer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In all the cases, it can be seen that the range of p where the student outperforms the teacher as per Theorem 5 falls within the actual range of p where the student outperforms the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The detailed proof of Theorem 5 can be found in Appendix G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we now outline the key steps in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Step 1 (Details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It can be shown that the teacher’s learned parameter θ∗ T = arg minθfT(θ) = �2n i=1 αiφ(xi) for some real numbers {αi}2n i=1 which are known as the teacher’s dual-space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Lemma 2, we obtain expressions for {αi}2n i=1 which then enables us to obtain the teacher’s predicted soft labels � y(T) i �2n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, we get: y(T) i = � � � � � � � � � � � ˆλˆα for i ∈ S1,bad, 1 − ˆλα for i ∈ S1,good, 1 − ˆλˆα for i ∈ S0,bad, ˆλα for i ∈ S0,good, (19) 101(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, 1(z) = 1 if z is true and 0 if z is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 9 (a) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Derived bound in Theorem 5: p ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='423, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='475).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Derived bound in Theorem 5: p ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='461).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (c) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Derived bound in Theorem 5: p ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='378, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='444).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Figure 2: Comparison of student’s and teacher’s accuracies for different values of label corruption fraction p obtained by exactly solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (20) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (21) for the teacher and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (23) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (24) for the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We set c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 and n = 5000 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In all the cases, note that our predicted range of p where the student outperforms the teacher as per Theorem 5 falls within the actual range of p where the student outperforms the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' where α ≥ 0 and ˆα ≥ 0 are obtained by jointly solving: σ � cn � α − (α + ˆα)p) − (1 − c)ˆα � = ˆλˆα, (20) and σ � cn � α − (α + ˆα)p) + (1 − c)α � = 1 − ˆλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (21) We focus on the interesting case of: (a) p being large enough so that the teacher misclassifies the incorrectly labeled points (S1,bad ∪ S0,bad) because otherwise, there is no need for SD, and (b) ˆλ being chosen sensibly so that the teacher at least correctly classifies the correctly labeled points (S1,good ∪ S0,good) because otherwise, SD is hopeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Later in Step 3, we impose conditions on p (a lower bound) and ˆλ such that (a) and (b) hold by requiring ˆλˆα < 1 2 and ˆλα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Step 2 (Details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similar to the teacher in Step 1, in Lemma 3, we 10 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 100 90 Accuracy 80 70 60 Teacher Student 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 Label Corruption Fraction pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 100 90 Accuracy 80 70 60 Teacher Student 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 Label Corruption Fraction pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4 100 Teacher Student 90 Accuracy 80 70 60 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 Label Corruption Fraction pshow that the student’s predicted soft label for the ith sample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' y(S) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' turns out to be: y(S) i = � � � � � � � � � � � ˆλˆα + ˆλˆβ for i ∈ S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='bad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 1 − ˆλα − ˆλβ for i ∈ S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='good,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 1 − ˆλˆα − ˆλˆβ for i ∈ S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='bad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ˆλα + ˆλβ for i ∈ S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='good,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (22) where β ≥ 0 and ˆβ ≥ 0 (assuming ˆλˆα < 1 2 and ˆλα < 1 2) are obtained by jointly solving: σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � = ˆλˆα + ˆλˆβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (23) and σ � cn � β − (β + ˆβ)p) + (1 − c)β � = 1 − ˆλα − ˆλβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (24) Now note that if ˆλˆα + ˆλˆβ > 1 2 and ˆλα + ˆλβ < 1 2, then the student has managed to correctly classify all the points in the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we ensure this in Step 3 by upper bounding p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The tradeoff here is that the (1-0) accuracy of the student increases at the cost of decreased confidence in classifying the correctly labeled points compared to the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Step 3 (Details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now we come to the challenging part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To obtain a range for p, we need to analytically solve eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (20) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (21) for the teacher and then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (23) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (24) for the student, which is particularly challenging due to the non-linearity of the sigmoid function present in these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Our novel proof technique involves employing the first-order Maclaurin series expansion of the sigmoid function which enables us to bound α, ˆα, β and ˆβ as a function of p, ˆλ and c in a small range (while imposing some conditions on p and ˆλ to ensure the range is small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using this, we can bound the teacher’s and student’s predictions, and then impose conditions on p and ˆλ such that the teacher only correctly classifies the correctly labeled points and errs on all the incorrectly labeled points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ˆλα < 1 2 and ˆλˆα < 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Step 1) but the student correctly classifies all the points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ˆλα + ˆλβ < 1 2 and ˆλˆα + ˆλˆβ > 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Finally, since n → ∞, population accuracy → training accuracy (we formalize this at the end in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 Variability of Predictions of Student and Teacher Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 (Variability of predictions of points within the same class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Define ∆T := maxi̸=i′,yi=yi′ |y(T) i − y(T) i′ | as the teacher’s variability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the maximum difference between the teacher’s predictions on two points having the same ground truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similarly, ∆S := maxi̸=i′,yi=yi′ |y(S) i − y(S) i′ | is defined as the student’s variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Under the conditions of Theorem 5, ∆S < ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In other words, the student’s predictions are more homogeneous than the teacher’s predictions as per Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is analogous to SD mitigating the variance term due to label noise in linear regression (Remark 1) leading to smaller variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We prove Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 in Appendix H and corroborate it with empirical evidence in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5 Empirical Results For our experiments, we consider multi-class classification with the cross-entropy loss on sev- eral vision datasets available in PyTorch’s torchvision, namely, CIFAR-100 with 100 classes, Caltech-256 [Griffin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2007] with 257 classes, Food-101 [Bossard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2014] with 101 classes, 11 StanfordCars [Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2013] with 196 classes and Flowers-102 [Nilsback and Zisserman, 2008] with 102 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since Caltech-256 does not have any train/test split provided by default, we pick 25k random images from the full dataset to form the training set, while the remaining images form the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For all the datasets, we train a softmax layer on top of a pre-trained ResNet-34/VGG-16 model on ImageNet which is kept fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', we do linear probing on ResNet- 34/VGG-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' No data augmentation is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, we describe the different types of label corruption that we experiment on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Label Corruption Type 1 (Random Corruption): Suppose the set of labels is [C] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Consider a sample whose true label is c ∈ [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' A corruption level of 100p % means we observe this sample’s label as c with a probability of (1 − p) or some random i ∈ [C] \\ c with a probability of p/(C − 1) for each such i ̸= c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We call this random corruption11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Label Corruption Type 2 (Hierarchical Corruption [Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2018]): Here, the label corruption only occurs between semantically similar classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is a more realistic type of corruption compared to random corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' By default, CIFAR-100 comes with 20 super-classes each containing 5 semantically similar classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the super-class “fish” consists of aquarium fish, flatfish, ray, shark and trout, while the super-class “small mammals” consists of hamster, mouse, rabbit, shrew and squirrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Unfortunately, the other datasets do not have any semantically similar classes provided by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, we describe the exact corruption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Consider a sample whose true class is c and super-class is S = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , c|S|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' A corruption level of 100p % means we observe this sample’s label as c with a probability of (1 − p) or some random c′ ∈ S \\ c with a probability of p/(|S| − 1) (for each such c′ ̸= c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Following [Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2018], we call this hierarchical corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Label Corruption Type 3 (Adversarial Corruption): Instead of semantically similar classes, we determine “hard” classes for each class by looking at the output of the teacher in the noiseless case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', when there is no corruption) and induce label corruption only among these hard classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, in the noiseless case, for a sample x, let pT(x, c) be the teacher’s predicted probability of x belonging to class c ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, let Xc be the set of samples in the training set belonging to class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, for each class c, we compute νc = � 1 |Xc| � x∈Xc pT(x, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 1 |Xc| � x∈Xc pT(x, C) � ∈ RC, and define the k hardest classes for class c to be the indices in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , C} \\ c corresponding to the k largest values in νc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For our experiments, we take k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, we describe the corruption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Consider a sample whose true class is c and the set of hardest 5 classes for c is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' A corruption level of 100p % means we observe this sample’s label as c with a probability of (1 − p) or some random c′ ∈ S with a probability of p/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We call this adversarial corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Verifying Remark 2 In Remark 2, we advocated trying ξ > 1 in the high noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We shall now test our recommendation on several noisy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The teacher is trained with the ℓ2-regularized cross- entropy loss and the student’s per-sample loss is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (1) where ℓ is the ℓ2-regularized cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Following our theory setting, the teacher and student are both trained with the same ℓ2-regularization parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the common weight decay value (PyTorch’s ℓ2-regularization parameter) is set to 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that this weight decay value was the first one that we tried (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', it was not cherry-picked);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' in fact, we show results with other weight decay values in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We defer the remaining experimental details to Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 1, we list the 11This has been also called symmetric noise in prior work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] 12 student’s improvement over the teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test accuracy - teacher’s test accuracy)12 averaged across 3 different runs for different values of ξ with ResNet-34 and VGG-16 in the case of 50% random, hierarchical and adversarial corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In all these experiments, note that the value of ξ yielding the biggest improvement is > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Table 5 (in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1) shows results with 30% corruption in Stanford Cars and Flowers-102;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' even there, ξ > 1 does better than ξ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 Verifying Remark 3 In Remark 3, we claimed that the utility of the teacher’s predictions increases with the amount of label noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To demonstrate this, we train the student with ξ = 1 which corresponds to setting the teacher’s predicted soft labels as the student’s targets (just as we did in Section 4) and completely ignoring the provided labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' All other experimental details (including weight decay) are the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 2, we show the student’s improvement over the teacher averaged across 3 different runs for varying degrees and types of label corruption with ResNet-34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see the table caption for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 Verifying Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 We now provide empirical evidence for our claim of the student’s predictions being more homogeneous than the teacher’s predictions in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since our experiments are for the multi-class (and not binary) case, we look at a slightly different metric to quantify variability which we introduce next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For a sample x belonging to class c(x), let ˆpT(x) and ˆpS(x) be the teacher’s and student’s predicted probability of x belonging to c(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, let X ′ c be the set of samples in the test set belonging to class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' To quantify the variability of the teacher and student for class c, we look at maxx1,x2∈X ′c |ˆpT(x1) − ˆpT(x2)| and maxx1,x2∈X ′c |ˆpS(x1) − ˆpS(x2)|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the range of ˆpT(x) and ˆpS(x) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' x ∈ X ′ c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Figure 3, we plot the per-class variability as defined here for three of the cases of Table 2 covering all three types of label corruption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' please see the caption for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 12The individual accuracies of the teacher and student can be found in Appendix J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we omit them in the main text for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 13 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='55 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='73 % (a) 50% Random Corruption in Caltech-256 with ResNet-34 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 % (b) 50% Random Corruption in Caltech-256 with VGG-16 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % (c) 50% Hierarchical Corruption in CIFAR-100 with ResNet-34 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='42 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % (e) 50% Adversarial Corruption in Food-101 with ResNet-34 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 % (f) 50% Adversarial Corruption in Food-101 with VGG-16 Table 1: Average (± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test set accuracy teacher’s test set accuracy) with different values of the imitation parameter ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' recall that ξ = 0 corresponds to the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Observe that in all cases, the value of ξ yielding the biggest improvement is more than 1 (although in Food-101 with ResNet-34, ξ = 1 does just as well as ξ > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is consistent with our message in Remark 2, where we advocate trying ξ > 1 in the high noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 14 Corruption level Random corruption: Improvement of student Adversarial corruption: Improvement of student 0% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 10% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 50% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % (a) Caltech-256 (Random and Adversarial Corruption) Corruption level Random corruption: Improvement of student Hierarchical corruption: Improvement of student 0% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 30% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 50% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % (b) CIFAR-100 (Random and Hierarchical Corruption) Corruption level Random corruption: Improvement of student Adversarial corruption: Improvement of student 0% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 30% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 50% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % (c) Food-101 (Random and Adversarial Corruption) Table 2: ResNet-34 with ξ = 1: Average (± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test set accuracy - teacher’s test set accuracy) with different kinds and varying levels of label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Observe that as the corruption level increases, so does the improvement of the student over the teacher for all types of corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This shows that the utility of the teacher’s predictions (which is the core idea of SD) increases with the amount of label noise corroborating our claim in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 248 Class Number Teacher Student Caltech-256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='9 (a) Caltech-256 with 50% random corruption 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 Class Number Teacher Student CIFAR-100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='95 (b) CIFAR-100 with 50% hierarchical corruption 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 Class Number Teacher Student Food-101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='9 (c) Food-101 with 50% adversarial corruption Figure 3: ResNet-34 with ξ = 1: Comparison of the per-class variability of the teacher and student (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', range of the teacher’s and student’s predictions of belonging to the correct class, as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3) for three of the cases of Table 2 as a heat map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that a darker shade corresponds to a lower value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' in all the cases, the student’s heat map has a darker shade than the teacher’s heat map which means that the student has a smaller variability than the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is consistent with the claim in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 15 6 Conclusion In this work, we analyzed the utility of self-distillation (SD) in supervised learning with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Our main algorithmic contribution was introducing the idea of trying ξ > 1 in the high label noise regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' On the theoretical side, for a binary classification problem where some fraction of the sample’s labels are flipped, we quantified the range of label corruption fraction in which the student outperforms the teacher under some assumptions on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We also characterized when optimal SD is better than optimal regularization in linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' There are some limitations of our work which pave the way for interesting directions of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Our results in Section 4 for logistic regression are under Assumption 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' it would be nice to derive similar results under a weaker assumption such as in expectation (see Assumption 3′ in the discussion after Assumption 3) or by assuming that the feature inner products are bounded in some range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, our results for logistic regression are with ξ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' one could try to obtain results with a general ξ to shed some light on how to better tune ξ for noisy datasets, like we did for linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, our empirical results are with linear probing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' experiments with full network fine-tuning are left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 7 Acknowledgement This work was supported by NSF TRIPODS grant 1934932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' References [Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] Ahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Damianou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Lawrence, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', and Dai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (2019).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Knowledge distillation beyond model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In 2020 25th International Conference on Pattern Recognition (ICPR), pages 6136–6143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Stanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2021] Stanton, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Does knowledge distillation really work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34:6906–6919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' [Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2019] Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Cheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Gan, Z.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2020] Xie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Luong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', Hovy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', and Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Self-training with noisy student improves imagenet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10687–10698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 18 Appendix Contents Appendix A: Proof of Theorem 1 Appendix B: Behavior of ξ∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' γ2 Appendix C: Detailed Version and Proof of Theorem 2 Appendix D: Detailed Version and Proof of Theorem 3 Appendix E: Proof of Theorem 4 Appendix F: Empirical Motivation for Assumption 3 Appendix G: Proof of Theorem 5 Appendix H: Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Appendix I: More Empirical Results Appendix J: Detailed Empirical Results 19 A Proof of Theorem 1 With the SVD notation of X, we can rewrite ˆθS(ξ) (from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (9)) as: ˆθS(ξ) = r � j=1 ⟨θ∗, uj⟩ � 1 + λ/σ2 j � � 1 − ξ � λ/σ2 j 1 + λ/σ2 j �� uj + r � j=1 ⟨η, vj⟩/σj � 1 + λ/σ2 j � � 1 − ξ � λ/σ2 j 1 + λ/σ2 j �� uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (25) Also, since {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ud} forms an orthonormal basis for Rd, we have: θ∗ = d � j=1 ⟨θ∗, uj⟩uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (25): ϵS(ξ) = − r � j=1 ⟨θ∗, uj⟩ � λ/σ2 j 1 + λ/σ2 j �� 1 + ξ 1 + λ/σ2 j � uj − d � j=r+1 ⟨θ∗, uj⟩uj + r � j=1 ⟨η, vj⟩/σj � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j �� uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (26) Using Assumption 1, we have: Eη[ϵS(ξ)] = − r � j=1 ⟨θ∗, uj⟩ � λ/σ2 j 1 + λ/σ2 j �� 1 + ξ 1 + λ/σ2 j � uj − d � j=r+1 ⟨θ∗, uj⟩uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (27) Thus, using the orthonormality of {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ud}, we get: ��Eη[ϵS(ξ)] ��2 = r � j=1 � ⟨θ∗, uj⟩ �2 � λ/σ2 j 1 + λ/σ2 j �2� 1 + ξ 1 + λ/σ2 j �2 + d � j=r+1 � ⟨θ∗, uj⟩ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (28) Next: Eη ���ϵS(ξ) − Eη[ϵS(ξ)] ��2� = Eη ������ r � j=1 ⟨η, vj⟩/σj � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j �� uj ����� 2� (29) = r � j=1 Eη �� ⟨η, vj⟩ �2� σ2 j � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j ��2 (30) = r � j=1 vT j Eη � ηηT � vj σ2 j � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j ��2 (31) = γ2 � r � j=1 1 σ2 j � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (32) Equation (30) follows from the orthonormality of the uj’s, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (31) follows because the vj’s are independent of η from Assumption 1, and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (32) follows because Eη � ηηT � = γ2In from Assumption 1 and because vT j vj = 1 for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (32) slightly differently, we get: Eη ���ϵS(ξ) − Eη[ϵS(ξ)] ��2� = γ2 λ � r � j=1 λ/σ2 j � 1 + λ/σ2 j �2 � 1 − ξ � λ/σ2 j 1 + λ/σ2 j ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (33) 20 B Behavior of ξ∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' γ2 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ξ∗ (in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1) is an increasing function of γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let ρ = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then from Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1: ξ∗ = �r j=1 � ρ λ − θ∗ j � c2 j (1+cj)3 �r j=1 � ρ λcj + θ∗ j � c2 j (1+cj)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (34) Now, ∂ξ∗ ∂ρ = � �r j=1 c2 j (1+cj)3 �� �r j=1 θ∗ j c2 j (1+cj)4 � + � �r j=1 c3 j (1+cj)4 �� �r j=1 θ∗ j c2 j (1+cj)3 � λ � �r j=1 � ρ λcj + θ∗ j � c2 j (1+cj)4 �2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (35) Thus, ξ∗ is an increasing function of ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ C Detailed Version and Proof of Theorem 2 Theorem 6 (Detailed Version of Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The following hold with θ∗ j := � ⟨θ∗, uj⟩ �2 (and with ′ denoting the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' λ): esd(λ) = ereg(λ) − � e′ reg(λ) �2 h(λ) and e′ sd(λ) = e′ reg(λ) � 1 − 2e′′ reg(λ) h(λ) + e′ reg(λ)h′(λ) (h(λ))2 � , where ereg(λ) = r � j=1 λ2θ∗ j (λ + σ2 j )2 + d � j=r+1 θ∗ j+ r � j=1 γ2σ2 j (λ + σ2 j )2 and h(λ) = 4 r � j=1 �γ2 σ2 j + θ∗ j � σ4 j (λ + σ2 j )4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (36) Let λ∗ reg := arg minλereg(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, esd(λ∗ reg) = ereg(λ∗ reg) and e′ sd(λ∗ reg) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', λ = λ∗ reg is a stationary point of esd(λ) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It is a local maximum point of esd(λ) when: r � k=1 k−1 � j=1 σ2 j σ2 k(σ2 j − σ2 k)(θ∗ k − θ∗ j) (λ∗reg + σ2 j )4(λ∗reg + σ2 k)4 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (37) When the above holds13, optimal self-distillation is better than optimal ℓ2-regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that if λ = λ∗ reg is not a local maximum point of esd(λ), it could be a sub-optimal local minimum point or the global minimum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The other stationary points of esd(λ) are obtained by solving (this follows from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (36)): 1 − 2e′′ reg(λ) h(λ) + e′ reg(λ)h′(λ) (h(λ))2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (38) Unfortunately, it seems difficult to determine whether a root of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (38) or λ∗ reg will be the global minimum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' If λ∗ reg is the global minimum point of esd(λ), then optimal SD is not better than (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', does not yield any improvement over) optimal ℓ2-regularization as esd(λ∗ reg) = ereg(λ∗ reg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 13Also, assume that λ∗ reg ≥ 0 as the ℓ2-regularization parameter is supposed to be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (11) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (12) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (10) while using our notation of cj = λ/σ2 j and θ∗ j = � ⟨θ∗, uj⟩ �2 from Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, we get: e(λ, ξ) = r � j=1 θ∗ j � cj 1 + cj �2� 1 + ξ 1 + cj �2 + d � j=r+1 θ∗ j + γ2 λ � r � j=1 cj (1 + cj)2 � 1 − ξ � cj 1 + cj ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (39) Thus, ereg(λ) := e(λ, 0) = r � j=1 θ∗ j � cj 1 + cj �2 + d � j=r+1 θ∗ j + γ2 λ r � j=1 cj (1 + cj)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (40) Next, we compute esd(λ) := e(λ, ξ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' esd(λ) = ereg(λ) − � �r j=1 � θ∗ j − γ2 λ � c2 j (1+cj)3 �2 �r j=1 � γ2 λ cj + θ∗ j � c2 j (1+cj)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (41) Lemma 1 involves a little bit of algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' we prove it in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since the cj’s depend on λ, let us substitute cj in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (40) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (41) and rewrite them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ereg(λ) = r � j=1 λ2θ∗ j (λ + σ2 j )2 + d � j=r+1 θ∗ j + r � j=1 γ2σ2 j (λ + σ2 j )2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (42) esd(λ) = ereg(λ) − �� r � j=1 � λθ∗ j − γ2� σ2 j (λ + σ2 j )3 � �� � :=g(λ) �2��� r � j=1 �γ2 σ2 j + θ∗ j � σ4 j (λ + σ2 j )4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (43) Interestingly, it can be checked that g(λ) = 1 2e′ reg(λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' here ′ indicates the derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Plugging this in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (43), we get: esd(λ) = ereg(λ) − � e′ reg(λ) �2 h(λ) , where h(λ) = 4 r � j=1 �γ2 σ2 j + θ∗ j � σ4 j (λ + σ2 j )4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (44) Now note that: e′ sd(λ) = e′ reg(λ) � 1 − 2e′′ reg(λ) h(λ) + e′ reg(λ)h′(λ) (h(λ))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (45) Thus, e′ reg(λ) = 0 =⇒ e′ sd(λ) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', any stationary point of ereg(λ) is also a stationary point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, λ∗ reg := arg minλereg(λ) satisfies: e′ reg(λ∗ reg) = 2 r � j=1 � λ∗ regθ∗ j − γ2� σ2 j (λ∗reg + σ2 j )3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (46) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (45), e′ sd(λ∗ reg) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', λ = λ∗ reg is a stationary point of esd(λ) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We shall now show that λ = λ∗ reg can be a local maximum point of esd(λ) in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For that, we need to check the sign of e′′ sd(λ∗ reg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that: e′′ sd(λ∗ reg) = e′′ reg(λ∗ reg) � 1 − 2e′′ reg(λ∗ reg) h(λ∗reg) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (47) 22 The above follows by just differentiating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (45) and evaluating it at λ = λ∗ reg while using the fact that e′ reg(λ∗ reg) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also note that e′′ reg(λ∗ reg) > 0 as λ = λ∗ reg is a minimizer of ereg(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us now examine the sign of t = � 1 − 2e′′ reg(λ∗ reg) h(λ∗reg) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' After a bit of algebra: t = 1 − �r j=1 σ2 j (λ∗reg+σ2 j )4 � θ∗ jσ2 j + 3γ2 − 2λ∗ regθ∗ j � �r j=1 σ2 j (λ∗reg+σ2 j )4 � γ2 + θ∗ jσ2 j � = 2�r j=1 σ2 j (λ∗reg+σ2 j )4 � λ∗ regθ∗ j − γ2� �r j=1 σ2 j (λ∗reg+σ2 j )4 � γ2 + θ∗ jσ2 j � (48) The denominator of t is positive so we only need to analyze the sign of the numerator, �r j=1 σ2 j (λ∗reg+σ2 j )4 � λ∗ regθ∗ j − γ2� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' let us refer to it as t2 for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (46), we have that: λ∗ reg = γ2 �r j=1 σ2 j (λ∗reg+σ2 j )3 �r j=1 θ∗ j σ2 j (λ∗reg+σ2 j )3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (49) Using this, we get: t2 = � γ2 �r j=1 θ∗ j σ2 j (λ∗reg+σ2 j )3 � � �� � >0 � � j,k σ2 j σ2 k(θ∗ j − θ∗ k) (λ∗reg + σ2 j )4(λ∗reg + σ2 k)3 � � �� � :=t3 (50) Simplifying t3 a bit, we get: t3 = r � k=1 k−1 � j=1 σ2 j σ2 k(σ2 j − σ2 k)(θ∗ k − θ∗ j) (λ∗reg + σ2 j )4(λ∗reg + σ2 k)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (51) So, t3 < 0 =⇒ t2 < 0 =⇒ t < 0 =⇒ e′′ sd(λ∗ reg) < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' but this means λ = λ∗ reg is a local maximum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Proof of Lemma 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that e(λ, ξ) is a quadratic function of ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' specifically, it is of the form aξ2 + bξ + c, where: a = r � j=1 �γ2 λ cj + θ∗ j � c2 j (1 + cj)4 , b = 2 r � j=1 � θ∗ j − γ2 λ � c2 j (1 + cj)3 , and c = ereg(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (52) By simple differentiation, ξ∗ = arg minξ∈Re(λ, ξ) = − b 2a (which is what we obtained in Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' A little bit of algebra gives us: e(λ, ξ∗) = c − b2 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (53) Plugging in the values of a, b and c from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (52) in yields: esd(λ) := e(λ, ξ∗) = ereg(λ) − � �r j=1 � θ∗ j − γ2 λ � c2 j (1+cj)3 �2 �r j=1 � γ2 λ cj + θ∗ j � c2 j (1+cj)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (54) This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ 23 D Detailed Version and Proof of Theorem 3 Theorem 7 (Detailed Version of Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Without loss of generality, let ∥θ∗∥ = 1 and σ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Further, suppose σj ≤ δ for j ∈ {q + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , r} and θ∗ 1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' > θ∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, suppose λ∗ reg > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For any ν > 1, if δ ≤ 1 √ 2νr � mink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � and γ2 ≥ maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',r} θ∗ j ν−1 , then λ = λ∗ reg is a local maximum point of esd(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Theorem 3 is obtained by using ν = r in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Define vk := �k−1 j=1 σ2 j σ2 k(σ2 j −σ2 k)(θ∗ k−θ∗ j ) (λ∗reg+σ2 j )4(λ∗reg+σ2 k)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For λ = λ∗ reg to be a local maximum point of esd(λ), we must have �r k=1 vk < 0 as per Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us analyze vk for k > q first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using σk ≤ δ for k > q, (σ2 j − σ2 k) ≤ σ2 j ≤ σ2 1 = 1 for j < k and |θ∗ k − θ∗ j| ≤ ∥θ∗∥2 = 1, we get for k > q: |vk| ≤ δ2 k−1 � j=1 σ2 j (λ∗reg + σ2 j )4(λ∗reg + σ2 k)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (55) Now since λ∗ reg > 0, we can further simplify eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (55): |vk| ≤ δ2 k−1 � j=1 σ2 j (λ∗reg)8 = δ2 (λ∗reg)8 � q � j=1 σ2 j ���� ≤1 + r � j=q+1 σ2 j ���� ≤δ2 � ≤ δ2(q + rδ2) (λ∗reg)8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (56) Summing up eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (56) from k = q + 1 through to k = r, we get: r � k=q+1 vk ≤ r � k=q+1 |vk| ≤ rδ2(q + rδ2) (λ∗reg)8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (57) Let us now look at k ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since θ∗ 1 > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' > θ∗ q, we have that vk < 0 for all k ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that for each k ≤ q: vk ≤ σ2 k(1 − σ2 k)(θ∗ k − θ∗ 1) (λ∗reg + 1)4(λ∗reg + σ2 k)4 ≤ σ2 k(1 − σ2 k)(θ∗ k − θ∗ 1) (λ∗reg + 1)8 , (58) where the last step follows using λ∗ reg > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, q � k=1 vk ≤ 1 (λ∗reg + 1)8 q � k=1 σ2 k(1 − σ2 k)(θ∗ k − θ∗ 1) ≤ −q mink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � (λ∗reg + 1)8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (59) Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (57) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (59), we get: r � k=1 vk = q � k=1 vk + r � k=q+1 vk ≤ −q mink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � (λ∗reg + 1)8 + rδ2(q + rδ2) (λ∗reg)8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (60) So to ensure �r k=1 vk < 0, ensuring: rδ2(q + rδ2) (λ∗reg)8 < q mink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � (λ∗reg + 1)8 (61) suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This implies: λ∗ reg + 1 λ∗reg < � q mink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � rδ2(q + rδ2) �1/8 � �� � :=z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (62) 24 For any ν > 1, note that z > ν for δ2 < 1 2νrmink∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',q} � σ2 k(1 − σ2 k)(θ∗ 1 − θ∗ k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In that case, we must have λ∗ reg > 1 z−1, which can be ensured by having: λ∗ reg > 1 ν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (63) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (46), recall that λ∗ reg = γ2 �r j=1 σ2 j (λ∗reg+σ2 j )3 �r j=1 θ∗ j σ2 j (λ∗reg+σ2 j )3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now since λ∗ reg > 0, we have that: λ∗ reg ≥ γ2 �r j=1 σ2 j (λ∗reg+σ2 j )3 θ∗max �r j=1 σ2 j (λ∗reg+σ2 j )3 ≥ γ2 θ∗max , (64) where θ∗ max = maxj∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',r} θ∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using this, if γ2 > θ∗ max ν−1 , then λ∗ reg ≥ γ2 θ∗max > 1 ν−1 > 1 z−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ E Proof of Theorem 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We provide a 2-dimensional example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Take θ∗ = 1 √ 2(u1 + u2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' so, θ∗ 1 = θ∗ 2 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, suppose σ1 = 1 and σ2 = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For this case, we get (by using the formulas in Theorem 6): ereg(λ) = λ2 2 � 1 (λ + 1)2 + 16 (4λ + 1)2 � + γ2 � 1 (λ + 1)2 + 4 (4λ + 1)2 � , (65) and e′ reg(λ) = (λ − 2γ2) � 1 (λ + 1)3 + 16 (4λ + 1)3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (66) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (66), we have that λ∗ reg = arg minλ>0ereg(λ) = 2γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From Theorem 6, we have that: e′ sd(λ) = e′ reg(λ) � 1 − 2e′′ reg(λ) h(λ) + e′ reg(λ)h′(λ) (h(λ))2 � , where h(λ) = � 4γ2 + 2 (λ + 1)4 + 256γ2 + 32 (4λ + 1)4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (67) After a lot of algebraic heavy lifting, we get: e′ sd(λ) = 288(λ − 2γ2)3 (λ + 1)5(4λ + 1)5 � 2γ2+1 (λ+1)4 + 128γ2+16 (4λ+1)4 �2 � 1 (λ + 1)3 + 16 (4λ + 1)3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (68) Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (68), we can conclude that arg minλ>0esd(λ) = 2γ2 = λ∗ reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ 25 F Empirical Motivation for Assumption 3 We consider the same logistic regression setting as Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that the Gram matrix K ∈ R2n×2n (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=')) is of the form K = � K1 0n×n 0n×n K0 � , where 0n×n is the n × n matrix of all 0’s and K1 and K0 are both PSD matrices with diagonal entries = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For our simulations, the diagonal elements of K1 are set equal to 1 and the off-diagonal elements are set equal to the corresponding off-diagonal element of 1 nZ1ZT 1 , where each element of Z1 ∈ Rn×n is drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' from (i) Unif[0, 1], and (ii) Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8)14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' K0 is constructed in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that K is PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In the case of (i) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', (ii)), the expected off-diagonal element of both K1 and K0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='64), and so we compare against Assumption 3 with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, for our two Gram matrices, we compare the average predictions (average being over the training set) of our logistic regression model against the corresponding predictions under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We consider four values of n, namely, 1000, 5000, 10000 and 50000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 3, we show results for (i) when p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 (top) and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='35 (bottom) with ˆλ = 1 − c (recall that ˆλ ∈ � 1−c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16, 1−c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 � as per Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 4, we show results for (ii) when p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 (top) and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 (bottom) with ˆλ = 1−c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='50 = 2(1 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Please see the table captions for a detailed discussion, but in summary, we conclude that Assumption 3 is a reasonable assumption to analyze the average behavior of a linear model on a large dataset under random label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 14If X ∼ Bernoulli(p), then P(X = 1) = p and P(X = 0) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 26 Teacher n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6400 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5287 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7200 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7195 10k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7198 50k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7200 Student n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6636 50k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6640 (b) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='35 Table 3: (i) Unif[0, 1]: Results (up to fourth decimal point) for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 (top) and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='35 (bottom) with ˆλ = 1 − c on points with true label = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' points with true label = 0 follow the same trend by symmetry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In the table, “bad” (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', “good”) points mean incorrectly (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', correctly) labeled points, and A3 is Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, “pred.” is the predicted probability of the label being 1 and “Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points” (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', “Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points”) is the empirical average over all bad (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', good) points with true label = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' please note that this is with the actual Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Under Assumption 3, all bad/good points have the same prediction (see Equations (19) and (22) or Lemmas 2 and 3) due to which the corresponding columns do not have the word “Avg.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Observe that as n increases, the average prediction for both good and bad points (with the actual Gram matrix) matches the corresponding predictions under Assumption 3 (and n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, Assumption 3 is a reasonable assumption to analyze the average behavior of a linear model on a large dataset under random label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 27 Teacher n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7333 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6220 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7332 10k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7332 50k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7333 Student n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7037 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7033 10k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7035 50k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='6913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7037 (a) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 Teacher n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8222 5k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8219 10k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8221 50k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8222 Student n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for bad points under A3 & n → ∞ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' for good points under A3 & n → ∞ 1k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7872 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8024 (b) p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 Table 4: (ii) Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='8): Same as Table 3 except for p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 (top) and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 (bottom) with ˆλ = 2(1−c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Just like in Table 3, as n increases, the average prediction for both good and bad points (with the actual Gram matrix) matches the corresponding predictions under Assumption 3 (and n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, Assumption 3 is a reasonable assumption to analyze the average behavior of a linear model on a large dataset under random label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 28 G Proof of Theorem 5 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Step 1 in Detail The teacher’s estimated parameter θ∗ T := arg minθfT(θ) satisfies ∇fT(θ∗ T) = 1 2n �2n i=1 � σ(⟨θ∗ T, φ(xi)⟩)− ˆyi � φ(xi) + λθ∗ T = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From this, we get: θ∗ T = 2n � i=1 1 2nλ � ˆyi − σ(⟨θ∗ T, φ(xi)⟩) � � �� � :=αi φ(xi) = 2n � i=1 αiφ(xi), (69) for some real numbers {αi}2n i=1 which are known as the teacher’s dual-space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Recall that we defined ˆλ := 2nλ in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Lemma 2 (Teacher’s Dual-Space Coordinates and Predictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose Assumptions 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then: αi = � � � � � � � � � � � −ˆα for i ∈ S1,bad, α for i ∈ S1,good, ˆα for i ∈ S0,bad, −α for i ∈ S0,good, (70) where α ≥ 0 and ˆα ≥ 0 are obtained by jointly solving: σ � cn � α − (α + ˆα)p) − (1 − c)ˆα � = ˆλˆα, (71) and σ � cn � α − (α + ˆα)p) + (1 − c)α � = 1 − ˆλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (72) Also, the teacher’s prediction for the ith sample, y(T) i , turns out to be: y(T) i = � � � � � � � � � � � ˆλˆα for i ∈ S1,bad, 1 − ˆλα for i ∈ S1,good, 1 − ˆλˆα for i ∈ S0,bad, ˆλα for i ∈ S0,good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (73) Lemma 2 is proved next in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' As mentioned in the proof sketch in the main text, we shall focus on the interesting case of: (a) p being large enough so that the teacher misclassifies the incorrectly labeled points because otherwise, there is no need for SD, and (b) ˆλ being chosen sensibly so that the teacher at least correctly classifies the correctly labeled points because otherwise, SD is hopeless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Later in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5, we shall impose a lower bound on p (in terms of c and ˆλ) so that (a) is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, the teacher misclassifies the incorrectly labeled points (with indices S1,bad = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn} and S0,bad = {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn}) when ˆλˆα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (74) Moreover, in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5, we shall also restrict ˆλ (in terms of c) so that (b) is ensured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Specifically, the teacher correctly classifies the correctly labeled points (with indices S1,good = {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n} and S0,good = {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}) when 1 − ˆλα > 1 2 =⇒ ˆλα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (75) 29 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 Proof of Lemma 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (69), we have: 2nλαi = ˆyi − σ � 2n � j=1 αj⟨φ(xj), φ(xi)⟩ � , (76) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' For ease of notation, let us define vi := �2n j=1 αj⟨φ(xj), φ(xi)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, the above equation can be rewritten as: 2nλαi = ˆyi − σ(vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (77) Note here that the teacher’s predictions are: y(T) i := σ(vi) = ˆyi − 2nλαi, (78) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, using Assumptions 2 and 3, we have: vi = � αi + c � j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',n}\\i αj = αi(1 − c) + c �n j=1 αj for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, αi + c �n j∈{n+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=',2n}\\i αj = αi(1 − c) + c �2n j=n+1 αj for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (79) Let us focus on i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let S = �n j=1 αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, we have the following equations: 2nλαi = −σ(αi(1 − c) + cS) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, (80) and 2nλαi = 1 − σ(αi(1 − c) + cS) for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (81) Using the monotonicity of the sigmoid function, we conclude that: αi = � −ˆα for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn} α for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, (82) for some α, ˆα ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using a similar argument, we can conclude that for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}: αi = � ˆα2 for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn} −α2 for i ∈ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}, (83) for some α2, ˆα2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We further claim that: α2 = α and ˆα2 = ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (84) Let us verify if this indeed holds up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that with such a solution: n � j=1 αj = − 2n � j=n+1 αj = α(n − ˆn) − ˆαˆn = αn − (α + ˆα)ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (85) Plugging this back in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (79) for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n} and then in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (77), we get (after a bit of rewriting): σ � − (1 − c)ˆα + cαn − c(α + ˆα)ˆn � = 2nλˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (86) σ � (1 − c)α + cαn − c(α + ˆα)ˆn � = 1 − 2nλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (87) Doing the same but for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n} with α2 = α and ˆα2 = ˆα, we get (again, after a bit of rewriting): σ � (1 − c)ˆα − cαn + c(α + ˆα)ˆn � = 1 − 2nλˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (88) 30 σ � − (1 − c)α − cαn + c(α + ˆα)ˆn � = 2nλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (89) Now note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (86) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (88), and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (87) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (89) are the same – this is because σ(−z) = 1 − σ(z) for all z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, our claim in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (84) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Hence, we can consider only eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (86) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (87), and solve them to find the two unknown variables α and ˆα in order to obtain θ∗ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Recalling ˆn = np, we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (86) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (87) as follows: σ � cn � α − (α + ˆα)p) − (1 − c)ˆα � = 2nλˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (90) σ � cn � α − (α + ˆα)p) + (1 − c)α � = 1 − 2nλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (91) Thus, we have: αi = � � � � � � � � � � � −ˆα for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, α for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, ˆα for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn}, −α for i ∈ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}, (92) where α and ˆα are obtained by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (90) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (78), recall that the teacher’s predictions for the ith sample is: y(T) i := ˆyi − 2nλαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (93) Now using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (92) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (93), we get: y(T) i = � � � � � � � � � � � 2nλˆα for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, 1 − 2nλα for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, 1 − 2nλˆα for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn}, 2nλα for i ∈ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (94) Replacing 2nλ with ˆλ in equations (90), (91) and (94), and plugging in S1,bad = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, S1,good = {ˆn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, S0,bad = {n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n+ ˆn} and S0,good = {n+ ˆn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n} throughout finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='3 Step 2 in Detail Just like eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (69) for the teacher, it can be shown that: θ∗ S = 2n � i=1 βiφ(xi), (95) for some real numbers {βi}2n i=1 which are known as the student’s dual-space coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Lemma 3 (Student’s Dual-Space Coordinates and Predictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Suppose Assumptions 2 and 3 hold, and the teacher correctly classifies the correctly labeled points but misclassifies the incorrectly labeled points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ˆλα < 1 2 and ˆλˆα < 1 2 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then: βi = � � � � � � � � � � � −ˆβ for i ∈ S1,bad, β for i ∈ S1,good, ˆβ for i ∈ S0,bad, −β for i ∈ S0,good, (96) 31 where β ≥ 0 and ˆβ ≥ 0 are obtained by jointly solving: σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � = ˆλˆα + ˆλˆβ, (97) and σ � cn � β − (β + ˆβ)p) + (1 − c)β � = 1 − ˆλα − ˆλβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (98) Also, the student’s prediction for the ith sample, y(S) i , turns out to be: y(S) i = � � � � � � � � � � � ˆλˆα + ˆλˆβ for i ∈ S1,bad, 1 − ˆλα − ˆλβ for i ∈ S1,good, 1 − ˆλˆα − ˆλˆβ for i ∈ S0,bad, ˆλα + ˆλβ for i ∈ S0,good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (99) We prove Lemma 3 in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now note that if ˆλˆα + ˆλˆβ > 1 2 and ˆλα + ˆλβ < 1 2, then the student has managed to cor- rectly classify all the points in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We ensure this in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 by imposing an upper bound on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4 Proof of Lemma 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The student’s estimated parameter θ∗ S = arg minθfS(θ) satisfies ∇fS(θ∗ S) = ⃗0, from which we get: θ∗ S = 2n � i=1 1 2nλ � y(T) i − σ(⟨θ∗ S, φ(xi)⟩) � � �� � :=βi φ(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (100) Thus the student’s ith dual coordinate βi (as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (95)) satisfies: 2nλβi = y(T) i − σ(⟨θ∗ S, φ(xi)⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (101) By following the same approach as the one we took in the proof of Lemma 2 for the teacher (with hard labels replaced by soft labels), we can show that: βi = � � � � � � � � � � � −ˆβ for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, β for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, ˆβ for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn}, −β for i ∈ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}, (102) where β ∈ R and ˆβ ∈ R are obtained by solving the following two equations: σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � = 2nλˆα + 2nλˆβ, (103) and σ � cn � β − (β + ˆβ)p) + (1 − c)β � = 1 − 2nλα − 2nλβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (104) We shall now show that β ≥ 0 and ˆβ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' We shall prove this by contradiction – specifically, by showing that the other cases lead to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Case 1: β ≤ 0 and ˆβ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this case: cn � β − (β + ˆβ)p) − (1 − c)ˆβ ≥ cn � β − (β + ˆβ)p) + (1 − c)β, (105) 32 which implies (by the increasing nature of the sigmoid function): σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � � �� � =2nλˆα+2nλˆβ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (103) ≥ σ � cn � β − (β + ˆβ)p) + (1 − c)β � � �� � =1−2nλα−2nλβ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (104) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (106) Now using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (103) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (104), we get: 2nλˆα + 2nλˆβ ≥ 1 − 2nλα − 2nλβ =⇒ 2nλˆα ≥ 1 − 2nλα −2nλ(β + ˆβ) � �� � ≥0 =⇒ 2nλˆα ≥ 1 − 2nλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (107) But this is a contradiction because as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (74) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (75), we had: 2nλˆα < 1 2 and 1 − 2nλα > 1 2 =⇒ 2nλˆα < 1 − 2nλα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (108) Hence, β ≤ 0 and ˆβ ≤ 0 is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Case 2: β ≥ 0 and ˆβ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this case: cn � β − (β + ˆβ)p) − (1 − c)ˆβ ≥ 0 =⇒ σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � � �� � =2nλˆα+2nλˆβ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (103) ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (109) Using the above and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (103), we get that: 2nλˆα + 2nλˆβ � �� � ≤0 ≥ 1 2 =⇒ 2nλˆα ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (110) But this is again a contradiction as 2nλˆα < 1 2 as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Hence, β ≥ 0 and ˆβ ≤ 0 is also ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Case 3: β ≤ 0 and ˆβ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In this case: cn � β − (β + ˆβ)p) + (1 − c)β ≤ 0 =⇒ σ � cn � β − (β + ˆβ)p) + (1 − c)β � � �� � =1−2nλα−2nλβ from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (104) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (111) Using the above and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (104), we get that: 1 − 2nλα − 2nλβ � �� � ≤0 ≤ 1 2 =⇒ 1 − 2nλα ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (112) But this is also a contradiction as 1 − 2nλα > 1 2 as per eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Hence, β ≤ 0 and ˆβ ≥ 0 is also ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, only β ≥ 0 and ˆβ ≥ 0 is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Recall that β and ˆβ are solutions to: σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � = 2nλˆα + 2nλˆβ, (113) and σ � cn � β − (β + ˆβ)p) + (1 − c)β � = 1 − 2nλα − 2nλβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (114) 33 Just like we obtained the teacher’s predictions � y(T) i �2n i=1, the student’s predictions are: y(S) i = � � � � � � � � � � � 2nλˆα + 2nλˆβ for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, 1 − 2nλα − 2nλβ for i ∈ {ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, 1 − 2nλˆα − 2nλˆβ for i ∈ {n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n + ˆn}, 2nλα + 2nλβ for i ∈ {n + ˆn + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (115) Finally, replacing 2nλ with ˆλ in equations (113), (114) and (115), and plugging in S1,bad = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , ˆn}, S1,good = {ˆn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n}, S0,bad = {n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , n+ ˆn} and S0,good = {n+ ˆn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' , 2n} throughout gives us the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 Step 3 in Detail Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Here, we shall obtain analytical expressions for the teacher’s and student’s predictions by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (71) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (72) (in Lemma 2) for the teacher and then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (97) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (98) (in Lemma 3) for the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Our approach will involve employing the first-order Maclaurin series expansion of the sigmoid function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' specifically, we will use: σ(z) = 1 2 + z 4 + ε(z), (116) where ε(z) is the residual error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that: ε(z) � � � � � < 0 for z > 0 or equivalently when σ(z) > 1 2 = 0 for z = 0 or equivalently when σ(z) = 1 2 > 0 for z < 0 or equivalently when σ(z) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (117) It also holds that ε(z) is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, sup z∈[−1,0] ε(z) = ε(−1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 or equivalently sup z:σ(z)∈[σ(−1),0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5] ε(z) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02, (118) and inf z∈[0,1] ε(z) = ε(1) > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 or equivalently inf z:σ(z)∈[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5,σ(1)] ε(z) > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (119) Let us start with the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (71) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (72) while using the Maclaurin series expansion of the sigmoid function (from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (116)) and the fact that σ(−z) = 1 − σ(z) ∀ z ∈ R, we have: ˆλˆα = σ � cn � α − (α + ˆα)p) − (1 − c)ˆα � = 1 2 + � cn � α − (α + ˆα)p) − (1 − c)ˆα 4 � + ε1, (120) and ˆλα = σ � − cn � α − (α + ˆα)p) − (1 − c)α � = 1 2 − � cn � α − (α + ˆα)p) + (1 − c)α 4 � + ε2, (121) for some real numbers ε1, ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Solving the above two equations in the limit of n → ∞, when c = Θ(1) and ˆλ < O(n) (this will be ensured subsequently), gives us: lim n→∞ α = p(1 + ε1 + ε2) ˆλ + 1−c 4 and lim n→∞ ˆα = (1 − p)(1 + ε1 + ε2) ˆλ + 1−c 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (122) Henceforth, we shall drop the limn→∞ notation, and it is implied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 34 Let us now bound ε1 + ε2 by imposing some more constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' First, recall from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (74) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (75) that we want ˆλˆα < 1 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the teacher does not correctly classify the incorrectly labeled points) and ˆλα < 1 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the teacher correctly classifies the correctly labeled points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now since we are solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (120) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (121), we must have ˆλˆα = σ � cn � α−(α+ ˆα)p)−(1−c)ˆα � < 1 2 and ˆλα = σ � − cn � α − (α + ˆα)p) − (1 − c)α � < 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' in this case, we must have that ε1 > 0 and ε2 > 0 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (117).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, we shall obtain upper bounds for ε1 and ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (121), if σ � − cn � α − (α + ˆα)p) − (1 − c)α � = ˆλα > σ(−1), then ε2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Note that since ε1 + ε2 > 0 and p < 1 2, ˆα > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So if ˆλα > σ(−1) holds, then so does ˆλˆα > σ(−1), in which case ε1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But using the fact that ε1 + ε2 > 0, having ˆλp ˆλ + 1−c 4 > σ(−1), (123) ensures ˆλα > σ(−1) (as well as, ˆλˆα > σ(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Recalling that r = (1−c)/4 ˆλ and using the fact that σ(−1) = 1 1+e, we get: p > 1 + r 1 + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (124) But we must also have p < 1 2 due to which we should have 1+r 1+e < 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' this holds when: r = (1 − c)/4 ˆλ < e − 1 2 =⇒ ˆλ > 1 − c 2(e − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (125) The above two conditions can be evaluated and simplified a bit more to get: p > 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 and r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 or ˆλ > 1 − c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4 , (126) and under these conditions, ε1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 and ε2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Combining all this, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (122) can be rewritten as (while also dropping the limn→∞ notation): α = p(1 + ζ) ˆλ + 1−c 4 and ˆα = (1 − p)(1 + ζ) ˆλ + 1−c 4 , (127) where ζ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, recall that we want ˆλα < 1 2 and ˆλˆα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since, ˆα > α, both these conditions can be satisfied by just ensuring ˆλˆα < 1 2 which itself can be ensured by imposing: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04ˆλ(1 − p) ˆλ + 1−c 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04(1 − p) 1 + r < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (128) The above is obtained by making use of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (127) and the fact that ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This gives us: p > 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (129) But again, we must have p < 1 2 due to which we should also have 1 − � 1+r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � < 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' this holds when: r = (1 − c)/4 ˆλ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 =⇒ ˆλ < 1 − c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (130) So to recap, for the teacher, we have: α = p(1 + ζ) ˆλ + 1−c 4 and ˆα = (1 − p)(1 + ζ) ˆλ + 1−c 4 , (131) 35 where ζ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04), with ˆλα < ˆλˆα < 1 2 for p > max � 1 − � 1+r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1+r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' All this is valid when r ∈ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 � or equivalently when ˆλ ∈ � 1−c 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='4 , 1−c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us do a sanity check to verify that the above range of p ensures ˆλα < ˆλˆα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' First, we shall show that ζ = ε1 +ε2 ≥ 0 by contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' so suppose ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (127), we have ˆλˆα = (1+ζ)(1−p) 1+r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04(1−p) 1+r < 1 2, where the last step follows because p > 1− � 1+r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But if ˆλˆα < 1 2, we must have ε1 > 0 (using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (117)) as we are solving ˆλˆα = σ � cn � α − (α + ˆα)p � − (1 − c)ˆα � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similarly, we must also have ε2 > 0 as ˆλα is also < 1 2 (which is easy to see because 0 < α < ˆα since p < 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But then ζ = ε1+ε2 > 0, which is a contradiction to our earlier supposition of ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Hence, we must have ζ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But then using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (127), we have ˆλα = (1+ζ)p 1+r > p 1+r > σ(−1), where the last step follows because p > 1+r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But if ˆλα > σ(−1), we must have ε2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 (using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (118)) as we are solving ˆλα = σ � − cn � α − (α + ˆα)p � − (1 − c)α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similarly, we must also have ε1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 as ˆλˆα is also > σ(−1) (again, because α < ˆα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Combining all this, we get ζ = ε1 +ε2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So, ˆλα < ˆλˆα = (1+ζ)(1−p) 1+r < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04(1−p) 1+r < 1 2, where the last step follows because p > 1 − � 1+r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So our prescribed range of p indeed ensures ˆλα < ˆλˆα < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us now move onto the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (97) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (98) while using the Maclaurin series expansion of the sigmoid function (from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (116)) and the fact that σ(−z) = 1 − σ(z) ∀ z ∈ R, we get: ˆλˆα + ˆλˆβ = σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � = 1 2 + � cn � β − (β + ˆβ)p) − (1 − c)ˆβ 4 � + ε3, (132) and ˆλα+ˆλβ = σ � −cn � β −(β + ˆβ)p)−(1−c)β � = 1 2 − � cn � β − (β + ˆβ)p) + (1 − c)β 4 � +ε4, (133) for some real numbers ε3 and ε4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Solving the above two equations in the limit of n → ∞ (when c = Θ(1) and ˆλ < O(n)) while using the values of α and ˆα from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (131), we get: lim n→∞ β = p ˆλ + 1−c 4 � − ˆλ(1 + ζ) ˆλ + 1−c 4 +(1+ζ′) � and lim n→∞ ˆβ = 1 − p ˆλ + 1−c 4 � − ˆλ(1 + ζ) ˆλ + 1−c 4 +(1+ζ′) � , (134) with ζ′ := ε3 + ε4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Again, we shall drop the limn→∞ notation subsequently, and it is implied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Next, we get: α + β = p ˆλ + 1−c 4 � ( 1−c 4 )(1 + ζ) ˆλ + 1−c 4 + (1 + ζ′) � , (135) and ˆα + ˆβ = 1 − p ˆλ + 1−c 4 � ( 1−c 4 )(1 + ζ) ˆλ + 1−c 4 + (1 + ζ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (136) Now, recall that if ˆλ(ˆα + ˆβ) > 1 2 and ˆλ(α + β) < 1 2, then the student has managed to correctly classify all the points in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us first impose ˆλ(ˆα + ˆβ) ∈ � 1 2, σ(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Then, since we are solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (132), σ � cn � β − (β + ˆβ)p) − (1 − c)ˆβ � ∈ � 1 2, σ(1) � , and so ε3 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02, 0) using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, we shall be imposing ˆλ(α + β) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Additionally, we ensured earlier that ˆλα > σ(−1) and showed in Lemma 3 that β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Therefore, we will have ˆλ(α + β) ∈ � σ(−1), 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 36 Since we are solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (133), σ � − cn � β − (β + ˆβ)p) − (1 − c)β � ∈ � σ(−1), 1 2 � , due to which ε4 ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02) using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, ζ′ = ε3 + ε4 ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Now, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (135) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (136), and plugging in r = (1−c)/4 ˆλ , we get: ˆλ(α + β) = p 1 + r �r(1 + ζ) 1 + r + (1 + ζ′) � , (137) and ˆλ(ˆα + ˆβ) = 1 − p 1 + r �r(1 + ζ) 1 + r + (1 + ζ′) � , (138) with ζ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04) and ζ′ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Let us first ensure ˆλ(ˆα + ˆβ) ∈ � 1 2, σ(1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using the bounds on ζ and ζ′, this can be ensured by having: 1 − p 1 + r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04r 1 + r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 � < σ(1) = e 1 + e, (139) and 1 − p 1 + r � r 1 + r + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='98 � > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (140) Solving and simplifying the above two equations gives us: p ∈ � 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1 + r)2 1 + 2r , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (141) Note that: 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1 + r)2 1 + 2r < 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r < 1 2 (142) for all r > 0, and so we are good here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' But recall that from the teacher’s analysis (see the discussion after eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (131)), we had p > max � 1− � 1+r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1+r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Combining everything, our current bound on p is: p ∈ � max � 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1 + r)2 1 + 2r � , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (143) But the above is only meaningful when the lower bound on p is smaller than the upper bound on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So we must find the range of r for which: 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � < 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r and 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 < 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 1− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1+r)2 1+2r is trivially smaller than 1− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1+r)2 1+2r so we do not need to worry about that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Combining the range of r obtained from the above equation with the previous range of r ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85) (that we obtained from the teacher), we get: r ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54] =⇒ ˆλ ∈ �1 − c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 , 1 − c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (144) Finally, we need to ensure ˆλ(α + β) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (137) and the bounds on ζ and ζ′, this can be ensured by imposing: p 1 + r �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04r 1 + r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 � < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (145) This can be simplified to: p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='485(1 + r)2 1 + 2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 37 But recall that we already have an upper bound on p of 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1+r)2 1+2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' It can be checked that 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1+r)2 1+2r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='485(1+r)2 1+2r for r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Thus, for r ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54] or ˆλ ∈ � 1−c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16, 1−c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 � , our bound on p remains the same as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (143), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', p ∈ � max � 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1 + r)2 1 + 2r � , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (146) Finally, to simplify our bound on p a bit, we consider r ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54], where: max � 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7(1 + r)2 1 + 2r � = max � 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (147) Thus, our final bound on p is: p ∈ � max � 1 − �1 + r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 � , 1 + r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 � , 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51(1 + r)2 1 + 2r � , (148) for r ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54] or ˆλ ∈ �1 − c 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 , 1 − c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (149) Finally, note that the prescribed range of ˆλ is < O(n) (as required in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (122) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (134)) since c = Θ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' So we are good here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Also, since n → ∞, the generalization gap (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', population accuracy - training accuracy) → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' see for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the margin bounds (with ℓ2-regularization) in [Kakade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 2008] where it is shown that the generalization gap goes down as O(1/√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Therefore, the population accuracy of the student (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', teacher) is the same as the training accuracy of the student (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', teacher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ H Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (73), we have: ∆T = 1 − ˆλ(α + ˆα), (150) where ˆλ = 2nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similarly, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (99), we have: ∆S = 1 − ˆλ(α + β + ˆα + ˆβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (151) Next, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (127) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (150), we get: ∆T = 1 − �1 + ζ 1 + r � , (152) where ζ ∈ (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04) and r = (1−c) 4ˆλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Similarly, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (137) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (138), we get: ∆S = 1 − 1 1 + r �r(1 + ζ) 1 + r + (1 + ζ′) � , (153) where ζ′ ∈ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (153) slightly, we get: ∆S = 1 − �1 + ζ 1 + r �� r 1 + r + 1 + ζ′ 1 + ζ � (154) ≤ 1 − �1 + ζ 1 + r ��0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 � (155) < 1 − �1 + ζ 1 + r � (156) = ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (157) In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (155), we have used the fact that r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 (from the condition of Theorem 5), ζ′ ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 and ζ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ■ 38 I More Empirical Results I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1 Verifying Remark 2 (Continued) In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, we compared the performance of different values of ξ with 50% corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 5, we show results with 30% corruption in Stanford Cars and Flowers-10215 with the same weight decay value as in Section 5 (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', 5 × 10−4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' even here, the improvement with ξ > 1 is more than that with ξ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Again, the individual accuracies of the teacher and student and the experimental details are in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ξ Improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ξ = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='24 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % (a) 30% Random Corruption in Stanford Cars with ResNet-34 ξ Improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', ξ = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % (b) 30% Adversarial Corruption in Flowers-102 with ResNet-34 Table 5: Average (± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test set accuracy teacher’s test set accuracy) with different values of the imitation parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Just like in Table 1, note that the value of ξ yielding the biggest improvement is more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 Results with Other Weight Decay Values All our previous results were with weight decay = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Here, we verify Remarks 2 and 3 for two other weight decay values which are 1 × 10−3 and 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (i) Verifying Remark 2: In Table 6, we list the student’s improvement over the teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test accuracy - teacher’s test accuracy) averaged across 3 different runs for different values of ξ in the case of (a) Caltech-256 with 50% random corruption & weight decay = 1 × 10−4 and (b) CIFAR-100 with 50% hierarchical corruption & weight decay = 1 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' As was the case with weight decay = 5 × 10−4 in Tables 1 and 5, note that the value of ξ yielding the biggest improvement here is also > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' (ii)Verifying Remark 3: The setup is the same as Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', the student is trained with ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' In Table 7, we show the student’s improvement over the teacher averaged across 3 different runs for varying degrees of label corruption in the case of (a) Caltech-256 with random corruption & weight decay = 1 × 10−4 and (b) CIFAR-100 with hierarchical corruption & weight decay = 1 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' As was the case with weight decay = 5 × 10−4 in Table 2, note that the improvement of the student (trained with ξ = 1) over the teacher increases as the corruption level increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The individual accuracies of the teacher and student and the experimental details appear in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 15For Flowers-102, we include the provided validation set in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 39 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 % (a) ResNet-34: 50% Random Corruption in Caltech-256 with weight decay = 1 × 10−4 ξ Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % (b) ResNet-34: 50% Hierarchical Corruption in CIFAR-100 with weight decay = 1 × 10−3 Table 6: Average (± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test set accuracy teacher’s test set accuracy) with different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Just like with weight decay = 5 × 10−4 (Tables 1 and 5), note that the value of ξ yielding the biggest improvement with both weight decay values here is more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is consistent with our message in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Corruption level Improvement of student over teacher 0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 10% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 50% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % (a) Random Corruption in Caltech-256 with weight decay = 1 × 10−4 Corruption level Improvement of student over teacher 0% −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % 50% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % (b) Hierarchical Corruption in CIFAR-100 with weight decay = 1 × 10−3 Table 7: ResNet-34 with ξ = 1: Average (± 1 std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=') improvement of student over teacher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', student’s test set accuracy - teacher’s test set accuracy) with varying levels of label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Just like with weight decay = 5 × 10−4 (Table 2), note that the improvement of the student over the teacher increases as the corruption level increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is consistent with our claim in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 40 J Detailed Empirical Results We list the individual accuracies of the teacher and student (along with the student’s improve- ment) corresponding to the results of Table 1 in Tables 8-13, Table 5 in Tables 14-15, Table 2 in Tables 16-21, Table 6 in Tables 22-23 and Table 7 in Tables 24-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Experimental Details: In all the cases, we use SGD with momentum = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='9 and batch size = 128 for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Since we are training only the softmax layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=', doing logistic regres- sion), we use an exponentially decaying learning rate scheme with decay parameter = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='98 (for every epoch) and the initial learning rate is tuned16 over {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' The maximum number of epochs is 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 % 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='58 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='55 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='73 % Table 8: Detailed Version of Table 1a (50% Random Corruption in Caltech-256 with ResNet-34) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 % Table 9: Detailed Version of Table 1b (50% Random Corruption in Caltech-256 with VGG-16) 16The tuning is done by picking the learning rate which yields the lowest training loss with the observed (noisy) labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' This is consistent with our theory setup where we assume convergence to the optimum of the training loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' the observed labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' 41 ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 50.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % Table 10: Detailed Version of Table 1c (50% Hierarchical Corruption in CIFAR-100 with ResNet- 34) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 % Table 11: Detailed Version of Table 1d (50% Hierarchical Corruption in CIFAR-100 with VGG-16) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % Table 12: Detailed Version of Table 1e (50% Adversarial Corruption in Food-101 with ResNet-34) 42 ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='46 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='42 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='71 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='76 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 % Table 13: Detailed Version of Table 1f (50% Adversarial Corruption in Food-101 with VGG-16) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='24 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % Table 14: Detailed Version of Table 5a (30% Random Corruption in Stanford Cars with ResNet-34) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='24 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='41 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='44 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='31 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='50 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % Table 15: Detailed Version of Table 5b (30% Adversarial Corruption in Flowers-102 with ResNet- 34) 43 Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 10% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 30% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 % 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 50% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % Table 16: Detailed Version of Random Corruption in Caltech-256 with ResNet-34 and ξ = 1 (Table 2a) Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 10% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 % 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='36 % 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 % 50% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % Table 17: Detailed Version of Adversarial Corruption in Caltech-256 with ResNet-34 and ξ = 1 (Table 2a) Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 10% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 % 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 30% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 50% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % Table 18: Detailed Version of Random Corruption in CIFAR-100 with ResNet-34 and ξ = 1 (Table 2b) Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 10% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 30% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 50% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % Table 19: Detailed Version of Hierarchical Corruption in CIFAR-100 with ResNet-34 and ξ = 1 (Table 2b) Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 10% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 30% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 50% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % Table 20: Detailed Version of Random Corruption in Food-101 with ResNet-34 and ξ = 1 (Table 2c) 44 Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 10% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 30% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 50% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % Table 21: Detailed Version of Adversarial Corruption in Food-101 with ResNet-34 and ξ = 1 (Table 2c) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='16 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='33 % 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='43 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 % 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='60 % Table 22: Detailed Version of Table 6a (50% Random Corruption in Caltech-256 w/ ResNet-34 and wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' decay = 1 × 10−4) ξ Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 (=Teacher) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='08 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % Table 23: Detailed Version of Table 6b (50% Hierarchical Corruption in CIFAR-100 w/ ResNet-34 and wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' decay = 1 × 10−3) 45 Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='02 % 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 10% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='13 % 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='11 % 50% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='18 % 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='06 % 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='17 % Table 24: Detailed Version of Table 7a (Random Corruption in Caltech-256 w/ ResNet-34, ξ = 1 and wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' decay = 1 × 10−4) Corruption level Teacher’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Student’s test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' Improvement of student over teacher 0% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='01 % −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 10% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='07 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='10 % 30% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='12 % 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='09 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='19 % 50% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='05 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content='04 % Table 25: Detailed Version of Table 7b (Hierarchical Corruption in CIFAR-100 w/ ResNet-34, ξ = 1 and wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} +page_content=' decay = 1 × 10−3) 46' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFQT4oBgHgl3EQfWTbR/content/2301.13304v1.pdf'} diff --git a/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/2301.11272v1.pdf.txt b/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/2301.11272v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbaf986e761a5c3e265fd53241fafdb0f8800c19 --- /dev/null +++ b/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/2301.11272v1.pdf.txt @@ -0,0 +1,1732 @@ +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +1 +Location-based Activity Behavior Deviation +Detection for Nursing Home using IoT Devices +Billy Pik Lik Lau, Member, IEEE, Zann Koh, Yuren Zhou, Member, IEEE, Benny Kai Kiat Ng, +Chau Yuen, Fellow, IEEE, and Mui Lang Low +Abstract—With the advancement of the Internet of Things(IoT) +and pervasive computing applications, it provides a better oppor- +tunity to understand the behavior of the aging population. How- +ever, in a nursing home scenario, common sensors and techniques +used to track an elderly living alone are not suitable. In this +paper, we design a location-based tracking system for a four-story +nursing home - The Salvation Army, Peacehaven Nursing Home +in Singapore. The main challenge here is to identify the group +activity among the nursing home’s residents and to detect if they +have any deviated activity behavior. We propose a location-based +deviated activity behavior detection system to detect deviated +activity behavior by leveraging data fusion technique. In order +to compute the features for data fusion, an adaptive method +is applied for extracting the group and individual activity time +and generate daily hybrid norm for each of the residents. Next, +deviated activity behavior detection is executed by considering +the difference between daily norm patterns and daily input data +for each resident. Lastly, the deviated activity behavior among the +residents are classified using a rule-based classification approach. +Through the implementation, there are 44.4% of the residents do +not have deviated activity behavior, while 37% residents involved +in one deviated activity behavior and 18.6% residents have two +or more deviated activity behaviors. +Index Terms—Internet of Things, Deviated Activity Behavior +Detection, Data Fusion, Location-based Sensing, Nursing Home +Monitoring +I. INTRODUCTION +Over the past few years, the advancement of the Internet +of Things (IoT) technology has opened up a lot of research +potential in the area of tracking and monitoring. Among +them, a wide variety of projects have been carried out to +monitor the behavior of the human being as shown in [1]–[5]. +These technologies made room for implementing convenient +applications for enhancing day to day living of urban residents. +With the increase of the world aging population as shown +in [6] and [7], research in monitoring the elderly has gained +attention from different research principles. While these +works [8]–[11] address the support of the elderly as indepen- +dent beings of the society, and others [12], [13] have provided +the facility for the nursing home to monitor the daily activity +behavior of the residents. The former method commonly +leverages boundary-less tracking and monitoring techniques +as shown in [4], [5], which include smartphones and smart +Billy Pik Lik Lau, Zann Koh, Yuren Zhou, Benny Kai Kiat Ng, +and Chau Yuen are with the Engineering Product Development, Sin- +gapore University of Technology and Design, Corresponding E-mail: +billy lau@mymail.sutd.edu.sg, yuenchau@sutd.edu.sg. +Mui Lang Low is with the Peacehaven Nursing Home Day Centre run by +The Salvation Army. +Manuscript received January 25, 2023 +wearable devices. The majority of the latter approaches [12], +[13] mostly provide tracking in a confined environment, where +the accuracy of the boundary-less approach is not ideal. Our +work focuses on the latter approach, where the constraints +of monitoring senior citizens are limited to a nursing home. +Traditionally, it is labor-intensive to take care of the day- +to-day life of an elderly resident, and it is not possible to +constantly track an individual across 24 hours. Therefore, +using a building-scale human monitoring approach, it can +assist the nursing home staff to monitor residents and better +understand their activity behavior. +With this challenge in mind, we design a system to monitor +the deviated activity behavior of nursing home’s residents +leveraging IoT technology. We use bluetooth low energy +(BLE) technology as backbone for collecting the elderly +data due to nature of low energy, and coverage suitable for +indoor application compared to WiFi, RFID, ZigBee, etc. The +deviated activity behavior denotes a nursing home’s resident +behaves irregularly compared to his/her normal routine of daily +life. This type of detection only can be achieved through fully +understanding a resident’s life routine. The main objectives of +such a system are to identify the residents’ activity behavior +and determine the irregular activity behaviors. The constraints +of monitoring residents’ activity behavior in a nursing home +are bounded by building structure, and also their daily activity +is influenced by the group activities or community. Therefore, +our aim is to differentiate their activity between private and +group activity when computing their deviated activity behav- +ior. Another constraint when designing this system is that we +do not have ground truth on the data collected, which resulting +the accuracy of system output cannot be validated. Moreover, +the identity of the nursing home residents is anonymized to +comply with Singapore Personal Data Protection Act [14]. +Thus, unsupervised knowledge extraction is more desired +when compared to the supervised knowledge extraction model. +To address the aforementioned challenges, in this paper, +we present a building-scale monitoring system to study 50 +residents’ activity behavior in the Peacehaven Nursing Home, +Singapore. Using the building-scale monitoring system, resi- +dents’ activity behavior based on their location are investigated +using a wearable card tag with a build-in Bluetooth beacon. +Each room is equipped with a receiver to detect the Bluetooth +beacon transmitted to perform the resident’s location detection. +Based on the detected location, we study the activity behavior +of residents over 6 months and cluster them based on their +common patterns. In order to identify the normal activity +behavior, we use a data fusion method to generate the hybrid +arXiv:2301.11272v1 [cs.CY] 25 Jan 2023 + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +2 +norm by combining the group and individual norm. Using +the hybrid norm, deviated activity behavior can be extracted, +which does not follow the normal daily pattern of a resident. +Afterward, we perform empirical analysis on the deviated +activity behavior and classify them. +The key contributions of this paper are as follows: +• We study the resident’s activity behavior in a nursing +home from a location-based implementation of a moni- +toring system. +• We propose a data fusion method to identify the daily +norm for each nursing home’s resident based on two data +sources, which are individual and group norm. +• Based on the daily norm generated, we perform empirical +analysis on the deviated activity behavior and identify the +types of them using rules-based classification. +Our paper can be detailed as follows: In Section II, we study +related work about existing methodologies in detecting devi- +ated activity behavior with their pros and cons. Subsequently, +in Section III, the system architecture and data processing +model is presented. Afterward, we describe the group activity +behavior clustering method in Section IV. Based on the group +detected, we compute the deviated activity behavior utilizing +the hybrid norm and analyze them in Section V. Lastly, we +conclude our work in Section VI. +II. RELATED WORK +In this section, we discuss some of the related works in +the field, which are types of monitoring techniques used to +achieve human monitoring and methodologies applied to study +deviated activity behavior. +A. Types of Monitoring Techniques +There are four common types of monitoring techniques in +the literature, which are (1) people-driven, (2) event-driven, +(3) location-driven, and (4) data-driven. +The people-driven monitoring technique uses humans as the +main source of generating information, which normally in- +volves sensors installed in smartphones, watch, bracelets, and +other types of wearable. It is commonly not restricted by loca- +tion and has a wider coverage of sensing capability. Examples +of smartphone monitoring techniques can be found in [4], [15], +[16], where common sensors used are accelerometer, GPS, +microphone, etc. Examples of other types of wearable devices +are belt equipped with sensing unit [12] and bracelets [11]. +Generally, these types of monitoring techniques are intrusive +but able to capture good accuracy data. +The event-driven monitoring technique uses the activity of +daily life (ADL) of the users and attempts to understand the +activity behavior of the targeted user. For instance, Alcala et. +al. [17] uses the hidden Markov model (HMM) to process +ADL and detect the deviated activity behavior from the daily +routine, while Zerkouk et.al. [18] use a long term short +term memory-based model to identify deviated routine among +senior citizen. A detailed review of the ADL monitoring +approaches can be found in [19]. The downside of the events- +driven monitoring technique is that detailed data is desired and +requires a lot of effort as incomplete information will mislead +the study outcome. +When monitoring techniques involve installing multiple +sensors at a particular location or building, often it is known as +a location-driven method of monitoring people. The coverage +of the monitoring often involves a building or a particular area, +and commonly used sensors include motion sensors [20], vi- +sion [21], RFID sensors [22], infra-red [23], WiFi-passive [24]. +However, the inconvenience of this monitoring approach is +limited to the area coverage since it is location-bound and only +limited study scenarios would benefit from such an approach. +The data-driven approach uses various information sources +and combines them to infer human activity behavior. It can be +a mixture of different data sources as described in [25] such +as physical sensors or cyber data sources such as social media. +Examples of data fusion driven approaches can be found in +these works [26], [27], where multiple sensors are fused to +study human behavior. The disadvantages of this approach are +due to the complexity of the model and domain knowledge +required to select relevant information sources to combine. +Besides, every data sources require different preprocessing +techniques, which can be rather tedious and challenging. +B. Methodologies in Deviated Activity Behavior Extraction +The common methodologies in studying the deviated ac- +tivity behavior of senior citizens can be categorized into the +following: (1) prediction model, (2) state estimation model, +and (3) clustering and exploration model. +The prediction model utilizes statistics to predict the po- +tential activity behavior of a particular user and if the pre- +dicted behavior does not match the predictive outcome, it will +be labeled as deviated activity behavior. Recent prediction +methods such as long short term memory (LSTM) can be +found in [18], which detect the deviated activity behaviors +among the senior citizens using a deep learning approach. +Other types of statistical predictive models also can be found +in [8], [20], [28], [29]. The predictive model generally requires +good quality and a large amount of data as it is not ideal to +perform a model with limited or noisy data. +State estimation modeling maps the state behavior of a +particular user into a system state, which can be used to model +the users’ activity behavior and detect any deviated activity +behavior. State estimation usually requires human intervention +to map the total state of the given system, which involves +domain experts to carry out such tasks. In [30], Novak et. +al. performed the Self Organizing Maps (SOM) and Makrov +prediction model onto ADL of a senior citizen to detect their +deviated activity behavior. Another example of state estimation +modeling can be found in [31], where they detect the deviated +activity behavior using HMM. Other state estimation examples +can be found in [32], [33]. The drawback of this approach can +be rather complex since it does not consider the relationship +between activities that happened in parallel. +The clustering and exploration model normally use four +steps to generate an insights extraction model, which the user’s +deviated activity behavior is studied during the exploration +phase. It is first proposed in [34] and commonly used when +there is no ground-truth available or no prior knowledge +regarding deviated activity behavior. An example of this +approach can be found in [11], where Kurnianingsih et. al. + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +3 +use hybrid k-means clustering and isolation forest to detect +the deviated vital signals among senior citizen. In [22], authors +also use k-means clustering to formulate normal pattern and +if any event does not fit into the cluster, it will be labeled as +deviated activity behavior. Meanwhile, authors [4] studied the +activity behavior of senior citizens using k-means clustering +approach and decision tree to generate features for analyzing +the users’ demographic. The main drawback of this method +is that it requires extensive knowledge in specific domains +when analyzing potential deviated activity behavior, however +it works effectively when there is no ground-truth available. +Bluetooth Beacon Receiver +× 138 sensors +... +Card 1 +Card 2 +Card n +(a) Card tags with Bluetooth beacon (Transmitter) and +Redbear Duo (Receiver). +Basement 1 +Level 1 +Level 2 +Level 3 +Residents staying area, activity/common area +Residents staying area, activity/common area +Nursing Home Building Story +Staff area (Restricted area) +activity/common area +(b) The nursing home building’s floor level, which can +be divided into 4 story. Note that Level 2 and 3 consists +of residential staying area and common area, where +Basement 1 only has common activity area. Level 1 is +the nursing staff area, where elderly normally do not +have access to that area. +Fig. 1: Hardware used to setup the monitoring system. +III. SYSTEM DESIGN +In this paper, we focus on the location-driven monitoring +techniques since the resident of the nursing home stay within +the premises. Therefore, building-wide monitoring is more +desirable in our studies. Given that collecting ground-truth +appears to be impossible in our problem, the insights explo- +ration approach and unsupervised machine learning method is +more appropriate. In this section, the overall system design of +the deviated activity behavior detection system is presented +followed by the data specification and data preprocessing +steps. +A. Hardware Setup +The proposed hardware setup comprises of two crucial +components installed in the nursing home, which are card +tags with BLE beacons and beacon receiver. The Bluetooth +beacon card model is shown in Fig. 1a, which is capable of +transmitting beacon every 1000ms using Nordic nRF52 chip +with a range of up to 40 meters. +Algorithm 1 Location Detection Algorithm +Data: mac Address List, RSSI List, loc ID +Result: user list, location list, timeStamp +function listenData() +1. Perform detection cycle as follows: +for detection cycle from 1 to 5 do +while timer < 3 sec do +loc ID, RSSI List +Filter mac Address List ≤ −70dBm +for unique resident in loc ID do +location list = getHighestRSSI(loc ID, RSSI List) +Store the userlist and location +2. Compute the final location after detection cycle ended +foreach unique resident in userlist do +determine the location based on the strongest RSSI +if computeLocation(location) ̸= NULL then +final location ← computeLocation(location) +else +Retrieve last location from database +final location ← last location +3. update the resident final location and current timestamp +function getHighestRSSI(loc ID, RSSIList) +last location ← initial location +last RSSI ← initial RSSI +index ← 0 +foreach RSSI in RSSI List do +if RSSI > Last RSSI then +last RSSI ← RSSI +last location ← loc ID[index] +index ← index +1 +return last location +function computeLocation(locationList) +initialize location dict +foreach location in locationList do +update location dict count with location +if max(count(location dict)) exists then +return location in max(count(location dict)) +else +return NULL +It is equipped with a battery and capable of transmitting +the data for up to 6 months without recharging. To provide +sufficient coverage for the nursing home, each room has a +beacon receiver installed and pick up the Bluetooth beacon +transmitted. To filter out the irrelevant Bluetooth devices, a +unique identifier is assigned to each of the card tags. The +building studied as illustrated in Fig. 1b consists of 4 levels +with the top 2 levels served as the residential area, while the +lower 2 levels are the staff area and basement level 1. Note that +the residential area and basement level consists of a common +and dining area, where the residents can interact and have +activity together. In total, there is over 138 Bluetooth beacon +receiver installed to provide sufficient coverage to monitor +residents’ activity behavior. There is a total of 50 residents +within the nursing home who are agreed to participate in this +study. The beacon receiver uses RedBear Duo as shown in +Fig. 1a, which later transmits the collected Bluetooth beacon +list to the local server for further processing. +After the local server received the broadcast messages, it +performs threshold filtering to remove any weak signal less +than −70dBm. Subsequently, we compute the residents’ stay + +55 mm +ww +9855 mm +ww +9855 mm +ww +98EO5 +AP62PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +4 +location using Algorithm 1. The Algorithm 1 undergoes a +cycle of 15 seconds to determine the location of the residents +based on the strongest RSSI signal. The complexity of the +Algorithm 1 is O(R), which it depends on the number of +RSSI signals received, R. +Resident Card 1 +Location Detection Algorithm +(refer to Algorithm 1) + Database +Origin Computation +Compute Similarity Matrics +between Residents +Spectral Clustering +Residents Data Aggregation +using Ordinal Ranking +Temporal Filtering +for Data Lost +Residents Clustered Data +Raw Trajectory Data +Residents Aggregated Data +Determine Individual +Start and End Time +Determine Group +Start and End Time +Compute Adaptive +Hybrid Norm +Filter Norm Data +Data Collection Phase +Data Preprocessing (Single User) +User Group Detection Module +(Multiple Users) +Behavior Deviation +Detection Module +(Mutliple Users) +Deviated Events +Resident Card 2 +Resident Card n +RSSI Lists +. . . . . . +. . . . . . +Location Encoding +TimeSync +Windows Fitting +Data Smoothing +Distribution based +Classification of +Deviated Events +Deviated Activity +Labeled Trajectory Data +Loc ID +timeStamp +RSSI Lists +Loc ID +timeStamp +Behavior +Fig. 2: Data Processing Pipeline +B. Data Specification and Processing pipeline +After the residents’ location data is stored in the database, +we perform a series of processing onto the residents’ trajectory +data to extract the deviated activity behaviors. In this paper, +6 months of the residents’ trajectory data is studied ranged +from 01 Sep 2019 until 01 March 2020. The overall data +processing architecture is shown in Fig. 2. Each resident +undergoes the following preprocessing steps to extract their +daily trajectory as well as residing room for further analysis. +The preprocess module consists of the following steps, which +are (1) time-sync, (2) location encoding, (3) windows fitting, +and (4) data smoothing. The time-sync process is used to +synchronize time for the data entry as previously proposed +in [35]. Subsequently, the residents’ locations are encoded into +discrete numerical values for easier location representation. +Subsequently, windows fitting is performed to fit the location +data into five-minute windows with the location with the +longest duration denoted as stay location. Lastly, the data +smoothing is performed to remove location with short duration +stay, which could be a potential noise. After the preprocessing +step, we would obtain a more structured residents’ trajectory +data. +We have performed the complexity analysis on the system +architecture to ensure the proposed system does not take +ages to detect the activity behavior of a given nursing home +resident. The computational complexity is O(n3), which the +most time-consuming part is during clustering phase of the +nursing home resident, n. On the other hand, the space +complexity of the proposed system is O(n2). +Based on the trajectory, we compute the origin for each +resident based on their longest stay duration and location from +11:00pm to 6:00am. Note that origin denotes the room that a +resident stayed in, while other rooms are denoted as private. +The encoded locations are divided into the following: (a) origin +level 2, (b) origin level 3, (c) private level 2, (d) private level +3, (e) public area basement level, (f) public area level 2, (g) +public area level 3, and (h) restricted area. +In order to detect the different types of activity behavior +among the nursing home’s residents, the clustering method +utilizing a custom kernel is applied to compute similarity +metrics across different elderly. Based on the daily trajectory +data, there are group activities among residents, where the +residents are divided into multiple groups. To find the common +patterns among residents, the clustering approach is utilized to +group residents with similar activity behavior. Further details +of the clustering will be elaborated in Section IV. +After that, we want to study the deviated activity behavior +of the residents in the nursing home. Using the aggregated +residents’ data and cluster data, there are two types of norm +data that can be computed, which are (1) individual, and (2) +group norm data. Based on these two norms, a data fusion +technique is used to generate daily hybrid norm for each +resident and from there further extract each resident’s deviated +activity behavior. Subsequently, the deviated activity behavior +of the nursing home’s residents is analyzed and categorized. +IV. GROUP ACTIVITY BEHAVIOR CLUSTERING +A. Clustering Algorithm +In this subsection, we aim to study residents’ activity behav- +ior by applying the clustering algorithm to group residents with +similar trajectory patterns. To study the similarity between +residents based on the location data, a custom similarity kernel +is proposed to measure the resemblance between residents’ +activity behavior. Each location is treated as categorical data +and perform the windows sliding method to determine the +similarity score between residents. Let’s denote each resident +as ui in a nursing home, where the number of residents +consists range of i ∈ {1, ..., n}. The resident ui is assigned a +location xt, which it consists of the spatial information x and +temporal information t over the days d. This formulates the +basic trajectory of residents in a nursing home. By collecting +the data over different days, a spatial-temporal matrix, Xi +consists of the temporal information t can be denoted such +as: +Xi = +� +���� +x1,1 +x2,1 +... +x1,t +x2,1 +x2,2 +... +x2,t +... +... +... +... +xd,1 +xd,2 +... +xd,t +� +���� , +(1) + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +5 +where the d denotes number of days for the data collected +and the t denotes timestamp for each location x. Based on the +matrix Xi, the residents’ location is aggregated to generate an +individual pattern, yi, which later will be used for clustering. +We consolidate the spatial-temporal matrix Xi into a vector +of 288 samples, which each time-slot represents 5 minutes +interval of an encoded location. The main motivation of +choosing 288 sample is we want to achieve between computing +speed and data processing size. This is computed using ordinal +ranking [36] based on the frequency of the location a resident +stays throughout the data collection period. The aggregation +trajectory for resident ui into yi can be shown in the following +Algorithm 2: +Algorithm 2 Trajectory Aggregate Function +Data: trajectory, Xi +Result: aggregated trajectory yi +1. Initialize vector V. +2. Store location of all same timeslot +for q from 0 to 288 do +foreach j do +V[q] ← Xj +3. Initialize the aggregated trajectory yi +4. Perform aggregate function for each timeslot +for q from 0 to 288 do +Perform ordinal ranking for location based on the frequency of V[q] +After obtaining the residents’ aggregated data, we calculate +the similarity metrics between different residents for clustering +purposes in order to generate a similar group. Therefore, +the Weighted Windowed Overlap (WWO) similarity kernel is +introduced to calculate the similarity between pairwise resident +ua and ub. The WWO function dist(ua, ub) can be defined +as following: +dist(ua, ub) = 1 +t +t +� +i=0 +1 +2h + 1 +� +� +i+h +� +j=i−h +� 1 +if ya,j ̸= yb,j +0 +if ya,j = yb,j +� +� , +(2) +where ρ denotes window sliding parameter over the time +t by comparing the location of pairwise residents (ua, ub). +Meanwhile, h represents the threshold of the windows sliding +mechanism. In this paper, the threshold of h is defined as +30 minutes. Subsequently, the similarity calculation, s can be +calculated using the following equation: +s(ua, ub) = W × +1 +1 + dist(ua, ub) +(3) +where W denotes the weight vector corresponding to the time +of the day. +To ensure the effectiveness of computing similarity, we use a +toy example of two different residents with the same origin but +different activity behavior patterns to study similarity measure +as shown in Fig 3(a) as follows: +Three different weight vectors are being considered when +computing the similarity kernels for resident ua and ub using +WWO, which are (1) uniform, (2) active-day focus, and (3) +only active-day. The illustration of varying weight is presented +in Fig 3(b). These three weight vectors try to highlight +different temporal parts of the day and weight is adjusted ac- +cordingly. For instance, active-day focus emphasizes the active +0 +2 +4 +6 +8 +Encoded Location ID +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u3 trajectory +u20 trajectory +(a) Two trajectory examples (users pair(u3, u20)) in +encoded location for 11 Nov and 25 Oct 2019. +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +Weight Value +active-day focus +uniform +only active-day +(b) Weights used for calculating the similarity function +Fig. 3: Toy examples for calculating similarity metric and +custom weight for similarity calculation. +period of the day (06:00am - 11:59pm), where only active-day +only consider the active period (07:00am - 08:00pm). Note +that the total value of weight is 1.0 and is allocated across +288 vectors depending on the weight characteristic. We choose +one of the weights depending on the emphasis of the outcome +of the desired similarity measurement. Using some common +similarity measurement as introduced in [37], a comparison of +the similarity measurement is shown in Table I. +TABLE I: Comparison of Similarity Measurement +Similarity Measurement Method +Similarity Score +WWO (active-day focus) +0.7250 +WWO (uniform) +0.7778 +WWO (only active-day) +0.5675 +Overlap [38] +0.8182 +Eskin [39] +0.9259 +Goodall [40] +0.9254 +Based on the observation, the variation of overlap methods +has similar results ranging from 0.7250 to 0.8182, where +the similarity score of the WWO with only active-day is +0.5675. Meanwhile, the other methods (Eskin and Goodall) +have a higher similarity score despite the visualization of +two residents’ trajectory in Fig. 3 shows different patterns. +Ideally, the overlap method presents the most straightforward +method of computing the similarity score, which is roughly +around 0.8182. However, the overlap method only highlights +the similarity between two users during night time (08:00pm - +06:00am), which represents a large amount of time when users +are inactive. Thus, after considering the different weightage +similarity scores, active-day focus weight is more desirable +for computing the similarity metric, where it emphasizes a +more active period on the day. +By iterating the similarity score through different pairwise +residents, we can obtain the similarity matrix, A as such: + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +6 +A = +� +���� +0 +s(u1, u2) +. . . +s(u1, un−1) +s(u1, un) +s(u2, u1) +0 +. . . +s(u2, un−1) +s(u2, un) +... +... +... +... +... +s(un−1, u1) +s(un−1, u2) +· · · +0 +s(un−1, un) +s(un, u1) +s(un, u2) +· · · +s(un, un−1) +0 +� +���� +(4) +In order to compute the laplacian matrix, degree matrix, D +can be computed as follows: +D = +n +� +i +� 1 +Ai,i ≥ 0 +0 +otherwise , +(5) +where it represents a non-zero affinity matrix. +Next, using Eqn. 4 and Eqn 5, the normalized Laplacian +matrix, L can be generated using following equation: +L = I − D−1/2AD−1/2, +(6) +where I denotes the identity matrix. +Algorithm 3 Spectral Clustering Algorithm +Data: Spatio-temporal Matrix, X +Result: Cluster List, C +1. Aggregate data into daily windows data. +2. Calculate the affinity matrix, A as follows: +for i ← 1 to n do +for j ← 1 to n do +calculate the similarity metric using Eqn. 2 for each i and j resident. +3. Compute the degree matrix, D using Eqn. 5. +4. Calculate the Laplacian Matrix, L using Eqn. 6. +5. Calculate the Eigenvector, U. +6. Determine k value based on sum of squared distance (SSD). +7. Perform k-means and obtain cluster list, C. +return C. +Subsequently, we compute the k generalized eigenvectors +using the normalized Laplacian matrix, L as follows: +Lu = λDu +(7) +where vector, u is the calculated using the smallest k eigen- +value. By combining the afore-mentioned equation, we formu- +late the spectral clustering in the Algorithm 3. The computa- +tional complexity of the spectral clustering is O(n3), which is +the most time consuming part of the proposed system. Based +on the proposed algorithm, we perform group detection and +show the result in next sub-section. +B. Results and Validation +In order to study the optimal k for deciding the number +of clusters in the nursing home, we apply the sum of squared +distance (SSD) for different number of k values. The SSD can +be defined as follows: +k +� +n +� +a=1 +a−1 +� +b=1 +(dist(ua, ub))2, +(8) +where it utilizes dist() from Eqn. 2. It calculates the summa- +tion of squared distance for different k values in clustering, +which lower value indicates higher similarity between resi- +dents in the same cluster. Ideally, we want to find clusters +within the range of 2 to 7 clusters out from 50 residents. The +SSD result is presented in Fig 4. +Sum of Squared Distance +2250 +2000 +1750 +1500 +1250 +1000 +750 +500 +2 +3 +4 +5 +6 +7 +Number of Cluster, k +Lowest +Distance +Fig. 4: SSD for different number of k. +Based on observation, k=5 is an ideal number for the +clusters as the SSD value is the lowest compared to other +choices of k. Using k=5, the clustering algorithm is formulated +as shown in Algorithm 3, and the clustering result is presented +in Fig. 5. +We observed clusters are separated by building level and +have their own characteristic. The level 3 residents generally +can be divided into three different groups, where level 2 can +be divided into 2 different groups. The residents from Cluster +1 tend to have a longer visit duration at level 3 public area +compared to Cluster 2 and Cluster 3. Meanwhile, resident u15 +in Cluster 3 spends his/her lunchtime in the level 3 public area +instead of the basement public area. +Meanwhile, residents from Cluster 2 tend to spend less time +in public places, where Cluster 3’s residents spend more time +in level 2 public areas. In Cluster 2, resident u48 and resident +u49 spend time in their room without going to public spaces, +which is quite peculiar. Level 2’s residents in Cluster 4 mostly +spend their time in the public area, while residents in Cluster +5 spend less time in the public area and spend most of their +time in their respective origin. +Based on these 5 clusters, we will generate daily activity +based on their group to compute their daily activity behavior. +Further details of fusing individual activity behavior and group +activity behavior are elaborated in the next section. +V. RESIDENTS’ DEVIATED ACTIVITY BEHAVIOR STUDY +A. Residents’ Hybrid Norm Computation +Despite there is group activity behavior among the nursing +home’s residents in their daily routine, there exist cases of +some residents who behave differently than the others. The +trajectory of the residents may vary day to day depending +on the group schedule for the day. Therefore, it is crucial +to consider the residents’ groups daily trajectory to provide +additional information to generate daily norms for studying +potential deviated activity behavior. To show an example of +deviated activity behavior detection, we use resident u21’s +norm and the group norm (Cluster 3) as an example to +demonstrate the working of extracting the deviated locations +from the data. The example of input data and hybrid norm +generation are shown in following Fig. 6: +Using the example, we extract two different norms from the +resident u21 to generate a hybrid norm for 10 October 2019. + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +7 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u0 +List of users +u1 +u5 +u2 +u6 +u31 +u34 +u36 +u39 +u40 +u41 +u46 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u9 +List of users +u12 +u18 +u17 +u19 +u20 +u22 +u23 +u25 +u33 +u48 +u49 +(a) Cluster 1 +(b) Cluster 2 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u3 +List of users +u4 +u14 +u13 +u15 +u21 +u43 +u44 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u10 +List of users +u11 +u24 +u16 +u26 +u27 +u28 +u30 +u32 +u35 +u38 +u42 +u46 +(c) Cluster 3 +(d) Cluster 4 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u7 +List of users +u8 +u37 +u29 +u47 +Data Lost +Origin Level 3 +Origin Level 2 +Private Area +Level 3 +Private Area +Level 2 +Public Area +Level 3 +Public Area +Level 2 +Public Area +Basement Level +Restricted Area +Location Legends +(e) Cluster 5 +Fig. 5: Group Clustering Result based on k=5. +Both norms used as data input to generate a hybrid norm can +be defined as follows: +• Individual Norm, normind - The individual norm rep- +resents the regular pattern of the resident over the data +collection period. The regular pattern is generated through +the aggregated of the resident’s valid data, which can +be obtained using the user group detection module’s +Algorithm 2. Note that the individual norm is obtained +using aggregated trajectory yi for each resident, ui. +• Group Norm, normgrp - Group norm denotes the reg- +ular pattern of all clustered residents obtained using the +clustering algorithm in the previous section. Note that +this pattern changes every day as a nursing home have +different activities/events arranged during the active hour. +Therefore, normgrp is calculated every day for each +cluster. +Utilizing these norms, we propose a data fusion method +to generate a hybrid norm norm. The main challenge here +is to address the transition method between normind and +normgrp, where overlapping between two norms is possible +when combining both norms. +There are two transition periods in the day of study, which +are the group activity start transition period and group activity +end transition period. One needs to decide the exact timing +for transition based on the starting period of the group using +p2 and ending period p3. Moreover, similar consideration +applied for the ending time of the private period, p1 and the +starting period of private period p4. The p5 denotes the starting +of the group activity, while p6 represents the group activity +ending period. To compute the transition period [p5, p6] using +[p1, p2, p3, p4], the following Eqn. 9 is used and 10 to decide +the transition period for group starting and group ending +period: +p5 = +�p1 +if ∆t|p1 − p2| ≤ h or ∆t(p1 − p2) ≥ 0 +p2 +otherwise +, +(9) +p6 = +�p4 +if ∆t|p3 − p4| ≤ h or ∆t(p3 − p4) < 0 +p3 +otherwise +, +(10) +where h denotes the time gap limit between 2 time slots such +as (p1, p2) and (p3, p4). Note that if the time overlap such as +∆t(p1 − p2) ≥ 0 or ∆t(p3 − p4) < 0, the earliest time is +considered as transition period as default overlapping time. +After deciding the transition period, we compute the hybrid +norm, norm by combining the individual and group norm +of the nursing home residents. The merging process between + +:PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +8 +Aggregate to compute individual norm +Aggregate to compute group norm +Generate p1 and p4 +Generate p2 and p3 +Determine the transition period p5 and p6 +Generate hyrbid norm for day data +Individual Data (Different Days) +Group Data (Same Day - 10 Oct 2019) +Determine Individual Start and End Time +Hybrid Norm +Compute Adative +Determine Group Start and End Time +Individual Norm +Group Norm +Δt|p1-p2| +{ +p1 +p2 +p3 +p4 +Δt|p3-p4| +{ +Hybrid Norm +p5 +p6 +Merge ++ +Group Day Start +Transition Period +Transition Period +Group Day End +normgrp +normind +norm +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Encoded +0 +2 +4 +6 +8 +Location ID +Hybrid Norm +p5 +p6 +Encoded +0 +2 +4 +6 +8 +Location ID +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Group Norm +p3 +p2 +Encoded +0 +2 +4 +6 +8 +Location ID +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Individual Norm +p1 +p4 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +Group +norm, +normgrp +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +Individual +norm, +normind +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +Date +06 Jan +2020 +2019 +27 Nov +2019 +10 Oct +2019 +22 Oct +2019 +05 Dec +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +u4 +Cluster Member Data +u13 +u21 +u15 +u14 +Fig. 6: Toy example for extracting deviated activity behavior using resident u21’s trajectory data. +normind and normgrp can be formulated as follows: +normd,i[t = 0 : p5] +← normind[t = 0 : p5] +normd,i[p5 : p6] +← normgrp[p5 : p6] +normd,i[p6 : t = 288] +← normind[t = p6 : 288] +(11) +where the period between t=0 to p5 and t=p6 to 288 uses +norm from normind. The p5 to p6 denotes the group time. +In addition, d represents the particular day we are studying, +while i denotes each of the residents in the nursing home. +By combining the aforementioned equations, the hybrid norm +normd,i for each resident ui can be computed using Algo- +rithm 4 and the computational complexity is O(n2). +B. Residents’ Deviated Activity Behavior Identification +After computing the adaptive hybrid norm for each user, +we extract the deviated locations using a filtering function +based on the norm and daily input data, Xd. Since the encoded +location is categorical data, the binary comparison method is +adopted to compare whether the input data is the same as +the generated adaptive hybrid norm. The Eqn. 12 describe the +afore-mentioned process: +filter(xd,i, normi) = +�null +if xd,i = normi +xd,i +if xd,i ̸= normi +(12) +where the filter() function is applied to the daily input data +to remove norm data, while preserving the deviated locations. +Therefore, the following Algorithm 5 is proposed to compute +for daily trajectory by iterating the filter function every day. +Using the example stated earlier in Fig. 6, the deviated +activity behaviors of resident u21 can be computed for 10 +October 2019. The deviated activity behaviors are highlighted +as illustrated in Fig. 7. +In the upper part of Fig. 7, we show an example of +obtaining a hybrid norm based on the data fusion approach by +fusing group features and individual features. After performing +the filtering function, the input trajectory is highlighted in +red color, which the users’ trajectory is different than the +computed hybrid norm. Here, three different deviated activity +behaviors are detected across different locations. The first +deviated activity behavior happened around 7:30am, where the +resident u21 visit level 3 public area for 15 minutes. After that, +resident u21 return to his origin and stayed from 10:45am to +2:00pm. Based on the hybrid norm, the resident from cluster +4 went back to their room after 11:45pm, which is conflicted +with the resident u21’s hybrid norm. Lastly, the last deviated +activity behavior occurred around 4:50pm and continued until +the end of the day. Supposedly, the resident u21 would spend + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +9 +Algorithm 4 Generating Adaptive Hybrid Norm +Data: group label, Day Data Xd, origin xori, threshold h +Result: Hybrid Norm normd,i +function generateHybridNorm() +1. Obtain normind using Algorithm 2. +2. Obtain normgrp using getGroupNorm(group label, Xd). +3. Compute p1 and p4 using determineDayStartEnd(normind, xori, h). +4. Compute p2 and p3 using determineDayStartEnd(normgrp, xori, h). +5. Compute p5 and p6 using Eqn. 9 and Eqn. 10. +6. Merge the data to become adaptive hybrid norm using Eqn. 11 +function getGroupNorm(group label, Xd) +1. Initialize the same group dictionary +2. Add the residents’ data, X if they are from the same group +foreach Xd,j do +if resident ∈ group label then +same group dictionary append Xd,j +3. Check whether the same group dictionary has more than 2 entries. +if length( same group dictionary) > 2 then +return aggregate same group dictionary using Algorithm 2 +else +return NULL +function determineDayStartEnd(X, xori, h) +1. Compute day start pointer +for t, 0 to 144 do +if xt ̸= xori for h interval then +break +2. Store the value, DayStart ← t +3. Compute day end pointer +for t, 288 to 144 do +if xt ̸= xori for h interval then +break +4. Store the value, DayEnd ← t +return DayStart, DayEnd +Algorithm 5 Compute Deviated Activity Behaviors +Data: norm +Result: Hybrid Norm normd,i +function ComputeDeviatedEvents(X, normd) +1. Initialize Ed +2. Compute each inputted day using filter function in Eqn. 12 +for i, from 0 to d do +Ed append filter(xi ∈ X, normi) +return Ed +most of his/her time in origin, but instead, went to public +spaces around level 3 and level 2 private area. This shows the +process of detecting deviated activity behaviors for a particular +user by integrating the group norm and individual norm. By +fusing both information, the deviated activity behaviors can be +computed for every resident in the nursing home and further +study the types of deviated activity behavior. +C. Analysis and Classification of Deviated Activity Behaviors +After obtaining deviated activity behaviors, we proceed to +identify the types of it among the nursing home’s residents. +We found out some residents suffered from sleep irregularity, +and some of them absent during the group activity. Based +on observation of the data, three different deviated activity +behaviors can be generalized, which are (1) sleep irregularity, +(2) awake irregularity, and (3) private visiting. The definition +of deviated activity behaviors are listed as follows: +• (1) sleep irregularity - The sleep irregularity denotes the +resident goes to another location instead of staying back +to his/her origin at the usual timing during night time. +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +0 +1 +2 +3 +4 +5 +6 +7 +8 +Encoded Location ID +Input Data, xi +0 +1 +2 +3 +4 +5 +6 +7 +8 +Encoded Location ID +Deviated +Event 1 +Deviated +Event 2 +Deviated +Event 3 +Hybrid Norm, norm +Fig. 7: Example of extracting deviated activity behaviors for +resident u21 on 10 October 2019. +• (2) awake irregularity - The awake irregularity denotes the +resident either leaves the room late or early compared to +their routine schedule. +• (3) private visiting - The resident went to another staying +room (staying room for level 2 or level 3) instead of the +common places cluster or he/she normally will present. +Subsequently, a rules-based classification method is introduced +to define the features for identifying deviated activity behavior. +The rationality of using such an approach is that labels can +be assigned based on the predefined rules using a statistical +approach given there is no ground-truth available. Note that +each user can be assigned multiple classification labels. +TABLE II: Definition of the Transition Period +Temporal Period +Time Range of the Day +Midnight +day start → ( p5 +2 ) +Morning +p5 +2 → p5 +Pre-group +p5 → (p5 + 60mins) +Group +(p5 + 60mins) → (p5 − 60mins) +Post-group +(p6 − 60mins) → p6 +Evening +p6 → ( t=288−p6 +2 +) +Midnight +( p6 +2 ) → day end +To generate features for the classification, we further break +down the time of the day into seven different finer temporal +periods as shown in Fig. 8. By breaking down into different +periods, the time before and after group activity can be +investigated to generate features for studying deviated activity +behavior. Note that p5 and p6 shown here are based on the +average time start and time end throughout the data collection +period for each user. Using Fig. 8 as a reference, the finer +temporal periods are introduced as shown in Table II. +Next, the probabilities of deviated locations’ occurrence are +calculated based on the daily filtered deviated location for each +Group +Midnight +Midnight +Evening +Morning +Pre-Group +Post-Group +{ +{ +p5 +p6 +Fig. 8: Illustration of the finer temporal periods. + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +10 +finer temporal period. This can be used to define rules using +the probabilistic calculation based on the deviated activity +behavior statistic. The location probability, P(Ei,x) can be +computed using following Eqn. 13 +P(Ei,x) = q(Ei,x) +c +(13) +where i denotes the timeslot of the day and x represents the +location resident went when a deviated behavior occurred. The +q() counts the occurrence number of deviated activity behavior +Ei,x and c denotes the total number of the valid time slot. +By combining the probability of deviated locations of every +user, we can study the potential timeslot for each user, where +the deviated activity behaviors occurred most. To simplify +the locations of where deviated activity behaviors occurred, +encoded locations with similar functionalities are combined +and study the probability for each timeslot. The combined +encoded list are listed as follows: (c1) null, (c2) origin, (c3) +public area, (c4) private area, and (c5) restricted location. +After computing the probabilities of deviated locations for +the nursing home’s residents, we illustrate the distribution of +deviated locations using a violin plot as shown in Fig. 9. +The category c1 represents the probability of residents not +involving in deviated activity behavior at a different time of +the day, where category c2 to c5 represents the location for a +deviated activity behavior to be detected. The probability of +deviated activity behavior occur at other locations c2 to c5 is +small which the value ranges from 0.0 to 0.4. One can observe +a lot of deviated activity behaviors that happened mostly in +the public area (c3) and follow by origin (c2). That being +said, to further understand those involved in sleep irregularity +and social visiting events, attention is given to the residents +who yield a higher probability of deviated activity behaviors. +Specifically, we are interested in the distribution of residents, +whose probability is higher than the median for the encoded +location c2 to c5. Those residents have higher chances of +showing deviated activity behaviors compared to the median +of the distribution. Using the characteristic of a violin plot, +the classification threshold is defined based on the upper +adjustment value (UAV) and lower adjustment value (LAV), +which are computed through the first or third quantile ±1.5× +interquartile range. Based on the specific timing’s distribution, +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Deviated Events Distribution, P(Ex) +Deviated Location, Ex +Null (Normal) +Origin Level 2, 3 +Public Area B1 +Private Level 2, 3 +Restricted Area +Level 2, Level 3 +temporal period +midnight +morning +pre-group +group +post-group +evening +C1 +C2 +C3 +C4 +C5 +Fig. 9: Violin plot for the deviated location w.r.t. different +temporal periods. Note that c1 denotes probability of residents +being normal while c2 to c5 represent the deviated locations +based on their functionalities. +44.4% +37.0% +14.8% +3.7% +0 classification +label +1 classification +label +2 classification +labels +3 classification +labels +Nursing home's resident classification labels distribution +classification label +sleep irregularity +awake irregularity +private social visiting +Percentage +40% +30% +30% +classification label +awake irregularity +private social visiting +Percentage +sleep irregularity + +private social visiting +awake irregularity + +sleep irregularity + +50% +25% +25% +Fig. 10: Distribution of residents based on the number of +classification labels. +the UAV and LAV values are extracted as shown in Table III +to define the threshold values for the classification rules. +TABLE III: UAV and LAV Extracted for Classification Rules +Deviated Location +c1 +c2 +c4 +LAV +UAV +UAV +midnight +0.8762 +- +- +Morning +0.7323 +- +0.0118 +pre-group +- +0.2717 +0.0312 +group +- +- +0.0521 +post-group +- +- +0.0223 +evening +0.7522 +- +0.0174 +Based on the input from Table III, we define the classifica- +tion rules as presented in Table IV. +After defining the classification rules, we classify every +nursing home’s residents based on their probability for the +finer hybrid temporal period. A general statistic for the clas- +sified labels after performing the rules-based classification is +presented in Fig. 10. Based on the result, 44.4% of residents +do not have a classification label, which indicates the major- +ity of the residents do not have deviated activity behavior. +The remaining 37% of the residents are associated with one +classification label only, of which 40% of them are having +TABLE IV: Deviated Activity Behavior Classification Rules +Deviated Activity Behavior +Classification Rules +Sleep Irregularity +c1 midnight < LAV or c1 Evening < LAV +Awake Irregularity +c1 morning < LAV or c2 pre-group < LAV +Private Visiting +c4 morning > UAV or +c4 pre-group > UAV or +c4 group > UAV or +c4 post-group > UAV +or c4 evening > UAV + +PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 +11 +(a) Example of awake irregularity detected for resident u35 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +Deviated +Common +Activity Behavior +Activity Behavior +midnight morning pre-group group post-group evening midnight +Awake Irregularity +(b) Example of sleep irregularity detected for resident u5 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +midnight morning pre-group group post-group evening midnight +Sleep Irregularity +Deviated +Common +Activity Behavior +Activity Behavior +(c) Example of private visiting detected for resident u21 +00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 +Time, from 0000 to 2359 +midnight morning pre-group group post-group evening midnight +Private Social Visiting +Deviated +Common +Activity Behavior +Activity Behavior +Data Lost +Origin Level 3 +Origin Level 2 +Private Area +Level 3 +Private Area +Level 2 +Public Area +Level 3 +Public Area +Level 2 +Public Area +Basement Level +Restricted Area +Location Legends +Fig. 11: Example of the deviated activity behavior study. +Note that the y-axis consists of sample data extracted user +trajectory on selected day. The common activity behavior +denotes trajectory data without any deviated activity behavior, +while the deviated activity behavior category represents the +opposite. +private social visiting irregularity. On the other hand, half of +the residents with two classification labels mainly consist of +sleep and awake irregularity. +Based on the classification labels generated, we examine +three residents as case studies to study normal daily activ- +ity behavior and deviated activity behavior. Three different +deviated activity behaviors are illustrated in Fig. 11. From +Fig. 11(a), the resident u35 is not waking up based on his/her +daily schedule. Based on observation, there are two types of +awake irregularity, which are wake up earlier or wake up way +later than the normal wake up time (around 7:30 am). Next, +we investigate the sleep irregularity of resident u5 as depicted +in Fig. 11(b). The resident u5 normally goes back to the origin +after 9:00pm, but in the deviated activity behavior’s extracted +days, he/she remained at the public area until the end of the +day. These activity behaviors normally do not occur in his/her +normal routine, and this is a good representation of the sleep +irregularity. Subsequently in Fig. 11(c), one can discover that +the resident u21 absent during the group activity for few days, +where he/she went to a private area instead. This phenomenon +indicates the resident u21 either is having social interaction +with other residents in their private space or avoiding group +activity. This could be potentially a deviated activity behavior. +By utilizing the proposed hybrid deviated activity behavior +classification, we managed to obtain types of residents’ de- +viated activity behaviors in a nursing home. This provides +insight into the nursing home’s management regarding the +residents’ deviated activity behavior and attention can be +provided to the individual with needs. +VI. CONCLUSION +In this paper, we present a location-based deviated activity +behavior detection system for the Salvation Army, Peacehaven +Nursing Home, Singapore. The 50 residents can be segmented +into different groups based on their activity behavior, which +contributes to formulating group norms. By combining group +and individual norms, we can generate a hybrid norm to +identify the common behavior for each resident. By under- +standing residents’ normal activity behavior, the deviated ac- +tivity behavior can be differentiated and extracted from normal +activity behavior to study it. Based on the types of deviated +activity behavior, three categories of deviated activity behavior +are proposed, which are (1) sleep irregularity, (2) awake +irregularity, and (3) private visiting. 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Available: +http://www.jstor.org/stable/2528080 + diff --git a/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/load_file.txt b/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1599139376322ac87fd8e9d9b1ecffc4c421830a --- /dev/null +++ b/OdFIT4oBgHgl3EQfeCsB/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf,len=1039 +page_content='PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 1 Location-based Activity Behavior Deviation Detection for Nursing Home using IoT Devices Billy Pik Lik Lau, Member, IEEE, Zann Koh, Yuren Zhou, Member, IEEE, Benny Kai Kiat Ng, Chau Yuen, Fellow, IEEE, and Mui Lang Low Abstract—With the advancement of the Internet of Things(IoT) and pervasive computing applications, it provides a better oppor- tunity to understand the behavior of the aging population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' How- ever, in a nursing home scenario, common sensors and techniques used to track an elderly living alone are not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In this paper, we design a location-based tracking system for a four-story nursing home - The Salvation Army, Peacehaven Nursing Home in Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The main challenge here is to identify the group activity among the nursing home’s residents and to detect if they have any deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We propose a location-based deviated activity behavior detection system to detect deviated activity behavior by leveraging data fusion technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In order to compute the features for data fusion, an adaptive method is applied for extracting the group and individual activity time and generate daily hybrid norm for each of the residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Next, deviated activity behavior detection is executed by considering the difference between daily norm patterns and daily input data for each resident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Lastly, the deviated activity behavior among the residents are classified using a rule-based classification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Through the implementation, there are 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4% of the residents do not have deviated activity behavior, while 37% residents involved in one deviated activity behavior and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='6% residents have two or more deviated activity behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Index Terms—Internet of Things, Deviated Activity Behavior Detection, Data Fusion, Location-based Sensing, Nursing Home Monitoring I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' INTRODUCTION Over the past few years, the advancement of the Internet of Things (IoT) technology has opened up a lot of research potential in the area of tracking and monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Among them, a wide variety of projects have been carried out to monitor the behavior of the human being as shown in [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' These technologies made room for implementing convenient applications for enhancing day to day living of urban residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' With the increase of the world aging population as shown in [6] and [7], research in monitoring the elderly has gained attention from different research principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' While these works [8]–[11] address the support of the elderly as indepen- dent beings of the society, and others [12], [13] have provided the facility for the nursing home to monitor the daily activity behavior of the residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The former method commonly leverages boundary-less tracking and monitoring techniques as shown in [4], [5], which include smartphones and smart Billy Pik Lik Lau, Zann Koh, Yuren Zhou, Benny Kai Kiat Ng, and Chau Yuen are with the Engineering Product Development, Sin- gapore University of Technology and Design, Corresponding E-mail: billy lau@mymail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='sg, yuenchau@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Mui Lang Low is with the Peacehaven Nursing Home Day Centre run by The Salvation Army.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Manuscript received January 25, 2023 wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The majority of the latter approaches [12], [13] mostly provide tracking in a confined environment, where the accuracy of the boundary-less approach is not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Our work focuses on the latter approach, where the constraints of monitoring senior citizens are limited to a nursing home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Traditionally, it is labor-intensive to take care of the day- to-day life of an elderly resident, and it is not possible to constantly track an individual across 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, using a building-scale human monitoring approach, it can assist the nursing home staff to monitor residents and better understand their activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' With this challenge in mind, we design a system to monitor the deviated activity behavior of nursing home’s residents leveraging IoT technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We use bluetooth low energy (BLE) technology as backbone for collecting the elderly data due to nature of low energy, and coverage suitable for indoor application compared to WiFi, RFID, ZigBee, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The deviated activity behavior denotes a nursing home’s resident behaves irregularly compared to his/her normal routine of daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This type of detection only can be achieved through fully understanding a resident’s life routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The main objectives of such a system are to identify the residents’ activity behavior and determine the irregular activity behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The constraints of monitoring residents’ activity behavior in a nursing home are bounded by building structure, and also their daily activity is influenced by the group activities or community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, our aim is to differentiate their activity between private and group activity when computing their deviated activity behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Another constraint when designing this system is that we do not have ground truth on the data collected, which resulting the accuracy of system output cannot be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Moreover, the identity of the nursing home residents is anonymized to comply with Singapore Personal Data Protection Act [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Thus, unsupervised knowledge extraction is more desired when compared to the supervised knowledge extraction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To address the aforementioned challenges, in this paper, we present a building-scale monitoring system to study 50 residents’ activity behavior in the Peacehaven Nursing Home, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using the building-scale monitoring system, resi- dents’ activity behavior based on their location are investigated using a wearable card tag with a build-in Bluetooth beacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Each room is equipped with a receiver to detect the Bluetooth beacon transmitted to perform the resident’s location detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the detected location, we study the activity behavior of residents over 6 months and cluster them based on their common patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In order to identify the normal activity behavior, we use a data fusion method to generate the hybrid arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='11272v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='CY] 25 Jan 2023 PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 2 norm by combining the group and individual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using the hybrid norm, deviated activity behavior can be extracted, which does not follow the normal daily pattern of a resident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Afterward, we perform empirical analysis on the deviated activity behavior and classify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The key contributions of this paper are as follows: We study the resident’s activity behavior in a nursing home from a location-based implementation of a moni- toring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We propose a data fusion method to identify the daily norm for each nursing home’s resident based on two data sources, which are individual and group norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the daily norm generated, we perform empirical analysis on the deviated activity behavior and identify the types of them using rules-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Our paper can be detailed as follows: In Section II, we study related work about existing methodologies in detecting devi- ated activity behavior with their pros and cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, in Section III, the system architecture and data processing model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Afterward, we describe the group activity behavior clustering method in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the group detected, we compute the deviated activity behavior utilizing the hybrid norm and analyze them in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Lastly, we conclude our work in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' RELATED WORK In this section, we discuss some of the related works in the field, which are types of monitoring techniques used to achieve human monitoring and methodologies applied to study deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Types of Monitoring Techniques There are four common types of monitoring techniques in the literature, which are (1) people-driven, (2) event-driven, (3) location-driven, and (4) data-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The people-driven monitoring technique uses humans as the main source of generating information, which normally in- volves sensors installed in smartphones, watch, bracelets, and other types of wearable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' It is commonly not restricted by loca- tion and has a wider coverage of sensing capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Examples of smartphone monitoring techniques can be found in [4], [15], [16], where common sensors used are accelerometer, GPS, microphone, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Examples of other types of wearable devices are belt equipped with sensing unit [12] and bracelets [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Generally, these types of monitoring techniques are intrusive but able to capture good accuracy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The event-driven monitoring technique uses the activity of daily life (ADL) of the users and attempts to understand the activity behavior of the targeted user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' For instance, Alcala et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' [17] uses the hidden Markov model (HMM) to process ADL and detect the deviated activity behavior from the daily routine, while Zerkouk et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' [18] use a long term short term memory-based model to identify deviated routine among senior citizen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' A detailed review of the ADL monitoring approaches can be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The downside of the events- driven monitoring technique is that detailed data is desired and requires a lot of effort as incomplete information will mislead the study outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' When monitoring techniques involve installing multiple sensors at a particular location or building, often it is known as a location-driven method of monitoring people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The coverage of the monitoring often involves a building or a particular area, and commonly used sensors include motion sensors [20], vi- sion [21], RFID sensors [22], infra-red [23], WiFi-passive [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' However, the inconvenience of this monitoring approach is limited to the area coverage since it is location-bound and only limited study scenarios would benefit from such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The data-driven approach uses various information sources and combines them to infer human activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' It can be a mixture of different data sources as described in [25] such as physical sensors or cyber data sources such as social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Examples of data fusion driven approaches can be found in these works [26], [27], where multiple sensors are fused to study human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The disadvantages of this approach are due to the complexity of the model and domain knowledge required to select relevant information sources to combine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Besides, every data sources require different preprocessing techniques, which can be rather tedious and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Methodologies in Deviated Activity Behavior Extraction The common methodologies in studying the deviated ac- tivity behavior of senior citizens can be categorized into the following: (1) prediction model, (2) state estimation model, and (3) clustering and exploration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The prediction model utilizes statistics to predict the po- tential activity behavior of a particular user and if the pre- dicted behavior does not match the predictive outcome, it will be labeled as deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Recent prediction methods such as long short term memory (LSTM) can be found in [18], which detect the deviated activity behaviors among the senior citizens using a deep learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Other types of statistical predictive models also can be found in [8], [20], [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The predictive model generally requires good quality and a large amount of data as it is not ideal to perform a model with limited or noisy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' State estimation modeling maps the state behavior of a particular user into a system state, which can be used to model the users’ activity behavior and detect any deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' State estimation usually requires human intervention to map the total state of the given system, which involves domain experts to carry out such tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In [30], Novak et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' performed the Self Organizing Maps (SOM) and Makrov prediction model onto ADL of a senior citizen to detect their deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Another example of state estimation modeling can be found in [31], where they detect the deviated activity behavior using HMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Other state estimation examples can be found in [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The drawback of this approach can be rather complex since it does not consider the relationship between activities that happened in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The clustering and exploration model normally use four steps to generate an insights extraction model, which the user’s deviated activity behavior is studied during the exploration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' It is first proposed in [34] and commonly used when there is no ground-truth available or no prior knowledge regarding deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' An example of this approach can be found in [11], where Kurnianingsih et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 3 use hybrid k-means clustering and isolation forest to detect the deviated vital signals among senior citizen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In [22], authors also use k-means clustering to formulate normal pattern and if any event does not fit into the cluster, it will be labeled as deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Meanwhile, authors [4] studied the activity behavior of senior citizens using k-means clustering approach and decision tree to generate features for analyzing the users’ demographic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The main drawback of this method is that it requires extensive knowledge in specific domains when analyzing potential deviated activity behavior, however it works effectively when there is no ground-truth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Bluetooth Beacon Receiver × 138 sensors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Card 1 Card 2 Card n (a) Card tags with Bluetooth beacon (Transmitter) and Redbear Duo (Receiver).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Basement 1 Level 1 Level 2 Level 3 Residents staying area, activity/common area Residents staying area, activity/common area Nursing Home Building Story Staff area (Restricted area) activity/common area (b) The nursing home building’s floor level, which can be divided into 4 story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that Level 2 and 3 consists of residential staying area and common area, where Basement 1 only has common activity area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Level 1 is the nursing staff area, where elderly normally do not have access to that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 1: Hardware used to setup the monitoring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' SYSTEM DESIGN In this paper, we focus on the location-driven monitoring techniques since the resident of the nursing home stay within the premises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, building-wide monitoring is more desirable in our studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Given that collecting ground-truth appears to be impossible in our problem, the insights explo- ration approach and unsupervised machine learning method is more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In this section, the overall system design of the deviated activity behavior detection system is presented followed by the data specification and data preprocessing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Hardware Setup The proposed hardware setup comprises of two crucial components installed in the nursing home, which are card tags with BLE beacons and beacon receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The Bluetooth beacon card model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 1a, which is capable of transmitting beacon every 1000ms using Nordic nRF52 chip with a range of up to 40 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Algorithm 1 Location Detection Algorithm Data: mac Address List, RSSI List, loc ID Result: user list, location list, timeStamp function listenData() 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Perform detection cycle as follows: for detection cycle from 1 to 5 do while timer < 3 sec do loc ID, RSSI List Filter mac Address List ≤ −70dBm for unique resident in loc ID do location list = getHighestRSSI(loc ID, RSSI List) Store the userlist and location 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute the final location after detection cycle ended foreach unique resident in userlist do determine the location based on the strongest RSSI if computeLocation(location) ̸= NULL then final location ← computeLocation(location) else Retrieve last location from database final location ← last location 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' update the resident final location and current timestamp function getHighestRSSI(loc ID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' RSSIList) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='last location ← initial location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='last RSSI ← initial RSSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='index ← 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='foreach RSSI in RSSI List do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='if RSSI > Last RSSI then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='last RSSI ← RSSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='last location ← loc ID[index] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='index ← index +1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='return last location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='function computeLocation(locationList) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='initialize location dict ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='foreach location in locationList do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='update location dict count with location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='if max(count(location dict)) exists then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='return location in max(count(location dict)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='return NULL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='It is equipped with a battery and capable of transmitting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='the data for up to 6 months without recharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To provide sufficient coverage for the nursing home, each room has a beacon receiver installed and pick up the Bluetooth beacon transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To filter out the irrelevant Bluetooth devices, a unique identifier is assigned to each of the card tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The building studied as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 1b consists of 4 levels with the top 2 levels served as the residential area, while the lower 2 levels are the staff area and basement level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that the residential area and basement level consists of a common and dining area, where the residents can interact and have activity together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In total, there is over 138 Bluetooth beacon receiver installed to provide sufficient coverage to monitor residents’ activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' There is a total of 50 residents within the nursing home who are agreed to participate in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The beacon receiver uses RedBear Duo as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 1a, which later transmits the collected Bluetooth beacon list to the local server for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After the local server received the broadcast messages, it performs threshold filtering to remove any weak signal less than −70dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, we compute the residents’ stay 55 mm ww 9855 mm ww 9855 mm ww 98EO5 AP62PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 4 location using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The Algorithm 1 undergoes a cycle of 15 seconds to determine the location of the residents based on the strongest RSSI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The complexity of the Algorithm 1 is O(R), which it depends on the number of RSSI signals received, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Resident Card 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Location Detection Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='(refer to Algorithm 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Database ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Origin Computation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Compute Similarity Matrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='between Residents ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Spectral Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Residents Data Aggregation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='using Ordinal Ranking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Temporal Filtering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='for Data Lost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Residents Clustered Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Raw Trajectory Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Residents Aggregated Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Determine Individual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Start and End Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Determine Group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Start and End Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Compute Adaptive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Hybrid Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Filter Norm Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Data Collection Phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Data Preprocessing (Single User) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='User Group Detection Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='(Multiple Users) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Behavior Deviation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Detection Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='(Mutliple Users) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Deviated Events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Resident Card 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Resident Card n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='RSSI Lists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Location Encoding TimeSync Windows Fitting Data Smoothing Distribution based Classification of Deviated Events Deviated Activity Labeled Trajectory Data Loc ID timeStamp RSSI Lists Loc ID timeStamp Behavior Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2: Data Processing Pipeline B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Data Specification and Processing pipeline After the residents’ location data is stored in the database, we perform a series of processing onto the residents’ trajectory data to extract the deviated activity behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In this paper, 6 months of the residents’ trajectory data is studied ranged from 01 Sep 2019 until 01 March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The overall data processing architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Each resident undergoes the following preprocessing steps to extract their daily trajectory as well as residing room for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The preprocess module consists of the following steps, which are (1) time-sync, (2) location encoding, (3) windows fitting, and (4) data smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The time-sync process is used to synchronize time for the data entry as previously proposed in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, the residents’ locations are encoded into discrete numerical values for easier location representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, windows fitting is performed to fit the location data into five-minute windows with the location with the longest duration denoted as stay location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Lastly, the data smoothing is performed to remove location with short duration stay, which could be a potential noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After the preprocessing step, we would obtain a more structured residents’ trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We have performed the complexity analysis on the system architecture to ensure the proposed system does not take ages to detect the activity behavior of a given nursing home resident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The computational complexity is O(n3), which the most time-consuming part is during clustering phase of the nursing home resident, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' On the other hand, the space complexity of the proposed system is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the trajectory, we compute the origin for each resident based on their longest stay duration and location from 11:00pm to 6:00am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that origin denotes the room that a resident stayed in, while other rooms are denoted as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The encoded locations are divided into the following: (a) origin level 2, (b) origin level 3, (c) private level 2, (d) private level 3, (e) public area basement level, (f) public area level 2, (g) public area level 3, and (h) restricted area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In order to detect the different types of activity behavior among the nursing home’s residents, the clustering method utilizing a custom kernel is applied to compute similarity metrics across different elderly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the daily trajectory data, there are group activities among residents, where the residents are divided into multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To find the common patterns among residents, the clustering approach is utilized to group residents with similar activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Further details of the clustering will be elaborated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After that, we want to study the deviated activity behavior of the residents in the nursing home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using the aggregated residents’ data and cluster data, there are two types of norm data that can be computed, which are (1) individual, and (2) group norm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on these two norms, a data fusion technique is used to generate daily hybrid norm for each resident and from there further extract each resident’s deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, the deviated activity behavior of the nursing home’s residents is analyzed and categorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' GROUP ACTIVITY BEHAVIOR CLUSTERING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Clustering Algorithm In this subsection, we aim to study residents’ activity behav- ior by applying the clustering algorithm to group residents with similar trajectory patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To study the similarity between residents based on the location data, a custom similarity kernel is proposed to measure the resemblance between residents’ activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Each location is treated as categorical data and perform the windows sliding method to determine the similarity score between residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Let’s denote each resident as ui in a nursing home, where the number of residents consists range of i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The resident ui is assigned a location xt, which it consists of the spatial information x and temporal information t over the days d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This formulates the basic trajectory of residents in a nursing home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By collecting the data over different days, a spatial-temporal matrix, Xi consists of the temporal information t can be denoted such as: Xi = � ���� x1,1 x2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' x1,t x2,1 x2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' x2,t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' xd,1 xd,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' xd,t � ���� , (1) PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 5 where the d denotes number of days for the data collected and the t denotes timestamp for each location x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the matrix Xi, the residents’ location is aggregated to generate an individual pattern, yi, which later will be used for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We consolidate the spatial-temporal matrix Xi into a vector of 288 samples, which each time-slot represents 5 minutes interval of an encoded location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The main motivation of choosing 288 sample is we want to achieve between computing speed and data processing size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This is computed using ordinal ranking [36] based on the frequency of the location a resident stays throughout the data collection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The aggregation trajectory for resident ui into yi can be shown in the following Algorithm 2: Algorithm 2 Trajectory Aggregate Function Data: trajectory, Xi Result: aggregated trajectory yi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Initialize vector V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Store location of all same timeslot for q from 0 to 288 do foreach j do V[q] ← Xj 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Initialize the aggregated trajectory yi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Perform aggregate function for each timeslot for q from 0 to 288 do Perform ordinal ranking for location based on the frequency of V[q] After obtaining the residents’ aggregated data, we calculate the similarity metrics between different residents for clustering purposes in order to generate a similar group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, the Weighted Windowed Overlap (WWO) similarity kernel is introduced to calculate the similarity between pairwise resident ua and ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The WWO function dist(ua, ub) can be defined as following: dist(ua, ub) = 1 t t � i=0 1 2h + 1 � � i+h � j=i−h � 1 if ya,j ̸= yb,j 0 if ya,j = yb,j � � , (2) where ρ denotes window sliding parameter over the time t by comparing the location of pairwise residents (ua, ub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Meanwhile, h represents the threshold of the windows sliding mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In this paper, the threshold of h is defined as 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, the similarity calculation, s can be calculated using the following equation: s(ua, ub) = W × 1 1 + dist(ua, ub) (3) where W denotes the weight vector corresponding to the time of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To ensure the effectiveness of computing similarity, we use a toy example of two different residents with the same origin but different activity behavior patterns to study similarity measure as shown in Fig 3(a) as follows: Three different weight vectors are being considered when computing the similarity kernels for resident ua and ub using WWO, which are (1) uniform, (2) active-day focus, and (3) only active-day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The illustration of varying weight is presented in Fig 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' These three weight vectors try to highlight different temporal parts of the day and weight is adjusted ac- cordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' For instance, active-day focus emphasizes the active 0 2 4 6 8 Encoded Location ID 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time, from 0000 to 2359 u3 trajectory u20 trajectory (a) Two trajectory examples (users pair(u3, u20)) in encoded location for 11 Nov and 25 Oct 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time, from 0000 to 2359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='010 Weight Value active-day focus uniform only active-day (b) Weights used for calculating the similarity function Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 3: Toy examples for calculating similarity metric and custom weight for similarity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' period of the day (06:00am - 11:59pm), where only active-day only consider the active period (07:00am - 08:00pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that the total value of weight is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 and is allocated across 288 vectors depending on the weight characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We choose one of the weights depending on the emphasis of the outcome of the desired similarity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using some common similarity measurement as introduced in [37], a comparison of the similarity measurement is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' TABLE I: Comparison of Similarity Measurement Similarity Measurement Method Similarity Score WWO (active-day focus) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='7250 WWO (uniform) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='7778 WWO (only active-day) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='5675 Overlap [38] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8182 Eskin [39] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='9259 Goodall [40] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='9254 Based on the observation, the variation of overlap methods has similar results ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='7250 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8182, where the similarity score of the WWO with only active-day is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='5675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Meanwhile, the other methods (Eskin and Goodall) have a higher similarity score despite the visualization of two residents’ trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 3 shows different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Ideally, the overlap method presents the most straightforward method of computing the similarity score, which is roughly around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' However, the overlap method only highlights the similarity between two users during night time (08:00pm - 06:00am), which represents a large amount of time when users are inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Thus, after considering the different weightage similarity scores, active-day focus weight is more desirable for computing the similarity metric, where it emphasizes a more active period on the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By iterating the similarity score through different pairwise residents, we can obtain the similarity matrix, A as such: PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 6 A = � ���� 0 s(u1, u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' s(u1, un−1) s(u1, un) s(u2, u1) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' s(u2, un−1) s(u2, un) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' s(un−1, u1) s(un−1, u2) · · 0 s(un−1, un) s(un, u1) s(un, u2) · · s(un, un−1) 0 � ���� (4) In order to compute the laplacian matrix, degree matrix, D can be computed as follows: D = n � i � 1 Ai,i ≥ 0 0 otherwise , (5) where it represents a non-zero affinity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Next, using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 4 and Eqn 5, the normalized Laplacian matrix, L can be generated using following equation: L = I − D−1/2AD−1/2, (6) where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Algorithm 3 Spectral Clustering Algorithm Data: Spatio-temporal Matrix, X Result: Cluster List, C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Aggregate data into daily windows data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Calculate the affinity matrix, A as follows: for i ← 1 to n do for j ← 1 to n do calculate the similarity metric using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2 for each i and j resident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute the degree matrix, D using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Calculate the Laplacian Matrix, L using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Calculate the Eigenvector, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Determine k value based on sum of squared distance (SSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Perform k-means and obtain cluster list, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' return C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, we compute the k generalized eigenvectors using the normalized Laplacian matrix, L as follows: Lu = λDu (7) where vector, u is the calculated using the smallest k eigen- value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By combining the afore-mentioned equation, we formu- late the spectral clustering in the Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The computa- tional complexity of the spectral clustering is O(n3), which is the most time consuming part of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the proposed algorithm, we perform group detection and show the result in next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Results and Validation In order to study the optimal k for deciding the number of clusters in the nursing home, we apply the sum of squared distance (SSD) for different number of k values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The SSD can be defined as follows: k � n � a=1 a−1 � b=1 (dist(ua, ub))2, (8) where it utilizes dist() from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' It calculates the summa- tion of squared distance for different k values in clustering, which lower value indicates higher similarity between resi- dents in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Ideally, we want to find clusters within the range of 2 to 7 clusters out from 50 residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The SSD result is presented in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Sum of Squared Distance 2250 2000 1750 1500 1250 1000 750 500 2 3 4 5 6 7 Number of Cluster, k Lowest Distance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 4: SSD for different number of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on observation, k=5 is an ideal number for the clusters as the SSD value is the lowest compared to other choices of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using k=5, the clustering algorithm is formulated as shown in Algorithm 3, and the clustering result is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We observed clusters are separated by building level and have their own characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The level 3 residents generally can be divided into three different groups, where level 2 can be divided into 2 different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The residents from Cluster 1 tend to have a longer visit duration at level 3 public area compared to Cluster 2 and Cluster 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Meanwhile, resident u15 in Cluster 3 spends his/her lunchtime in the level 3 public area instead of the basement public area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Meanwhile, residents from Cluster 2 tend to spend less time in public places, where Cluster 3’s residents spend more time in level 2 public areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In Cluster 2, resident u48 and resident u49 spend time in their room without going to public spaces, which is quite peculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Level 2’s residents in Cluster 4 mostly spend their time in the public area, while residents in Cluster 5 spend less time in the public area and spend most of their time in their respective origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on these 5 clusters, we will generate daily activity based on their group to compute their daily activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Further details of fusing individual activity behavior and group activity behavior are elaborated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' RESIDENTS’ DEVIATED ACTIVITY BEHAVIOR STUDY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Residents’ Hybrid Norm Computation Despite there is group activity behavior among the nursing home’s residents in their daily routine, there exist cases of some residents who behave differently than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The trajectory of the residents may vary day to day depending on the group schedule for the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, it is crucial to consider the residents’ groups daily trajectory to provide additional information to generate daily norms for studying potential deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To show an example of deviated activity behavior detection, we use resident u21’s norm and the group norm (Cluster 3) as an example to demonstrate the working of extracting the deviated locations from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The example of input data and hybrid norm generation are shown in following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6: Using the example, we extract two different norms from the resident u21 to generate a hybrid norm for 10 October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' PUBLISHED IN ELSEVIER INTERNET OF THINGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 25 JANUARY 2023 7 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u0 List of users u1 u5 u2 u6 u31 u34 u36 u39 u40 u41 u46 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u9 List of users u12 u18 u17 u19 u20 u22 u23 u25 u33 u48 u49 (a) Cluster 1 (b) Cluster 2 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u3 List of users u4 u14 u13 u15 u21 u43 u44 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u10 List of users u11 u24 u16 u26 u27 u28 u30 u32 u35 u38 u42 u46 (c) Cluster 3 (d) Cluster 4 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u7 List of users u8 u37 u29 u47 Data Lost Origin Level 3 Origin Level 2 Private Area Level 3 Private Area Level 2 Public Area Level 3 Public Area Level 2 Public Area Basement Level Restricted Area Location Legends (e) Cluster 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 5: Group Clustering Result based on k=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Both norms used as data input to generate a hybrid norm can be defined as follows: Individual Norm, normind - The individual norm rep- resents the regular pattern of the resident over the data collection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The regular pattern is generated through the aggregated of the resident’s valid data, which can be obtained using the user group detection module’s Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that the individual norm is obtained using aggregated trajectory yi for each resident, ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Group Norm, normgrp - Group norm denotes the reg- ular pattern of all clustered residents obtained using the clustering algorithm in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that this pattern changes every day as a nursing home have different activities/events arranged during the active hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, normgrp is calculated every day for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Utilizing these norms, we propose a data fusion method to generate a hybrid norm norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The main challenge here is to address the transition method between normind and normgrp, where overlapping between two norms is possible when combining both norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' There are two transition periods in the day of study, which are the group activity start transition period and group activity end transition period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' One needs to decide the exact timing for transition based on the starting period of the group using p2 and ending period p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Moreover, similar consideration applied for the ending time of the private period, p1 and the starting period of private period p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The p5 denotes the starting of the group activity, while p6 represents the group activity ending period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To compute the transition period [p5, p6] using [p1, p2, p3, p4], the following Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 9 is used and 10 to decide the transition period for group starting and group ending period: p5 = �p1 if ∆t|p1 − p2| ≤ h or ∆t(p1 − p2) ≥ 0 p2 otherwise , (9) p6 = �p4 if ∆t|p3 − p4| ≤ h or ∆t(p3 − p4) < 0 p3 otherwise , (10) where h denotes the time gap limit between 2 time slots such as (p1, p2) and (p3, p4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that if the time overlap such as ∆t(p1 − p2) ≥ 0 or ∆t(p3 − p4) < 0, the earliest time is considered as transition period as default overlapping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After deciding the transition period, we compute the hybrid norm, norm by combining the individual and group norm of the nursing home residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The merging process between :PUBLISHED IN ELSEVIER INTERNET OF THINGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 25 JANUARY 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Aggregate to compute individual norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Aggregate to compute group norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Generate p1 and p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Generate p2 and p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Determine the transition period p5 and p6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Generate hyrbid norm for day data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Individual Data (Different Days) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Group Data (Same Day - 10 Oct 2019) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Determine Individual Start and End Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Hybrid Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Compute Adative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Determine Group Start and End Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Individual Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Group Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Δt|p1-p2| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='{ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Δt|p3-p4| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='{ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Hybrid Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Merge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Group Day Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Transition Period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Transition Period ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Group Day End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='normgrp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='normind ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='15:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='21:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='23:59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Encoded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Location ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Hybrid Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Encoded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Location ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='15:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='21:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='23:59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Group Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Encoded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Location ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='15:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='21:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='23:59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Individual Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='03:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='09:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='15:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='21:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='23:59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 Group norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' normgrp 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 Individual norm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' normind 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 Date 06 Jan 2020 2019 27 Nov 2019 10 Oct 2019 22 Oct 2019 05 Dec 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 u4 Cluster Member Data u13 u21 u15 u14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6: Toy example for extracting deviated activity behavior using resident u21’s trajectory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' normind and normgrp can be formulated as follows: normd,i[t = 0 : p5] ← normind[t = 0 : p5] normd,i[p5 : p6] ← normgrp[p5 : p6] normd,i[p6 : t = 288] ← normind[t = p6 : 288] (11) where the period between t=0 to p5 and t=p6 to 288 uses norm from normind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The p5 to p6 denotes the group time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In addition, d represents the particular day we are studying, while i denotes each of the residents in the nursing home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By combining the aforementioned equations, the hybrid norm normd,i for each resident ui can be computed using Algo- rithm 4 and the computational complexity is O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Residents’ Deviated Activity Behavior Identification After computing the adaptive hybrid norm for each user, we extract the deviated locations using a filtering function based on the norm and daily input data, Xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Since the encoded location is categorical data, the binary comparison method is adopted to compare whether the input data is the same as the generated adaptive hybrid norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 12 describe the afore-mentioned process: filter(xd,i, normi) = �null if xd,i = normi xd,i if xd,i ̸= normi (12) where the filter() function is applied to the daily input data to remove norm data, while preserving the deviated locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Therefore, the following Algorithm 5 is proposed to compute for daily trajectory by iterating the filter function every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using the example stated earlier in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6, the deviated activity behaviors of resident u21 can be computed for 10 October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The deviated activity behaviors are highlighted as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' In the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 7, we show an example of obtaining a hybrid norm based on the data fusion approach by fusing group features and individual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After performing the filtering function, the input trajectory is highlighted in red color, which the users’ trajectory is different than the computed hybrid norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Here, three different deviated activity behaviors are detected across different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The first deviated activity behavior happened around 7:30am, where the resident u21 visit level 3 public area for 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After that, resident u21 return to his origin and stayed from 10:45am to 2:00pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the hybrid norm, the resident from cluster 4 went back to their room after 11:45pm, which is conflicted with the resident u21’s hybrid norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Lastly, the last deviated activity behavior occurred around 4:50pm and continued until the end of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Supposedly, the resident u21 would spend PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 9 Algorithm 4 Generating Adaptive Hybrid Norm Data: group label, Day Data Xd, origin xori, threshold h Result: Hybrid Norm normd,i function generateHybridNorm() 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Obtain normind using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Obtain normgrp using getGroupNorm(group label, Xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute p1 and p4 using determineDayStartEnd(normind, xori, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute p2 and p3 using determineDayStartEnd(normgrp, xori, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute p5 and p6 using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 9 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Merge the data to become adaptive hybrid norm using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11 function getGroupNorm(group label, Xd) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Initialize the same group dictionary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Add the residents’ data, X if they are from the same group foreach Xd,j do if resident ∈ group label then same group dictionary append Xd,j 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Check whether the same group dictionary has more than 2 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' if length( same group dictionary) > 2 then return aggregate same group dictionary using Algorithm 2 else return NULL function determineDayStartEnd(X, xori, h) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute day start pointer for t, 0 to 144 do if xt ̸= xori for h interval then break 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Store the value, DayStart ← t 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute day end pointer for t, 288 to 144 do if xt ̸= xori for h interval then break 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Store the value, DayEnd ← t return DayStart, DayEnd Algorithm 5 Compute Deviated Activity Behaviors Data: norm Result: Hybrid Norm normd,i function ComputeDeviatedEvents(X, normd) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Initialize Ed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Compute each inputted day using filter function in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 12 for i, from 0 to d do Ed append filter(xi ∈ X, normi) return Ed most of his/her time in origin, but instead, went to public spaces around level 3 and level 2 private area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This shows the process of detecting deviated activity behaviors for a particular user by integrating the group norm and individual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By fusing both information, the deviated activity behaviors can be computed for every resident in the nursing home and further study the types of deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Analysis and Classification of Deviated Activity Behaviors After obtaining deviated activity behaviors, we proceed to identify the types of it among the nursing home’s residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' We found out some residents suffered from sleep irregularity, and some of them absent during the group activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on observation of the data, three different deviated activity behaviors can be generalized, which are (1) sleep irregularity, (2) awake irregularity, and (3) private visiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The definition of deviated activity behaviors are listed as follows: (1) sleep irregularity - The sleep irregularity denotes the resident goes to another location instead of staying back to his/her origin at the usual timing during night time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time, from 0000 to 2359 0 1 2 3 4 5 6 7 8 Encoded Location ID Input Data, xi 0 1 2 3 4 5 6 7 8 Encoded Location ID Deviated Event 1 Deviated Event 2 Deviated Event 3 Hybrid Norm, norm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 7: Example of extracting deviated activity behaviors for resident u21 on 10 October 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' (2) awake irregularity - The awake irregularity denotes the resident either leaves the room late or early compared to their routine schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' (3) private visiting - The resident went to another staying room (staying room for level 2 or level 3) instead of the common places cluster or he/she normally will present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently, a rules-based classification method is introduced to define the features for identifying deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The rationality of using such an approach is that labels can be assigned based on the predefined rules using a statistical approach given there is no ground-truth available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that each user can be assigned multiple classification labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' TABLE II: Definition of the Transition Period Temporal Period Time Range of the Day Midnight day start → ( p5 2 ) Morning p5 2 → p5 Pre-group p5 → (p5 + 60mins) Group (p5 + 60mins) → (p5 − 60mins) Post-group (p6 − 60mins) → p6 Evening p6 → ( t=288−p6 2 ) Midnight ( p6 2 ) → day end To generate features for the classification, we further break down the time of the day into seven different finer temporal periods as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By breaking down into different periods, the time before and after group activity can be investigated to generate features for studying deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that p5 and p6 shown here are based on the average time start and time end throughout the data collection period for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 8 as a reference, the finer temporal periods are introduced as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Next, the probabilities of deviated locations’ occurrence are calculated based on the daily filtered deviated location for each Group Midnight Midnight Evening Morning Pre-Group Post-Group { { p5 p6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 8: Illustration of the finer temporal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' PUBLISHED IN ELSEVIER INTERNET OF THINGS, 25 JANUARY 2023 10 finer temporal period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This can be used to define rules using the probabilistic calculation based on the deviated activity behavior statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The location probability, P(Ei,x) can be computed using following Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 13 P(Ei,x) = q(Ei,x) c (13) where i denotes the timeslot of the day and x represents the location resident went when a deviated behavior occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The q() counts the occurrence number of deviated activity behavior Ei,x and c denotes the total number of the valid time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By combining the probability of deviated locations of every user, we can study the potential timeslot for each user, where the deviated activity behaviors occurred most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' To simplify the locations of where deviated activity behaviors occurred, encoded locations with similar functionalities are combined and study the probability for each timeslot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The combined encoded list are listed as follows: (c1) null, (c2) origin, (c3) public area, (c4) private area, and (c5) restricted location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After computing the probabilities of deviated locations for the nursing home’s residents, we illustrate the distribution of deviated locations using a violin plot as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The category c1 represents the probability of residents not involving in deviated activity behavior at a different time of the day, where category c2 to c5 represents the location for a deviated activity behavior to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The probability of deviated activity behavior occur at other locations c2 to c5 is small which the value ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' One can observe a lot of deviated activity behaviors that happened mostly in the public area (c3) and follow by origin (c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' That being said, to further understand those involved in sleep irregularity and social visiting events, attention is given to the residents who yield a higher probability of deviated activity behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Specifically, we are interested in the distribution of residents, whose probability is higher than the median for the encoded location c2 to c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Those residents have higher chances of showing deviated activity behaviors compared to the median of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Using the characteristic of a violin plot, the classification threshold is defined based on the upper adjustment value (UAV) and lower adjustment value (LAV), which are computed through the first or third quantile ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='5× interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the specific timing’s distribution, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0 Deviated Events Distribution, P(Ex) Deviated Location, Ex Null (Normal) Origin Level 2, 3 Public Area B1 Private Level 2, 3 Restricted Area Level 2, Level 3 temporal period midnight morning pre-group group post-group evening C1 C2 C3 C4 C5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 9: Violin plot for the deviated location w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' different temporal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that c1 denotes probability of residents being normal while c2 to c5 represent the deviated locations based on their functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content="7% 0 classification label 1 classification label 2 classification labels 3 classification labels Nursing home's resident classification labels distribution classification label sleep irregularity awake irregularity private social visiting Percentage 40% 30% 30% classification label awake irregularity private social visiting Percentage sleep irregularity + private social visiting awake irregularity + sleep irregularity + 50% 25% 25% Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 10: Distribution of residents based on the number of classification labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' the UAV and LAV values are extracted as shown in Table III to define the threshold values for the classification rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' TABLE III: UAV and LAV Extracted for Classification Rules Deviated Location c1 c2 c4 LAV UAV UAV midnight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='8762 Morning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='7323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0118 pre-group 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='2717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0312 group 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0521 post-group 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0223 evening 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='7522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='0174 Based on the input from Table III, we define the classifica- tion rules as presented in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' After defining the classification rules, we classify every nursing home’s residents based on their probability for the finer hybrid temporal period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' A general statistic for the clas- sified labels after performing the rules-based classification is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the result, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content='4% of residents do not have a classification label, which indicates the major- ity of the residents do not have deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The remaining 37% of the residents are associated with one classification label only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' of which 40% of them are having TABLE IV: Deviated Activity Behavior Classification Rules Deviated Activity Behavior Classification Rules Sleep Irregularity c1 midnight < LAV or c1 Evening < LAV Awake Irregularity c1 morning < LAV or c2 pre-group < LAV Private Visiting c4 morning > UAV or c4 pre-group > UAV or c4 group > UAV or c4 post-group > UAV or c4 evening > UAV PUBLISHED IN ELSEVIER INTERNET OF THINGS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 25 JANUARY 2023 11 (a) Example of awake irregularity detected for resident u35 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 Deviated Common Activity Behavior Activity Behavior midnight morning pre-group group post-group evening midnight Awake Irregularity (b) Example of sleep irregularity detected for resident u5 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 midnight morning pre-group group post-group evening midnight Sleep Irregularity Deviated Common Activity Behavior Activity Behavior (c) Example of private visiting detected for resident u21 00:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 23:59 Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' from 0000 to 2359 midnight morning pre-group group post-group evening midnight Private Social Visiting Deviated Common Activity Behavior Activity Behavior Data Lost Origin Level 3 Origin Level 2 Private Area Level 3 Private Area Level 2 Public Area Level 3 Public Area Level 2 Public Area Basement Level Restricted Area Location Legends Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11: Example of the deviated activity behavior study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Note that the y-axis consists of sample data extracted user trajectory on selected day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The common activity behavior denotes trajectory data without any deviated activity behavior, while the deviated activity behavior category represents the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' private social visiting irregularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' On the other hand, half of the residents with two classification labels mainly consist of sleep and awake irregularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the classification labels generated, we examine three residents as case studies to study normal daily activ- ity behavior and deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Three different deviated activity behaviors are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11(a), the resident u35 is not waking up based on his/her daily schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on observation, there are two types of awake irregularity, which are wake up earlier or wake up way later than the normal wake up time (around 7:30 am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Next, we investigate the sleep irregularity of resident u5 as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The resident u5 normally goes back to the origin after 9:00pm, but in the deviated activity behavior’s extracted days, he/she remained at the public area until the end of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' These activity behaviors normally do not occur in his/her normal routine, and this is a good representation of the sleep irregularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Subsequently in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' 11(c), one can discover that the resident u21 absent during the group activity for few days, where he/she went to a private area instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This phenomenon indicates the resident u21 either is having social interaction with other residents in their private space or avoiding group activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This could be potentially a deviated activity behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By utilizing the proposed hybrid deviated activity behavior classification, we managed to obtain types of residents’ de- viated activity behaviors in a nursing home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' This provides insight into the nursing home’s management regarding the residents’ deviated activity behavior and attention can be provided to the individual with needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' CONCLUSION In this paper, we present a location-based deviated activity behavior detection system for the Salvation Army, Peacehaven Nursing Home, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' The 50 residents can be segmented into different groups based on their activity behavior, which contributes to formulating group norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By combining group and individual norms, we can generate a hybrid norm to identify the common behavior for each resident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' By under- standing residents’ normal activity behavior, the deviated ac- tivity behavior can be differentiated and extracted from normal activity behavior to study it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Based on the types of deviated activity behavior, three categories of deviated activity behavior are proposed, which are (1) sleep irregularity, (2) awake irregularity, and (3) private visiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Next, three users’ normal and deviated activity behavior are studied after performing rules-based classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' For future works, we plan to incorporate more data sources to generate more accurate deviated activity behavior detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Also, a real-time deviated activity behavior system is part of the future research direction as the dynamic formation of group activity is yet another issue to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Another aspect that can be improved is getting ground-truth for the data collected to further enhancing deviated activity behavior detection module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' Moreover, a more throughout study on the compatibility with different countries’ data protection rules would be part of the research interest to reach out to more nursing homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFIT4oBgHgl3EQfeCsB/content/2301.11272v1.pdf'} +page_content=' L.' 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+Maxime W. Lafarge and Viktor H. Koelzer +Department of Pathology and Molecular Pathology, University Hospital and University of Zurich, Zurich, Switzerland +Abstract +Making histopathology image classifiers robust to a wide range of real-world variability is a challenging task. Here, we +describe a candidate deep learning solution for the Mitosis Domain Generalization Challenge 2022 (MIDOG) to address the +problem of generalization for mitosis detection in images of hematoxylin-eosin-stained histology slides under high variability +(scanner, tissue type and species variability). Our approach consists in training a rotation-invariant deep learning model +using aggressive data augmentation with a training set enriched with hard negative examples and automatically selected +negative examples from the unlabeled part of the challenge dataset. +To optimize the performance of our models, we investigated a hard negative mining regime search procedure that lead +us to train our best model using a subset of image patches representing 19.6% of our training partition of the challenge +dataset. Our candidate model ensemble achieved a F1-score of .697 on the final test set after automated evaluation on +the challenge platform, achieving the third best overall score in the MIDOG 2022 Challenge. +Introduction +To support the research community with the development +of new mitosis detection algorithms that are robust to +scanner variability, the MIDOG 2021 Challenge [1] lead +to an overview of efficient approaches towards solving this +task. To further encourage the development of models that +can generalize beyond inter-scanner variability, the MIDOG +2022 Challenge was initiated [2], offering a unique oppor- +tunity to compare the generalization ability of mitosis de- +tectors in a blind manner, as the challenge organizers inde- +pendently evaluate candidate solutions on held-out sets of +images from undisclosed and unseen domains. +This opportunity motivated us to revisit the pioneer +methodology proposed by Cire¸san et al. [3], and to assess +the relative performance of standard methods in the context +of modern deep convolutional neural network architectures +and large high-variability mitosis datasets as the one pro- +vided for this challenge. +Our approach consists of a patch-based training proce- +dure that we used to train deep learning models that can +then be applied to detect mitoses on unseen images, build- +ing upon the strategy we employed for the MIDOG 2021 +challenge [4]. We incrementally made changes to this train- +ing procedure over the development phase of the challenge +and monitored performances on our validation partition of +the challenge dataset in order to select and submit our best +performing model. In this paper, we describe the different +components that constitute our submitted solution, includ- +ing a fine-grained assessment of a hard negative mining pro- +cedure that we consider to be the main contributing part of +our solution. + + +BN + ReLU +P4 Convolution +4×c×c×1×1 +Input +4×c×h×w +BN + ReLU +P4 Convolution +4×c×c×k×k +BN + ReLU +P4 Convolution +4×c’×c×1×1 +Shape Matching +& Addition +Output +4×c’×(h-k+1)×(w-k+1) +Residual Block +(parameters: c’, k) +Input +3×(78+2m)×(78+2m’) +Hidden Representation +4×8×(37+m)×(37+m’) +Hidden Representation +64×(1+m)×(1+m’) +Output +1×(1+m)×(1+m’) +P4 Max Projection +Residual Block +c’=64, k=1 +Residual Blocks +c’=8, k=3 +×9 +Residual Blocks +c’=16, k=3 +×9 +Residual Blocks +c’=32, k=1 +×3 +Lifting P4 +Convolution +32×3×5×5 +BN + ReLU +P4 Convolution +4×8×32×1×1 +Max Pooling (2×2) +Convolution +64×64×1×1 +BN + ReLU +Sigmoid +Convolution +1×64×1×1 +P4 ResNet-70 +Raw Input Image +Output Probability Map +Dense Application of +Trained Model +Detected Objects +(local maximum detection & thresholding) +Figure 1: +Architecture of the 70-layer ResNet used in this work and +example of its application to a raw input H&E histopathology image. +The shape of output tensors is written with the following format: +(Orientations×)Channels×Height×Width. The +shape of trainable operator tensors is written with the following format: +(Orientations×)Out.Channels×In.Channels×Ker.Height×Ker.Width. +Model Architecture +We implemented a customized 70-layer ResNet architecture +[5] to model the confidence probability for input images to +be centered on a mitotic figure within a receptive field of +78×78 pixels. We replaced standard convolutional layers by +1 +arXiv:2301.01079v1 [eess.IV] 3 Jan 2023 + +.MIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining +Table 1: +Data augmentation protocol: for each input image patch, the following list of transformations is scanned and applied with a given +probability. Transformation parameters are randomly sampled in a given interval. The two variants of the protocol used to train our +submitted model ensemble are detailed here. +Transformation +Policy A +Policy B +Coefficients +Probability +Coefficients +Probability +Transposition +– +50% +– +50% +Elastic Deformation +– +100% +– +50% +Spatial Shift (∆x,y) +[−12px, 12px] +100% +[−12px, 12px] +100% +Spatial Zoom (α) +[−10%, 20%] +50% +[−10%, 20%] +50% +(HLS) Hue Rotation (h) +[0◦, 360◦] +80% +[−60◦, 60◦] +50% +Color Shift (cr,g,b) +[−51, 51] +80% +[−51, 51] +50% +Contrast Correction (µr,g,b) +[0.8, 1.2] +80% +[0.8, 1.2] +50% +Gamma Correction (γr,g,b) +– +0% +[0.8, 1.2] +50% +P4-group convolutional layers [6] to guarantee invariance of +our models to 90-degree rotations without requiring train- +time or test-time rotation augmentations, with a low com- +putational overhead. This change was further motivated by +the improvment of performance across multiple histopathol- +ogy image classification tasks for models using this type of +operation reported in the literature [7–9]. This architecture +was adjusted to enable dense application of the models to +arbitrarily large input images. A detailed flowchart of this +architecture and an example of the application of a trained +model to an input image are shown in Figure 1. +Dataset Partitioning +To train models and evaluate their performance, we exclu- +sively used the data provided for the track 1 of the MIDOG +2022 Challenge [2]. We split the provided annotated images +according to a 80-20 training-validation scheme such that +labels and domains were stratified (training set: 7588 mi- +toses from 283 images; validation set: 1913 mitoses from 71 +images). +Given the provided ground-truth locations of mitotic fig- +ures in these images, we derived a set of image patches of +positive examples centered on mitotic figures and a set of +image patches of all negative examples whose center is suffi- +ciently distant from the center of annotated mitotic figures. +Training Procedure +All models were trained by minimizing the cross-entropy +loss via stochastic gradient descent with momentum (initial +learning rate 0.03 and momentum 0.9) using input mini- +batches of size 128. We used a cyclic learning rate schedul- +ing [10] with a cycle period of 10k iterations and applied +weight decay regularization (coefficient 10−4). Mini-batches +were generated with randomly sampled image patches of size +78×78, balanced between positive and negative examples. +All image patches were randomly transformed according to +an augmentation protocol (including channel-wise intensity +distortions) whose operations and parameter ranges are de- +tailed in Table. 1. For evaluation purposes, we saved the +weights of the model that achieved the lowest validation loss +within 150k training iterations. +Figure 2: +Average precision (AP) on the validation set of models +trained with subsets of non-mitosis examples of different sizes. These +subsets were either generated via random sampling or by hard +negative mining. During training, the positive class is oversampled to +ensure balance of classes in training batches. Circles and bars +represent the mean and standard deviation of AP for three repeated +experiments with different random seeds. The dashed line indicates +the mean AP obtained when training using all the possible negative +examples of our training partition. +Fine-Grained Hard Negative Mining +Hard negative mining (HNM) has become a standard pro- +cedure for the development of mitosis detectors since the +solution proposed by [3], and aims at improving the model +performance by using a well-chosen subset of ”hard” nega- +tive examples for training instead of using randomly sampled +negative examples. This procedure typically requires train- +ing a first model with all the available positive and negative +examples, then ranking all negative examples based on the +confidence score output by this trained model, and finally +keeping the negative examples with a score above a fixed +cutoff threshold as a set of ”hard negatives” to be used to +repeat training and improve performances. +For this challenge, we considered this threshold as a hyper- +parameter and searched for an optimal number of hard neg- +2 + +hard negative mining +0.84 +random sampling +0.82 +0.80 +0.78 +0.74 +0.72 +0.70 +104 +105 +106 +Number of Non-Mitosis Training Examples (log scale)MIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining +ative examples to select that would maximize performance +on the validation set. A summary of the binary search we +conducted for this parameter is shown in Figure 2. +This fine-grained assessment enabled the selection of an +optimal subset of 121 738 hard negative examples of our +training partition, which improved validation performances +in comparison to using randomly sampled training exam- +ples. Our submitted model ensemble was trained using this +optimal subset of image patches along with all positive ex- +amples whose joint total pixel count represents 19.6% of the +overall pixel count of our training partition. +Automated Enrichment +with Negative Examples +To further enrich the dataset with additional negative ex- +amples we implemented a stain unmixing algorithm derived +from [11] to first separate the hematoxylin, eosin and resid- +ual components in the unlabeled images of the dataset, and +then automatically extract image patches with high optical +density in the estimated residual component to enrich the +training set with extra negative examples. This procedure +resulted in the selection of 72 negative examples after appli- +cation of an optimal hard negative mining selection rule as +described in the previous section. A batch of such selected +examples is shown in Figure 3. This approach was motivated +by the observation that true positive mitotic events reside +in defined stain vectors and are separable from background +events such as pigmentation and ink that form common im- +postors. +Figure 3: +Example of automatically selected image patches from +the unlabeled part of the MIDOG 2022 dataset based on the optical +density of their residual component after application of a stain +unmixing procedure. These selected image patches were used to +enrich the training set. +Inference Pipeline +Once our models were trained, we produced prediction maps +by applying them densely on test images, and then derived +candidate detection locations as the set of local maxima. We +then considered all candidate locations with a prediction +score above a threshold value as final detection locations +(this threshold was chosen as the one maximizing the F1- +score on the validation set). +For our final submission, we created a model ensemble +by taking the agreement between the detection sets of two +models trained under two variants of our augmentation pro- +tocol as detailed in Table 1. A comparison of performance +of our models against baselines is summarized in Table 2. +Discussion +We present a candidate algorithm that achieves moderate +generalization performance with an overall F1-score of .696 +on the final test set of the challenge (set of images from +ten unseen domains), while relying mostly on conventional +methods (patch-based training procedure, ResNet architec- +ture, cross-entropy loss minimization, hard negative mining, +standard augmentation protocol). +Performance on the different domains of the test sets were +heterogeneous indicating that the investigated pipeline only +enables generalization to a subset of the held-out test do- +mains of the challenge. +Hard Negative Mining played an important role in achiev- +ing the final performance of our submitted solution: the +search for an optimal amount of hard negative training ex- +amples helped improving performances on the validation set +with an increase of Average Precision from .760 to .826. Our +comparative analysis suggests that this level of performance +could not have been achieved from training using only arbi- +trary negative examples. +Furthermore, we investigated the use of a conventional +stain unmixing method to automatically extract potential +negative examples in unlabeled images. As this approach +did not decrease the performances on the validation set, we +assumed this helped our models better generalizing. +We +thus used the resulting enriched training set to train our +final models, yet, because of the very small number of train- +ing examples selected this way, we cannot further conclude +whether the effect of this strategy was beneficial or not re- +garding the performance on the final test set. Because this +enrichment procedure relies on the strong assumption that +extracted examples belong to the non-mitosis class, it should +be kept in mind that positive examples could still poten- +tially be extracted. Here, we recommend that the examples +extracted by such a method should ideally be reviewed by +expert annotators to ensure the correctness of the class of +these additional training examples. +Although the reported variations of our augmentation +protocol produced similar results on the validation set, we +observed that they produced heterogeneous performances +on the preliminary test set (Table 2). This suggests that +generalization to the domains of the preliminary test set +is sensitive to variations of the augmentation protocol used +for training. Indeed, the two reported augmentation policies +(Table 1) helped generalizing to different parts of the unseen +variability of the preliminary test set (Policy A enabled bet- +ter generalization for Tumor 1 whereas Policy B enabled +3 + +Original Image +Eosin +Component +Hematoxylin +Component +Residual +ComponentMIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining +Table 2: +Comparison of F1-scores of our trained models (two different augmentation policies and their ensemble) and baselines of the +MIDOG 2022 Challenge on the four tumor types (T1,2,3,4) of the hidden preliminary test set and on the final test set of the challenge. +Model +Internal +Preliminary Test +Final Test +Validation +Overall +T1 +T2 +T3 +T4 +Overall +ours +(Policy A) +.784 +.646 +.783 +.726 +.548 +.708 +– +ours +(Policy B) +.787 +.571 +.757 +.775 +.428 +.571 +– +ours +(Ensemble) +.791 +.690 +.758 +.735 +.653 +.610 +.696 +MIDOG 2022 +(Baseline 2) +– +.715 +.744 +.732 +.692 +.711 +.714 +MIDOG 2022 +(Baseline 1) +– +.629 +.753 +.608 +.585 +.743 +– +better generalization for Tumor 2). These results corrob- +orate the known association between chosen augmentation +policies and generalization performance: this motivates us +to further study how to best design and select augmentation +protocols to improve domain generalization. +References +[1] Marc Aubreville, +Nikolas Stathonikos, +Christof A Bertram, +Robert Klopleisch, Natalie ter Hoeve, Francesco Ciompi, Frauke +Wilm, Christian Marzahl, Taryn A Donovan, Andreas Maier, +et al. +Mitosis domain generalization in histopathology images +– the MIDOG challenge. Medical Image Analysis, 2022. +[2] Marc Aubreville, Christof Bertram, Katharina Breininger, Samir +Jabari, Nikolas Stathonikos, and Mitko Veta. MItosis DOmain +Generalization challenge 2022. In Proceedings of the International +Conference on Medical Image Computing and Computer-Assisted +Intervention (MICCAI), 2022. doi: 10.5281/zenodo.6362337. +[3] Dan C Cire¸san, Alessandro Giusti, Luca M Gambardella, and +J¨urgen Schmidhuber. Mitosis detection in breast cancer histology +images with deep neural networks. In Proceedings of the Interna- +tional Conference on Medical Image Computing and Computer- +Assisted Intervention (MICCAI), 2013. +[4] Maxime W Lafarge and Viktor H Koelzer. Rotation invariance +and extensive data augmentation: A strategy for the MItosis DO- +main Generalization (MIDOG) challenge. In International Con- +ference on Medical Image Computing and Computer-Assisted In- +tervention, 2021. +[5] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Iden- +tity mappings in deep residual networks. In European Conference +on Computer Vision (ECCV), pages 630–645, 2016. +[6] Taco Cohen and Max Welling. Group equivariant convolutional +networks. In Proceedings of the International Conference on Ma- +chine Learning (ICML), pages 2990–2999, 2016. +[7] Bastiaan S Veeling, Jasper Linmans, Jim Winkens, Taco Cohen, +and Max Welling. Rotation equivariant CNNs for digital pathol- +ogy. +In Proceedings of the International Conference on Medi- +cal Image Computing and Computer-Assisted Intervention (MIC- +CAI), 2018. +[8] Maxime W Lafarge, Erik J Bekkers, Josien PW Pluim, Remco +Duits, and Mitko Veta. Roto-translation equivariant convolutional +networks: Application to histopathology image analysis. Medical +Image Analysis, 68:101849, 2021. +[9] Simon Graham, David Epstein, and Nasir Rajpoot. Dense steer- +able filter CNNs for exploiting rotational symmetry in histology +images. IEEE Transactions on Medical Imaging, 39:4124–4136, +2020. +[10] Ilya Loshchilov and Frank Hutter. SGDR: Stochastic gradient de- +scent with warm restarts. International Conference on Learning +Representations (ICLR), 2017. +[11] Marc Macenko, Marc Niethammer, JS Marron, David Borland, +John T Woosley, Xiaojun Guan, Charles Schmitt, and Nancy E +Thomas. A method for normalizing histology slides for quantita- +tive analysis. In Proceedings of the IEEE International Sympo- +sium on Biomedical Imaging (ISBI), 2009. +4 + diff --git a/S9AzT4oBgHgl3EQfJfsf/content/tmp_files/load_file.txt b/S9AzT4oBgHgl3EQfJfsf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eec1667dd968b84ef76ad211636e674da2b14434 --- /dev/null +++ b/S9AzT4oBgHgl3EQfJfsf/content/tmp_files/load_file.txt @@ -0,0 +1,189 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf,len=188 +page_content='Fine-Grained Hard Negative Mining: Generalizing Mitosis Detection with a Fifth of the MIDOG 2022 Dataset Maxime W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Lafarge and Viktor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Koelzer Department of Pathology and Molecular Pathology, University Hospital and University of Zurich, Zurich, Switzerland Abstract Making histopathology image classifiers robust to a wide range of real-world variability is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Here, we describe a candidate deep learning solution for the Mitosis Domain Generalization Challenge 2022 (MIDOG) to address the problem of generalization for mitosis detection in images of hematoxylin-eosin-stained histology slides under high variability (scanner, tissue type and species variability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Our approach consists in training a rotation-invariant deep learning model using aggressive data augmentation with a training set enriched with hard negative examples and automatically selected negative examples from the unlabeled part of the challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' To optimize the performance of our models, we investigated a hard negative mining regime search procedure that lead us to train our best model using a subset of image patches representing 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='6% of our training partition of the challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Our candidate model ensemble achieved a F1-score of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='697 on the final test set after automated evaluation on the challenge platform, achieving the third best overall score in the MIDOG 2022 Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Introduction To support the research community with the development of new mitosis detection algorithms that are robust to scanner variability, the MIDOG 2021 Challenge [1] lead to an overview of efficient approaches towards solving this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' To further encourage the development of models that can generalize beyond inter-scanner variability, the MIDOG 2022 Challenge was initiated [2], offering a unique oppor- tunity to compare the generalization ability of mitosis de- tectors in a blind manner, as the challenge organizers inde- pendently evaluate candidate solutions on held-out sets of images from undisclosed and unseen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This opportunity motivated us to revisit the pioneer methodology proposed by Cire¸san et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' [3], and to assess the relative performance of standard methods in the context of modern deep convolutional neural network architectures and large high-variability mitosis datasets as the one pro- vided for this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Our approach consists of a patch-based training proce- dure that we used to train deep learning models that can then be applied to detect mitoses on unseen images, build- ing upon the strategy we employed for the MIDOG 2021 challenge [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We incrementally made changes to this train- ing procedure over the development phase of the challenge and monitored performances on our validation partition of the challenge dataset in order to select and submit our best performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' In this paper, we describe the different components that constitute our submitted solution, includ- ing a fine-grained assessment of a hard negative mining pro- cedure that we consider to be the main contributing part of our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' BN + ReLU P4 Convolution 4×c×c×1×1 Input 4×c×h×w BN + ReLU P4 Convolution 4×c×c×k×k BN + ReLU P4 Convolution 4×c’×c×1×1 Shape Matching & Addition Output 4×c’×(h-k+1)×(w-k+1) Residual Block (parameters: c’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' k) Input 3×(78+2m)×(78+2m’) Hidden Representation 4×8×(37+m)×(37+m’) Hidden Representation 64×(1+m)×(1+m’) Output 1×(1+m)×(1+m’) P4 Max Projection Residual Block c’=64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' k=1 Residual Blocks c’=8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' k=3 ×9 Residual Blocks c’=16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' k=3 ×9 Residual Blocks c’=32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' k=1 ×3 Lifting P4 Convolution 32×3×5×5 BN + ReLU P4 Convolution 4×8×32×1×1 Max Pooling (2×2) Convolution 64×64×1×1 BN + ReLU Sigmoid Convolution 1×64×1×1 P4 ResNet-70 Raw Input Image Output Probability Map Dense Application of Trained Model Detected Objects (local maximum detection & thresholding) Figure 1: Architecture of the 70-layer ResNet used in this work and example of its application to a raw input H&E histopathology image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' The shape of output tensors is written with the following format: (Orientations×)Channels×Height×Width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' The shape of trainable operator tensors is written with the following format: (Orientations×)Out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Channels×In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Channels×Ker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Height×Ker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Model Architecture We implemented a customized 70-layer ResNet architecture [5] to model the confidence probability for input images to be centered on a mitotic figure within a receptive field of 78×78 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We replaced standard convolutional layers by 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='01079v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='IV] 3 Jan 2023 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='MIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining Table 1: Data augmentation protocol: for each input image patch, the following list of transformations is scanned and applied with a given probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Transformation parameters are randomly sampled in a given interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' The two variants of the protocol used to train our submitted model ensemble are detailed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Transformation Policy A Policy B Coefficients Probability Coefficients Probability Transposition – 50% – 50% Elastic Deformation – 100% – 50% Spatial Shift (∆x,y) [−12px, 12px] 100% [−12px, 12px] 100% Spatial Zoom (α) [−10%, 20%] 50% [−10%, 20%] 50% (HLS) Hue Rotation (h) [0◦, 360◦] 80% [−60◦, 60◦] 50% Color Shift (cr,g,b) [−51, 51] 80% [−51, 51] 50% Contrast Correction (µr,g,b) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='2] 80% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='2] 50% Gamma Correction (γr,g,b) – 0% [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='2] 50% P4-group convolutional layers [6] to guarantee invariance of our models to 90-degree rotations without requiring train- time or test-time rotation augmentations, with a low com- putational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This change was further motivated by the improvment of performance across multiple histopathol- ogy image classification tasks for models using this type of operation reported in the literature [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This architecture was adjusted to enable dense application of the models to arbitrarily large input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' A detailed flowchart of this architecture and an example of the application of a trained model to an input image are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Dataset Partitioning To train models and evaluate their performance, we exclu- sively used the data provided for the track 1 of the MIDOG 2022 Challenge [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We split the provided annotated images according to a 80-20 training-validation scheme such that labels and domains were stratified (training set: 7588 mi- toses from 283 images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' validation set: 1913 mitoses from 71 images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Given the provided ground-truth locations of mitotic fig- ures in these images, we derived a set of image patches of positive examples centered on mitotic figures and a set of image patches of all negative examples whose center is suffi- ciently distant from the center of annotated mitotic figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Training Procedure All models were trained by minimizing the cross-entropy loss via stochastic gradient descent with momentum (initial learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='03 and momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='9) using input mini- batches of size 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We used a cyclic learning rate schedul- ing [10] with a cycle period of 10k iterations and applied weight decay regularization (coefficient 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Mini-batches were generated with randomly sampled image patches of size 78×78, balanced between positive and negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' All image patches were randomly transformed according to an augmentation protocol (including channel-wise intensity distortions) whose operations and parameter ranges are de- tailed in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' For evaluation purposes, we saved the weights of the model that achieved the lowest validation loss within 150k training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Figure 2: Average precision (AP) on the validation set of models trained with subsets of non-mitosis examples of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' These subsets were either generated via random sampling or by hard negative mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' During training, the positive class is oversampled to ensure balance of classes in training batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Circles and bars represent the mean and standard deviation of AP for three repeated experiments with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' The dashed line indicates the mean AP obtained when training using all the possible negative examples of our training partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Fine-Grained Hard Negative Mining Hard negative mining (HNM) has become a standard pro- cedure for the development of mitosis detectors since the solution proposed by [3], and aims at improving the model performance by using a well-chosen subset of ”hard” nega- tive examples for training instead of using randomly sampled negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This procedure typically requires train- ing a first model with all the available positive and negative examples, then ranking all negative examples based on the confidence score output by this trained model, and finally keeping the negative examples with a score above a fixed cutoff threshold as a set of ”hard negatives” to be used to repeat training and improve performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' For this challenge, we considered this threshold as a hyper- parameter and searched for an optimal number of hard neg- 2 hard negative mining 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='84 random sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='70 104 105 106 Number of Non-Mitosis Training Examples (log scale)MIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining ative examples to select that would maximize performance on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' A summary of the binary search we conducted for this parameter is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This fine-grained assessment enabled the selection of an optimal subset of 121 738 hard negative examples of our training partition, which improved validation performances in comparison to using randomly sampled training exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Our submitted model ensemble was trained using this optimal subset of image patches along with all positive ex- amples whose joint total pixel count represents 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='6% of the overall pixel count of our training partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Automated Enrichment with Negative Examples To further enrich the dataset with additional negative ex- amples we implemented a stain unmixing algorithm derived from [11] to first separate the hematoxylin, eosin and resid- ual components in the unlabeled images of the dataset, and then automatically extract image patches with high optical density in the estimated residual component to enrich the training set with extra negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This procedure resulted in the selection of 72 negative examples after appli- cation of an optimal hard negative mining selection rule as described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' A batch of such selected examples is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This approach was motivated by the observation that true positive mitotic events reside in defined stain vectors and are separable from background events such as pigmentation and ink that form common im- postors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Figure 3: Example of automatically selected image patches from the unlabeled part of the MIDOG 2022 dataset based on the optical density of their residual component after application of a stain unmixing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' These selected image patches were used to enrich the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Inference Pipeline Once our models were trained, we produced prediction maps by applying them densely on test images, and then derived candidate detection locations as the set of local maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We then considered all candidate locations with a prediction score above a threshold value as final detection locations (this threshold was chosen as the one maximizing the F1- score on the validation set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' For our final submission, we created a model ensemble by taking the agreement between the detection sets of two models trained under two variants of our augmentation pro- tocol as detailed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' A comparison of performance of our models against baselines is summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Discussion We present a candidate algorithm that achieves moderate generalization performance with an overall F1-score of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='696 on the final test set of the challenge (set of images from ten unseen domains), while relying mostly on conventional methods (patch-based training procedure, ResNet architec- ture, cross-entropy loss minimization, hard negative mining, standard augmentation protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Performance on the different domains of the test sets were heterogeneous indicating that the investigated pipeline only enables generalization to a subset of the held-out test do- mains of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Hard Negative Mining played an important role in achiev- ing the final performance of our submitted solution: the search for an optimal amount of hard negative training ex- amples helped improving performances on the validation set with an increase of Average Precision from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='760 to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Our comparative analysis suggests that this level of performance could not have been achieved from training using only arbi- trary negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Furthermore, we investigated the use of a conventional stain unmixing method to automatically extract potential negative examples in unlabeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' As this approach did not decrease the performances on the validation set, we assumed this helped our models better generalizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' We thus used the resulting enriched training set to train our final models, yet, because of the very small number of train- ing examples selected this way, we cannot further conclude whether the effect of this strategy was beneficial or not re- garding the performance on the final test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Because this enrichment procedure relies on the strong assumption that extracted examples belong to the non-mitosis class, it should be kept in mind that positive examples could still poten- tially be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Here, we recommend that the examples extracted by such a method should ideally be reviewed by expert annotators to ensure the correctness of the class of these additional training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Although the reported variations of our augmentation protocol produced similar results on the validation set, we observed that they produced heterogeneous performances on the preliminary test set (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' This suggests that generalization to the domains of the preliminary test set is sensitive to variations of the augmentation protocol used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' the two reported augmentation policies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='(Table 1) helped generalizing to different parts of the unseen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='variability of the preliminary test set (Policy A enabled bet- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='ter generalization for Tumor 1 whereas Policy B enabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Original Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Eosin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Hematoxylin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Component ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Residual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='ComponentMIDOG 2022 - Mitosis Detection under Fine-Grained Hard Negative Mining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Table 2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='Comparison of F1-scores of our trained models (two different augmentation policies and their ensemble) and baselines of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='MIDOG 2022 Challenge on the four tumor types (T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='4) of the hidden preliminary test set and on the final test set of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Model Internal Preliminary Test Final Test Validation Overall T1 T2 T3 T4 Overall ours (Policy A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='784 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='646 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='783 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='726 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='548 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='708 – ours (Policy B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='787 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='571 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='757 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='775 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='428 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='571 – ours (Ensemble) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='791 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='690 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='758 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='735 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='653 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='610 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='696 MIDOG 2022 (Baseline 2) – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='715 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='744 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='732 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='692 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='711 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='714 MIDOG 2022 (Baseline 1) – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='629 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='753 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='608 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='585 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content='743 – better generalization for Tumor 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' These results corrob- orate the known association between chosen augmentation policies and generalization performance: this motivates us to further study how to best design and select augmentation protocols to improve domain generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' References [1] Marc Aubreville, Nikolas Stathonikos, Christof A Bertram, Robert Klopleisch, Natalie ter Hoeve, Francesco Ciompi, Frauke Wilm, Christian Marzahl, Taryn A Donovan, Andreas Maier, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Mitosis domain generalization in histopathology images – the MIDOG challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' Medical Image Analysis, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' [2] Marc Aubreville, Christof Bertram, Katharina Breininger, Samir Jabari, Nikolas Stathonikos, and Mitko Veta.' metadata={'source': 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Niethammer, JS Marron, David Borland, John T Woosley, Xiaojun Guan, Charles Schmitt, and Nancy E Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' A method for normalizing histology slides for quantita- tive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' In Proceedings of the IEEE International Sympo- sium on Biomedical Imaging (ISBI), 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9AzT4oBgHgl3EQfJfsf/content/2301.01079v1.pdf'} diff --git a/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/2301.03856v1.pdf.txt b/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/2301.03856v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0b613b22cc18b1c07c4f8d875ce4e478bc78c85 --- /dev/null +++ b/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/2301.03856v1.pdf.txt @@ -0,0 +1,1368 @@ +Footprints of loop extrusion in statistics of intra-chromosomal distances: an +analytically solvable model +Sergey Belan∗ and Vladimir Parfenyev +Landau Institute for Theoretical Physics, Russian Academy of Sciences, +1-A Akademika Semenova av., 142432 Chernogolovka, Russia and +National Research University Higher School of Economics, +Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia +(Dated: January 11, 2023) +Active loop extrusion – the process of formation of dynamically growing chromatin loops due to +the motor activity of DNA-binding protein complexes – is firmly established mechanism responsible +for chromatin spatial organization at different stages of cell cycle in eukaryotes and bacteria. The +theoretical insight into the effect of loop extrusion on the experimentally measured statistics of +chromatin conformation can be gained with an appropriately chosen polymer model. +Here we +consider the simplest analytically solvable model of interphase chromosome which is treated as ideal +chain with disorder of sufficiently sparse random loops whose conformations are sampled from the +equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression +for the mean-squared distance between pairs of genomic loci which is valid beyond the one-loop +approximation in diagrammatic representation. Besides, we analyse the loops-induced deviation of +chain conformations from the Gaussian statistics by calculating kurtosis of probability density of +the pairwise separation vector. The presented results suggest the possible ways of estimating the +characteristics of the loop extrusion process based on the experimental data on the scale-dependent +statistics of intra-chromosomal pair-wise distances. +Introduction. A series of recent single-molecule exper- +iments have shown that the structural maintenance of +chromosomes proteins, such as condensin and cohesin, +when binding to DNA can exhibit ATP-dependent motor +activity leading to progressive growth of DNA loops [1–7]. +These works provided long-awaited direct evidence of ac- +tive loop extrusion – a hypothetical molecular mechanism +previously introduced to explain a broad range of data on +spatial organization of genome throughout the cell cycle +[8–10]. Incorporation of loop extrusion mechanism into +polymer models of chromatin folding has proven to be +successful in explaining the experimental data on three +dimensional genome organization in live cells available +due to explosion of super-resolution imaging methods and +sequencing-based techniques. In particular, the molecu- +lar dynamics simulation of chromatin folding accounting +for the motor units that randomly bind to chromatin fiber +and extrude chromatin loops until stochastically dissoci- +ating (see Fig. 1a) allows to reproduce the interphase +domains observed in the population-averaged Hi-C maps +[11–15]. +Besides, computational models indicate that +loop extrusion can explain condensin-mediated mitotic +chromosome compaction and segregation [16–19]. Taken +together, these results pave the way towards a better un- +derstanding of how 3d chromatin architecture regulates +the genome function [20]. +The growing body of experimental data calls for de- +velopment of analytical models that would give easily +interpretable predictions concerning effect of loop extru- +sion machinery on statistics of chromatin conformation +avoiding the need to perform computationally intensive +simulations. Recent theoretical work [21] has shown the +promise of the fractal polymer model with quenched dis- +FIG. 1. (a) A schematic of the loop extrusion model: over +time a motor protein (depicted in red) binds chromatin, ex- +trudes a loop, and unbinds. (b) Variants of mutual arrange- +ment of two neighboring cohesin-anchored loops. From left +to right: two loops separated a gap; blocking configuration; +nested configuration; two cohesins bypassing each other form +a Z-loop. In sufficiently low cohesion concentration one can +neglect the second and the third scenarios. (c) Polymer chain +with an array of sparse random loops as a model of interphase +chromosome (loop bases are depicted in red). +order of random loops for systematization of the experi- +mentally available statistical information on the pairwise +contacts in interphase genome of higher eukaryotes for +genomic scales up to several megabases. Here we exploit +the minimalistic version of this model where chromatin +is treated as ideal chain with loops disorder to describe +the expected footprints of cohesin-driven loop extrusion +in the statistics of the physical distances between pairs +of genomic loci in interphase chromosome, which can po- +tentially be extracted via state-of-art microscopy-based +techniques [22–32]. +arXiv:2301.03856v1 [cond-mat.stat-mech] 10 Jan 2023 + +2 +Model formulation. Let us list key assumptions under- +lying our theoretical analysis. First of all, based on esti- +mates presented in previous studies [21, 33], we will as- +sume that for interphase chromatin the fraction of nested, +blocking and Z-like loop configurations (see Fig. 1b) is +relatively small, so that most of the cohesin-mediated +loops are separated from each other by loops-free gaps as +shown in Fig 1c. Since both cohesin-chromatin binding +kinetics and ATP-consuming motor activity of cohesin +are inherently stochastic, the array of cohesion-mediated +loops should be characterized statistically. +Given the +previous assumption of a fairly low concentration of co- +hesin, one can treat the lengths of loops and of inter- +loops gaps as statistically independent. Assuming addi- +tionally a constant extrusion speed, Poisson kinetics of +cohesin binding/dissociation, uniform distribution of co- +hesin binding sites and neglecting distinct loop extrusion +barriers (see, e.g., Refs. [13, 34]), we adopt the expo- +nential probability densities for random lengths of loops +and gaps with parameters λ and g denoting the mean +loop length and mean gap length, respectively. The di- +mensionless ratio λ/g is less than or of order of unity +in interphase [21, 33]. Next, simple estimates show that +the characteristic time required for the cohesin complex +to extrude a chromatin loop corresponding to a DNA +region of ∼ 100 kbp, which corresponds to typical loop +length in interphase estimated from in vivo Hi-C data, +is long compared to the relaxation time of such a loop +[21]. Given this argument, in our analytical calculations +we will treat the loops disorder as frozen. Finally, com- +pletely neglecting steric effects and affinity interactions, +we will assume that chromatin is an ideal phantom chain +with the Kuhn segment leff [35]. +Summarizing the above assumptions, we arrive at a +model of an equilibrium ideal chain with quenched dis- +order of random loops, characterized by exponential +probability densities of statistically independent con- +tour lengths of loops and gaps. As shown in Ref. [21], +the semi-analytical calculations and asymptotic one-loop +analysis based on this model qualitatively reproduce spe- +cific shape of experimental contact probability curves +universal among mammalian cells. +Also, in the work +[36] this model has been used to extract one-loop predic- +tions regarding the scale-dependent conditional probabil- +ities of triple contacts, which can be measured with the +experimental techniques for detecting multiple contacts +between more than two chromatin regions [24, 27, 37–46]. +In this paper, we focus on the statistics of the physical +distances between pairs of genome regions rather than on +pairwise contact frequencies. Overcoming the method- +ological shortcomings of the semi-analytical and pertur- +bative approaches used in Refs. +[15, 21, 36], here we +present a method for exact summation of a diagrammatic +series which allows us to derive an analytical answer for +mean-squared distance between pair of loci and can po- +tentially be generalized to the statistical moments of ar- +FIG. 2. Four classes of diagrams contributing to the MSD +between two points of the ideal chain with disorder of random +loops: (a) both points reside beyond the cohesin-mediated +loops; (b) one point resides at a loop, while another point is +in inter-loop gap (t1 ≥ 0, 0 ≤ t2 ≤ s); (c) both points reside +in the same loop (t1 ≥ 0, 0 ≤ s ≤ t2); (d) the points belong +to two different loops (t1 ≥ 0, 0 ≤ t2 ≤ s, 0 ≤ τ ≤ s − t2, +˜T ≥ s − t2 − τ). Note that the dash-dotted lines in diagrams +(a), (b) and (d) may contain arbitrary number of random +loops. +bitrary order. In what follows, the key steps of derivation +are outlined, whereas the technical details can be found +in Appendix. +Outline of calculations. Let us denote as ⃗R(s) the vec- +tor between two points of the chain separated by the +contour distance s. The main metric of interest for us +is the mean-squared distance (MSD) defined as ⟨R2(s)⟩, +where angular brackets denote averaging of the statis- +tics of thermal noise and random loops. Clearly, there +are four scenarios for the relative arrangement of the se- +lected points and bases of the cohesin-mediated loops, +see Fig. 2. Given this, the average physical separation +can be represented as +⟨R2(s)⟩ = +� +α=a,b,c,d +⟨R2 +α(s|{A}α)⟩loops, +(1) +where α enumerates the diagrams according to Fig. 2, +R2 +α(s|{A}α) is the conditional MSD obtained by averag- +ing of R2(s) over thermal noise at fixed pattern of ran- +dom loops, {A}α represents the set of random variables +parametrising the corresponding diagram, and ⟨...⟩loops +denotes averaging over variables {A}α. Note that since +the loop disorder is quenched by assumption, the averag- +ing over thermal fluctuations precedes the averaging over +the statistics of random loops in Eq. (1). +To arrive at the MSD, one first needs to derive the con- +ditional expressions R2 +α(s|{A}α) associated with the dif- +ferent diagrams, depicted in Fig. 2. By virtue of the cen- +tral limit theorem, the large-scale conformational statis- +tics of the loop-free ideal chain is equivalent to that of the +Brownian particle trajectory, with time measured in the +units of the polymer contour length and diffusion coeffi- +cient D = leff/6 (see, e.g., Ref. [35]). Thus, if λ, g ≫ leff +and we are interested at scales s ≫ leff, then chromatin + +C3 +conformation can be thought of as alternating free Brow- +nian paths and Brownian bridges. +In the absence of random loops, the MSD between +two sites of an equilibrium Gaussian chain behaves as +R2(s) = leffs. +As follows from analysis presented in +Refs. [15, 21], the conditional MSD R2 +α(s|{A}α) associ- +ated with fixed configuration of random loops obeys the +same linear scaling law, but with an effective contour sep- +aration ˜sα[s, {A}α] substituted for s. More specifically, +one obtains (see Appendix for details) +R2 +α(s|{A}α) = leff˜sα[s, {A}α], +(2) +where +˜sa[s, xs] = (1 − xs)s, +(3) +˜sb[s, t1, t2, xs−t2] = (1 − xs−t2)(s − t2) + +t1t2 +t1+t2 , +(4) +˜sc[s, t1, t2] = +� +1 − +s +t1+t2 +� +s, +(5) +˜sd[s, t1, t2, τ, xτ, ˜T] = +t1t2 +t1+t2 + (1 − xτ)τ + +˜t1˜t2 +˜t1+˜t2 , (6) +and ˜t1 = ˜T + τ + t2 − s, ˜t2 = s − τ − t2. Here t1, t2, +˜T and τ represent the contour lengths of the segments +depicted in Figs. 2a, b, and d, while xs, xs−t2 and xτ are +the fractions of contour length occupied by loops in the +segments depicted by dotted lines in diagrams (a), (b) +and (d), respectively. Note, that this variables obey the +constraint 0 ≤ xs, xs−t2, xτ < 1. +Next, we should average conditional MSD R2 +α(s|{A}α) +over the statistics of random variables {A}α. In order to +derive the corresponding statistical weights, it is conve- +nient to introduce a two-state Markov jump process in +continuous time where time intervals are measured in +the units of the polymer contour length and stochas- +tic transitions between two states occur with the rates +αl = λ−1 and αg = g−1. Clearly, the statistics of alter- +nating loops and gaps in our original problem are equiv- +alent to the statistics of time intervals that this auxiliary +Markov process spends in different states in the course of +its stochastic dynamics. As shown in Ref. [21] (see also +Appendix), the exact analytical expressions for statisti- +cal weights Wα({A}α; s) can be derived from the basic +properties of two-state Markov chain: +Wa = pgπg→g(s)F(xs), +(7) +Wb = 2plα2 +l e−αl(t1+t2)πg→g(s − t2)F(xs−t2), +(8) +Wc = plα2 +l e−αl(t1+t2), +(9) +Wd = plα3 +l e−αl(t1+t2+ ˜T )αgπg→g(τ)F(xτ), +(10) +where pg = +αl +αg+αl and pl = +αg +αg+αl give the probabili- +ties that a starting point of the walker’s trajectory be- +longs to a free Brownian path and loop, respectively, +πg→g(s) = +1 +αg+αl (αl +αge−(αl+αg)s) is the probability to +find Markov process in the gap state after time s given +that initially it was in the same state, and F(xs) repre- +sents the probability density of xs. Exact expression for +F(xs) can be extracted from the Pendler’s work [47] on +the occupation time statistics of two-state Markov pro- +cess and is given by Eq. (29) in Appendix. +The loop-averaged conditional MSD ⟨R2 +α(s|{A}α)⟩loops +entering Eq. (1), is given by integration of R2 +α(s|{A}α) +with weight Wα({A}α; s) over the variables {A}α. The +main technical difficulties are associated with averaging +over the random variables xs, which parametrizes the ex- +pressions (3), (4) and (6). An exact probability density +F(xs), while efficient for numerical analysis, is incon- +venient for analytical calculations. Note, however, the +conditional MSDs defined by expression (2) are linear +with respect to the variable xs. Exploiting properties of +Markov bridge statistics one obtains (see Appendix) +⟨xs⟩ = 1 +s +� s +0 +dtπg→l(t)πl→g(s − t) +πg→g(s) +, +(11) +where πg→l(s) = +αg +αg+αl (1 − e−(αg+αl)s) and πl→g(s) = +αl +αg+αl (1 − e−(αg+αl)s). With Eq. (11) we can express +conditional MSDs without usage of the cumbersome for- +mula for F(xs). Note that such a trick does not work +in more sophisticated case of contact probability calcu- +lations where associated diagram contributions are non- +linear in ⟨xs⟩ and should be analysed numerically [21, 36]. +Results. Rather laborious calculation procedure finally +leads us to surprisingly elegant analytical expression for +the MSD +⟨R2(s)⟩ = +leffs +1 + λ/g +� +1 + λ +g fMSD +� s +λ +�� +, +(12) +where fMSD(z) = 2 +3(z−1(1 − e−z) + E3(z)) and En(z) = +� +∞ +1 +x−ne−zxdx is the exponential integral function. Im- +portantly, this result is non-perturbative in the sense that +it takes into account all zoo of diagrams in our model and, +thus, is formally valid for any value of the dimensionless +ratio λ/g. +Let us pass to the analysis of the asymptotic behavior +dictated by Eq. (12). Since limz→0 fMSD(z) = 1, we see +from Eq. +(12) that the well-known ideal-chain scaling +law, ⟨R2(s)⟩ = leffs, is recovered at s ≪ λ. Clearly, this +is because the sufficiently small segments of the chain are +non-sensitive to the loops constraints. In the opposite +limit one finds limz→∞ fMSD(z) = 0, so that ⟨R2(s)⟩ = +leffs +1 + λ/g < leffs at s ≫ λ if λ/g ≲ 1. This conclusion also +has rather transparent explanation: the random loops +compactify the large segments of ideal chain via effective +shortening of contour distance between their end points. +As expected, the compactification degree is stronger for +larger values of λ/g. +The double logarithmic scale graph of ⟨R2(s)⟩ is pre- +sented in Fig. 3a. We see that at λ ≲ g crossover between +small- and large-s linear asymptotic regimes takes place +at the scale s ∼ λ, whereas the mean inter-loop spacing + +4 +(a) +(b) +(c) +FIG. 3. (a) The MSD ⟨R2(s)⟩ (top panel), its log-log deriva- +tive d log10⟨R2(s)⟩ +d log10 s +(middle panel) and +d +ds +⟨R2(s)⟩ +s +(bottom panel) +in dependence on the contour separation s for different values +of λ/g. (b) Kurtosis coefficient K(s) as a function of contour +separation s for the same set of parameters. (c) The minimum +smin of the log-log derivative d log10⟨R2(s)⟩ +d log10 s +(top panel) and the +maximum smax of the kurtosis coefficient K(s) (bottom panel) +in their dependence on the dimensionless parameter λ/g. +g affects only the magnitude of disorder-induced pertur- +bation of the MSD profile. +This observation suggests +how it would be possible to estimate the average length +of the cohesin-anchored loops, having an experimentally +measured profile of MSD. Namely, analysis of Eq. (12) +indicate that the minimum of expression s d +ds[ ⟨R2(s)⟩ +s +] in +its dependence on the contour separation s, is determined +by λ and is equal to s∗ ≈ 1.14λ irrespectively of g, see +Fig. 3a. Also, it may be informative to analyse the log- +derivative d log10⟨R2(s)⟩ +d log10 s +which determines the slope of the +MSD in the log-log scale plot. +As we see in Fig. 3c, +the log-derivative exhibits local minimum whose position +smin is of the order of λ and it changes by only 50% with +a twenty-fold increase in g. +Beyond the MSD our model allows to explore how the +functional form of the probability density of separation +vector ⃗R(s) depends on the linear scale s. Qualitatively, +one may expect that cohesin-mediated random loops do +not destroy normality of statistics of sufficiently short +chain segments of contour length s ≪ λ which are not +affected by loops constraints. +Also, Gaussianity must +also restore at large scales, s ≫ λ, g. Indeed, for each +diagram in Fig. 2 the conditional probability density of +⃗R(s) is Gaussian (see Appendix) with an effective contour +separation whose fluctuations at s ≫ λ, g become small +compared to the average value due to the central limit +theorem. +To quantify the possible deviations of scale-dependent +two-point statistics from Gaussianity we calculate the +kurtosis coefficient defined as K(s) = +⟨R4(s)⟩ +⟨R2(s)⟩2 . Clearly, +the value 5/3 corresponds to the normal statistics of +three-dimensional ideal chain. The generalization of non- +perturbative calculations presented above to the case +of the fourth-order statistical moment ⟨R4(s)⟩, enter- +ing the definition of the kurtosis, is possible in prin- +ciple, but practically difficult to implement. +However, +if λ/g ≪ 1 and s ≪ g, one can neglect the diagrams +containing two or more cohesin-mediated loops due to +their vanishing statistical weights, and analytical calcu- +lations become feasible. +Expanding statistical weights +given by Eqs. (32)-(40) in linear order upon small param- +eters λ/g and s/g and using the relation R4 +α(s|{A}α) = +5 +3l2 +eff˜s2 +α[s, {A}α], which follows from the Gaussianity of +conditional statistics of vector ⃗R for each diagram, we +find the following asymptotic result (see Appendix) +K(s) = 5 +3 +� +α⟨˜s2 +α[s, {A}α]⟩loops +� +α⟨˜sα[s, {A}α]⟩2 +loops +≈ 5 +3 + λ +g fKurt +� s +λ +� +, +(13) +where fKurt(s) = +2 +3s2 ((9 + 4s − 3s2)e−s − 9 + 5s + s2(5 + +3s)E3(s)). +Equation (13) tells us that rare random loops produce +a linear correction in small parameter λ/g ≪ 1 to the +value 5/3 corresponding to normal statistics of three- +dimensional ideal chain in the absence of loops disorder. +The corresponding plot of the kurtosis coefficient K(s) +as a function of s is represented in Fig. 3b. +Data as- +sociated with the regime λ/g ∼ 1 were generated via +numerical integration of diagram contributions over ex- +act statistical weights. We found that the one-loop pre- +diction (dashed line) is rather accurate at λ/g ≲ 0.1, +but underestimates K(s) when λ/g ≳ 1. In agreement +with the general arguments discussed above, the kurto- +sis coefficient is close to 5/3 at s ≪ λ and s ≫ λ. At +intermediate scales of contour distances statistics of the +separation vector ⃗R exhibits deviation from Gaussianity, +and this effect is the more pronounced, the greater the +dimensionless parameter λ/g. Most importantly, the kur- +tosis coefficient is peaked at the point s = smax, whose +position is mainly determined by λ and changes by only +10% when g is changed by a factor of 20. Thus, we ex- +pect that measurement of the scale-dependent kurtosis +may provide an estimate for mean loop size λ along with +the analysis of experimental MSD profile. Note also that +the loops-induced violation of normality predicted by our +model cannot be reproduced in the framework of Hetero- +geneous Loop Model [48–50] since it postulates normal +statistics of chromatin at all genomic scales. The same +applies to the modelling approach based on inference of +the maximum entropy distribution of pair-wise distances +with experimental mean-squared distances as constraints +[51]. +Conclusion. To the best to our knowledge, the exist- +ing literature lacks the sufficient amount of relevant sta- +tistical information characterised by high genomic and +spatial resolution required to directly confront our pre- + +5 +dictions with experiment. Nevertheless, we believe that +the required data will become available in the coming +years due to modern tools for high-throughput super- +resolution imaging enabling direct visualization of the +spatial positions of many genomic loci at the single-cell +level [32, 52–55]. Noteworthy, while modelling the chro- +matin conformation by ideal chain seem to be reasonable +for some types of data [21, 54, 56], quantitative agreement +between theory and experiment in a wider range of sit- +uations may require more complex polymer models that +resist analytical treatment. In particular, further (mostly +numerical) work is required to establish how statistics of +pairwise distances in the presence of loop extrusion is +affected by excluded volume effects. +S.B. thanks Leonid A. Mirny, Hugo B. Brand˜ao and +Kirill Polovnikov for valuable discussions. The work was +supported by the Russian Science Foundation, project +no. 22-72-10052. +APPENDIX +The Appendix is structured as follows. In the first section, we remind the basic statistical properties of free Brownian +paths and Brownian bridges relevant for derivation of diagram contributions. Next, in the second section we discuss +the basic properties of two-state Markov jump process required to construct exact statistical weights of the diagrams. +In sections III-VI, we derive the integral expressions for the loop-averaged contributions coming from each type of +diagrams. Finally, in the section VII, we provide details of one-loop calculations of the kurtosis coefficient. +I. Basic Statistical properties of Brownian paths +In what follows we will heavily exploit the well-known analogy between a polymer and a random walk, see, e.g., +Refs. [35, 57]. Within this analogy, the coordinate along the polymer plays a role of time and the polymer contour is +thought of as the trajectory of a random walker, see Fig. 4a. Adopting this language, we, thus, obtain a random walk +whose trajectory represents the alternating free Brownian paths, which correspond to the gap regions of the polymer, +and the Brownian bridges corresponding to the cohesion-mediated loops in our original polymer model. Let us recall +the key statistical properties of Brownian motion. +The propagator of the free Brownian motion in three dimensions, +Gfree(⃗r, t|⃗r0, 0) = +1 +(4πDt)3/2 exp +� +−(⃗r − ⃗r0)2 +4Dt +� +, +(14) +describes the probability to find the Brownian particle having diffusivity D in the point ⃗r after time t if it starts in +⃗r0. In context of the polymer model, Eq. (14) represents the probability distribution of the separation vector ⃗r − ⃗r0 +between two monomers inside a gap region of the polymer provided that their contour separation is t. +The Brownian bridge is the Brownian trajectory subject to the condition that the particle must return to its starting +position after a certain amount of time. Propagator of a Brownian bridge of length T with a base in ⃗r0 is given by +Gbridge(⃗r, t|⃗r0, 0;⃗r0, T) = Gfree(⃗r, t|⃗r0, 0)Gfree(⃗r0, T|⃗r, t) +Gfree(⃗r0, T|⃗r0, 0) += +� +T +4πDt(T − t) +�3/2 +exp +� +− T(⃗r − ⃗r0)2 +4Dt(T − t) +� +, +(15) +where 0 ≤ t ≤ T. Eq. (15) describes the probability that the Brownian particle, which starts in ⃗r0 and returns to ⃗r0 +after time T, will be in ⃗r at the moment of time t. Equivalently, this equation defines the probability distribution of +the separation vector between the loop base and the monomer inside this loop given the contour separation t and the +loop length T. +More generally, the Brownian bridge pinned at two different points ⃗r1 and ⃗r2 at the moments of time t1 and t2, +respectively, is characterised by the following probability distribution +Gbridge(⃗r, t|⃗r1, t1;⃗r2, t2) = +� +t2 − t1 +4πD(t2 − t)(t − t1) +�3/2 +exp +� +− (⃗r2 − ⃗r)2 +4D(t2 − t) − (⃗r − ⃗r1)2 +4D(t − t1) + (⃗r2 − ⃗r1)2 +4D(t2 − t1) +� +, +(16) +where t1 ≤ t ≤ t2. +In what follows the propagators determined by Eqs. (14) and (15) play a role of building blocks of the diagram +calculations. But before proceeding to the corresponding calculations, we need to discuss the basic properties of the +two-state Markov chain that will be required to derive the statistical weights of the diagrams depicted in Fig. 2 in +main text. + +6 +FIG. 4. +(a) Based on the analogy between polymer conformation and random walk trajectory, we introduce a time axis with +time intervals measured in the units of the polymer contour length. (b) The continuous time Markov jump process with two +states, “Loop” and “Gap”, and transition rates αl = λ−1, αg = g−1. By construction, statistics of time intervals that this +auxiliary Markov process spends in different states coincides with the statistics of alternating loops and gaps in our polymer +model. +II. Basic Statistical Properties of Two-State Markov Process +Let us consider a Markov process with the transition rates αl = 1/λ and αg = 1/g between two states, “Gap” and +“Loop”, which dictates the duration of random time intervals which the random walker introduced in previous section +spends in the free and looped segments of its trajectory, see Fig. 4b. In other words, this auxiliary Markov process +generates the random length of gaps and loops in the original polymer model. +The stochastic dynamics of the two-state continuous-time Markov jump process is described by the following pair +of equations +dπg +ds = −αgπg + αlπl, +(17) +dπl +ds = αgπg − αlπl, +(18) +where πg(s) and πl(s) represent the probabilities that the monomer having contour coordinate s lies on the gap or +loop, respectively. It is straightforward to find the stationary solution of these equations +pg = +αl +αg + αl +, +pl = +αg +αg + αl +. +(19) +Clearly, pg (pl) gives the probability that a randomly chosen point of the polymer with disorder op loops belongs to +a gap (loop) region. +The propagator πA→B(s) of the Markov process is defined as the probability to find the process in the state “B” +after time s under the condition that it starts in the state “A”. It is easy to find from Eqs. (17) and (18) that +πg→g(s) = +1 +αg + αl +� +αl + αge−(αl+αg)s� +, +(20) +πg→l(s) = +αg +αg + αl +� +1 − e−(αg+αl)s� +, +(21) +πl→g(s) = +αl +αg + αl +� +1 − e−(αg+αl)s� +, +(22) +πl→l(s) = +1 +αg + αl +� +αg + αle−(αg+αl)s� +. +(23) +In the limit s → +∞, these expressions turn into statistically stationary probabilities pg and pl to find the process in +given states, i.e. lims→∞ πl→g(s) = lims→∞ πg→g(s) = pg, lims→∞ πl→l(s) = lims→∞ πg→l(s) = pl. +To perform averaging over the loop disorder (see below), we will also need to know the statistical moment ⟨xs⟩, +where xs is the time spent in the “Loop” state during the time interval [0, s] under the condition that the Markov +process occupies the “Gap” state at both ends of this interval. To calculate the expectation of xs, we introduce +the stochastic variable ζ(t), which can take two values: ζ(t) = l if at the moment t the Markov jump process is in +the “Loop” state, and ζ(t) = g if the process is currently in the “Gap” state. Then the random variable xs can be +represented as +xs = 1 +s +� s +0 +I[ζ(t) = l]dt, +(24) + +O +loop +gap +GAP +LOOP7 +where I[...] is an indicator variable equal to one if the condition in its argument is true, and equal to zero otherwise. +Performing averaging one obtains +⟨xs⟩ = 1 +s +� s +0 +⟨I[ζ(t) = l]⟩dt = 1 +s +� s +0 +Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g]dt, +(25) +where Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] is the probability of finding the Markov jump process in the “Loop” state at +time t given that it was in the “Gap” state both at time 0 and at time s. This probability can be easily calculated +due to the lack of memory of the past in a Markov process. Indeed, +Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] = Pr[ζ(s) = g|ζ(t) = l]Pr[ζ(t) = l|ζ(0) = g] +Pr[ζ(s) = g|ζ(0) = g] +, +(26) +and since Pr[ζ(s) = g|ζ(t) = l] = πl→g(s − t), Pr[ζ(t) = l|ζ(0) = g] = πg→l(t) and Pr[ζ(s) = g|ζ(0) = g] = πg→g(s) +we obtain +Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] = πl→g(s − t)πg→l(t) +πg→g(s) +. +(27) +Substituting this result into Eq. (25) yields +⟨xs⟩ = αgαl[2 + (αg + αl)s + e(αg+αl)s((αg + αl)s − 2)] +s(αg + αl)2[αg + αle(αg+αl)s] +. +(28) +Beyond the mean value, the full statistics of the random variable xs can be extracted from the results of Ref. [47]. +Namely, the probability density F(xs) is given by +F(xs) = +e−αgsδ(xs) + +� +αgαl(1−xs)s2 +xs +I1 +� +2 +� +αgαlxs(1 − xs)s2 +� +e−αg(1−xs)s−αlxss +αl +αg+αl + +αg +αg+αl e−(αg+αl)s +, +(29) +where I1(...) denotes the modified Bessel function of the first kind [58]. +III. Diagram A. Derivation of Eqs. (3) and (7) +We wish to calculate the mean-squared displacement (MSD) of the random walker after time s. Depending on the +modes of the walker motion at the initial and final moments of time we should distinguish four cases represented in +Fig. 2 of the main text. If the walker is in the free segments of its trajectory both initially and after time s, see +the diagram in Fig. 2a, then the probability density function of the walker’s displacement ⃗r is given by the Gaussian +distribution +Pa(⃗r|s, xs) = Gfree(⃗r, (1 − xs)s|⃗0, 0) = +1 +(4πD˜sa[s, xs])3/2 exp +� +− +r2 +4D˜sa[s, xs] +� +, +(30) +with the effective contour separation ˜sa[s, xs] = (1 − xs)s, where xs denotes the fraction of time that walker spent +performing Brownian bridges during the course of motion; 0 ≤ xs < 1. The intuition behind Eq. (30) is quite +transparent: since the closed Brownian paths don’t produce the walker’s displacement, the overall effect of loops in +diagram (a) is equivalent to reduction of the time allowed to the walker for exploration of the neighborhood. For the +mean-squared displacement we, thus, obtain +R2 +a(s|xs) = +� +d3rr2Pa(⃗r|s, xs) = 6D˜sa[s, xs]. +(31) +Next, using basic properties of two-state Markov jump process described in section II, we find that the diagram (a) +is characterized by the following statistical weight +Wa(xs; s) = pgπg→g(s)F(xs), +(32) +where pg = +αl +αg+αl gives the probability that a starting point of the walker’s trajectory belongs to a free Brownian +path, πg→g(s) = +1 +αg+αl (αl + αge−(αl+αg)s) is the probability to find the walker in the free segment of its trajectory + +8 +after time s under the condition that initially it is also in the free segment, and F(xs) is the probability distribution +of the random variable xs determined by Eq. (29). To average the contribution of the diagram (a) over the disorder +of random loops, we should integrate the product R2 +a(s|xs)Wa(xs; s) over xs from 0 up to 1, i.e. +⟨R2 +a(s|xs)⟩loops = +� 1 +0 +dxsR2 +a(s|xs)Wa(xs; s) = 6Dspgπg→g(s)(1 − ⟨xs⟩), +(33) +where ⟨xs⟩ is given by Eq. (28). +IV. Diagram B. Derivation of Eqs. (4) and (8) +Next, let us assume that the walker starts in the loop and finds itself in the free segment of its trajectory after +time s. As shown in Fig. 2b of the main text, the loop containing the starting point of the walker’s trajectory is +parameterized by the time intervals t1 and t2. After averaging over the position of the loop base ⃗r0, the probability +density function of the walker’s displacement becomes +Pb(⃗r|s, t1, t2, xs−t2) = +� +d3r0Gbridge(⃗0, 0|⃗r0, −t1;⃗r0, t2)Gfree(⃗r, t2 + (1 − xs−t2)(s − t2)|⃗r0, t2) = += +1 +(4πD˜sb[s, t1, t2, xs−t2])3/2 exp +� +− +r2 +4D˜sb[s, t1, t2, xs−t2] +� +, +(34) +where ˜sb[s, t1, t2, xs−t2] = (1 − xs−t2)(s − t2) + +t1t2 +t1+t2 , and 0 ≤ xs−t2 < 1, t1 ≥ 0, 0 ≤ t2 ≤ s. Now xs−t2 is the fraction +of time the walker spend performing Brownian bridges during the time interval between t2 and s. Therefore, the +mean-squared displacement of the walker is given by +R2 +b(s|t1, t2, xs−t2) = +� +d3rr2Pb(⃗r|s, t1, t2, xs−t2) = 6D˜sb[s, t1, t2, xs−t2]. +(35) +Next, for the statistical weight of the diagram (b) we obtain +Wb(t1, t2, xs−t2; s) = 2plα2 +l e−αl(t1+t2)πg→g(s − t2)F(xs−t2), +(36) +where pl = +αg +αg+αl gives the probability that a starting point of the walker’s trajectory belongs to a loop. Obviously, +the case when the walker starts in the free segment and finishes in the closed segment is completely equivalent to the +situation that we have just considered. This explains the origin of factor 2 in Eq. (36). +From Eqs. +(35) and (36) one obtains that the loops-averaged contribution of the diagram (b) is given by the +following integral +⟨R2 +b(s|t1, t2, xs−t2)⟩loops = +� ∞ +0 +dt1 +� s +0 +dt2 +� 1 +0 +dxs−t2R2 +b(s|t1, t2, xs−t2)Wb(t1, t2, xs−t2; s) = += 12Dplα2 +l +� ∞ +0 +dt1 +� s +0 +dt2 +� +(1 − ⟨xs−t2⟩)(s − t2) + +t1t2 +t1 + t2 +� +e−αl(t1+t2)πg→g(s − t2). +(37) +V. Diagram C. Derivation of Eqs. (5) and (9) +Now let us consider the scenario when the starting and the final points of the walker’s trajectory belong to the +same loop, see Fig. 2c in the main text. Performing averaging over the position of the loop base we find the following +result for the probability distribution of the walker’s displacement after time s +Pc(⃗r|s, t1, t2) = +� +d3r0Gbridge(⃗0, 0|⃗r0, −t1;⃗r0, t2)Gbridge(⃗r, s|⃗0, 0;⃗r0, t2) = += +1 +(4πD˜sc[s, t1, t2])3/2 exp +� +− +r2 +4D˜sc[s, t1, t2] +� +, +(38) + +9 +where ˜sc[s, t1, t2] = +� +1 − +s +t1+t2 +� +s, and t1 ≥ 0, t2 ≥ s. +From Eq. +(38) one obtains the mean-squared walker’s +displacement +R2 +c(s|t1, t2) = +� +d3rr2Pc(⃗r|s, t1, t2) = 6D˜sc[s, t1, t2], +(39) +whereas for the the statistical weight of the trajectories described by the diagram (c) we find +Wc(t1, t2; s) = plα2 +l e−αl(t1+t2). +(40) +Thus, the loop-averaged contribution of the diagram (c) to mean-squared displacement is determined by the fol- +lowing double integral +⟨R2 +c(s|t1, t2)⟩loops = +∞ +� +0 +dt1 +∞ +� +s +dt2R2 +c(s|t1, t2)Wc(t1, t2; s) = 6Dsplα2 +l +∞ +� +0 +dt1 +∞ +� +s +dt2 +� +1 − +s +t1 + t2 +� +e−αl(t1+t2). +(41) +VI. Diagram D. Derivation of Eqs. (6) and (10) +Finally, the probability distribution of the walker’s displacement in the situation when the initial and final point of +its trajectory belong to different loops is given by +Pd(⃗r|s, t1, t2, τ, xτ, T2) = +1 +(4πD˜sd[s, t1, t2, τ, xτ, ˜T] +exp +� +− +r2 +4D˜sd[s, t1, t2, τ, xτ, ˜T] +� +, +(42) +where ˜sd[s, t1, t2, τ, xτ, ˜T] = +t1t2 +t1+t2 + (1 − xτ)τ + +˜t1˜t2 +˜t1+˜t2 , and ˜t1 = ˜T + τ + t2 − s, ˜t2 = s − τ − t2, t1 ≥ 0, 0 ≤ t2 ≤ s, +0 ≤ τ ≤ s − t2, 0 ≤ xτ ≤ 1, ˜T ≥ s − t2 − τ. In this case, xτ denotes the fraction of time the walker spend in +”Loop” state during the time interval between t2 and s − ˜t2. From Eq. (42) one obtains the mean-squared walker’s +displacement +R2 +d(s|t1, t2, τ, xτ, ˜T) = +� +d3rr2Pc(⃗r|s, t1, t2) = 6D˜sd[s, t1, t2, τ, xτ, ˜T]. +(43) +Clearly, the statistical weight of the trajectories described by the diagram (d) is given by +Wd(t1, t2, τ, xτ, ˜T; s) = plα3 +l e−αl(t1+t2+ ˜T )αgπg→g(τ)F(xτ). +(44) +Thus, for the loops-averaged contribution of the diagram (d) we find +⟨R2 +d(s|t1, t2, τ, xτ, ˜T)⟩loops = +∞ +� +0 +dt1 +s +� +0 +dt2 +s−t2 +� +0 +dτ +∞ +� +s−t2−τ +d ˜T +1 +� +0 +dxτR2 +d(s|t1, t2, τ, xτ, ˜T)Wd(t1, t2, τ, xτ, ˜T; s) = +6Dplα3 +l αg +∞ +� +0 +dt1 +s +� +0 +dt2 +s−t2 +� +0 +dτ +∞ +� +s−t2−τ +d ˜T +� +(1 − ⟨xτ⟩)τ + +t1t2 +t1 + t2 ++ ( ˜T + t2 + τ − s)(s − t2 − τ) +˜T +� +e−αl(t1+t2+ ˜T )πg→g(τ). +(45) +Calculating the integrals in Eqs. (33), (37), (41) and (45) and summing the resulting expressions, we arrive at the +Eq. (12) in main text. +VII. One-loop approximation. Derivation of Eq. (13) +The kurtosis coefficient of the random vector ⃗R(s) is defined as +K(s) = ⟨R4(s)⟩ +⟨R2(s)⟩2 . +(46) + +10 +FIG. 5. For λ/g ≪ 1 and s/g ≪ 1 the two-point statistics of an ideal chain with a disorder of random loops can be computed +in the one-loop approximation, leaving only those diagrams containing at most one cohesin-mediated loop. In other words, +diagram (d) can simply be ignored, and the dash-dotted line in diagrams (a) and (b) can be replaced by a solid line. +We already know that the MSD ⟨R2(s)⟩ in our model is given by Eq. (12) in main text. As for the fourth order +statistical moment ⟨R4(s)⟩, taking into account the Gaussian form of the conditional distribution functions (30), (34), +(38), and (42), one readily obtains +⟨R4(s)⟩ = 60D2 +� +α=a,b,c,d +⟨˜s2 +α[s, {A}α]⟩loops. +(47) +Exact diagrammatic calculations accordingly to Eq. +(47) are possible in principle, but practically difficult to +implement. However, analytical derivation of the fourth moment ⟨R4(s)⟩ becomes feasible in the rare loops limit. +More specifically, if λ/g ≪ 1 and s/g ≪ 1, then one can neglect the realizations of diagrams where there is more than +one loop, see Fig. 5. Then, neglecting the diagram (d) and simplifying the formulas (3)-(5) from the main text, we +obtain +⟨R4(s)⟩ ≈ 60D2 +� +α=a,b,c +⟨˜s2 +α[s, {A}α]⟩one loop, +(48) +where +˜sa[s, xs] = (1 − xs)s, +(49) +˜sb[s, t1, t2] = s − t2 + +t1t2 +t1+t2 , +(50) +˜sc[s, t1, t2] = +� +1 − +s +t1+t2 +� +s. +(51) +When averaging over the disorder of the loops, it is convenient to pass to the new variables T and q defined as +t1 = (1 − q)T, +t2 = qT. +(52) +In terms of these variables, the diagrams (a), (b) and (c) depicted in Fig. +5 are characterized by the following +statistical weights +Wa(xs|s) = pgπg→g(s)F(xs), +for 0 ≤ xs < 1, +(53) +Wb(T, q, xs−qT |s) = 2pl˜ρl(T)πg→g(s − qT)F(xs−qT ), +for 0 ≤ q ≤ min[1, s +T ], T ≥ 0, 0 ≤ xs−qT < 1, +(54) +Wc(T, q|s) = pl˜ρl(T), +for +s +T ≤ q ≤ 1, T ≥ s, +(55) +where ˜ρl(T) denote the probability density of the random loop length in the statistical experiment where loops are +sampled by random choice of points along the polymer. Clearly, ˜ρl(T) = T +λ ρl(T), where ρl(T) = 1 +λ exp(− T +λ ) is the +actual loop length distribution. +Using the smallness of the dimensionless parameters λ/g ≪ 1 and s/g ≪ 1, we find from Eqs. (19), (20) and (29) +pg = +g +g + λ ≈ 1 − λ +g , +pl = +λ +g + λ ≈ λ +g , +(56) + +0() +0(1) +0() +(011 +and +πg→g(s)F(xs) ≈ (1 − s +g )δ(xs) + (1 − xs)s2 +g +ρl(xss). +(57) +By inserting Eqs. (56) and (57) into Eqs. (53,54,55) and neglecting the terms nonlinear in the parameter λ/g one +obtains +Wa(xs|s) ≈ δ(xs) + λ +g +� +−(1 + s +λ)δ(xs) + (1−xs)s2 +λ +ρl(xss) +� +, +for 0 ≤ xs < 1, +(58) +Wb(T, q|s) ≈ 2 λ +g +T +λ ρl(T), +for 0 ≤ q ≤ min[1, s +T ], T ≥ 0, +(59) +Wc(T, q|s) ≈ λ +g +T +λ ρl(T), +for +s +T ≤ q ≤ 1, T ≥ s. +(60) +Next, performing averaging over disorder of loops, we find that in the first order-approximation with respect to the +ratio λ/g, the fourth-order statistical moment of the random vector ⃗R(s) is given by +⟨R4(s)⟩ ≈ 60D2 +� 1 +0 +dxs˜s2 +a[s, xs]Wa(xs|s) + 60D2 +� ∞ +0 +dT +� min[1,s/T ] +0 +dq˜s2 +b[s, T, q]Wb(T, q|s) + +(61) ++60D2 +� ∞ +s +dT +� 1 +s/T +dq˜s2 +c[s, T, q]Wc(T, q|s) = +(62) += 60(Ds)2 +� +1 + λ +g +s2 +λ +�� 1 +0 +dxρl(xs)(−3 +5x3 + 5 +3x2 − 2x) + +� +∞ +1 +dxρl(xs)(− 3 +5x2 + 5 +3x − 2) +�� += +(63) += 60(Ds)2 +� +1 + λ +g f4( s +λ) +� +, +(64) +where +f4(s) = −54 − 96e−s − 10s(3s − 5) + 24(25 + 9s)E5(s) +15s2 +, +(65) +and En(s) = +� +∞ +1 +x−ne−sxdx is the exponential integral function. +As follows from Eq. (12) in the main text, the MSD in the same approximation is given by +⟨R2(s)⟩ ≈ 6Ds +� +1 + λ +d +�2λ(1 − e− s +λ ) +3s +− 1 + 2 +3E3( s +λ) +�� +(66) +Substituting Eqs. (61) and (66) into Eq. (46) finally yields +K(s) ≈ 5 +3 + λ +g fKurt( s +λ), +(67) +where +fKurt(s) = +2 +3s2 +� +(9 + 4s − 3s2)e−s − 9 + 5s + s2(5 + 3s)E3(s) +� +. +(68) +This result matches equation (13) from the main text. +As noted above, the one-loop approximation relies on smallness of two dimensionless parameters: λ/g and s/g. +However, by a happy coincidence the one-loop answer for MSD agrees with the exact result given by Eq. (12) in +the main text for arbitrary large value of s/g provided λ/g ≪ 1. In other words, the large-scale behaviour of MSD +obtained from one-loop calculations is accurate for any value of s, despite the one-loop approximation is justified only +if s ≪ g. 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Romer, Hand- +book of mathematical functions with formulas, graphs, +and mathematical tables (1988). + diff --git a/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/load_file.txt b/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c04bfefc767262b1337d7a4ebe8a5af89db46e07 --- /dev/null +++ b/XtE2T4oBgHgl3EQfYgfk/content/tmp_files/load_file.txt @@ -0,0 +1,896 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf,len=895 +page_content='Footprints of loop extrusion in statistics of intra-chromosomal distances: an analytically solvable model Sergey Belan∗ and Vladimir Parfenyev Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=', 142432 Chernogolovka, Russia and National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia (Dated: January 11, 2023) Active loop extrusion – the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes – is firmly established mechanism responsible for chromatin spatial organization at different stages of cell cycle in eukaryotes and bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Here we consider the simplest analytically solvable model of interphase chromosome which is treated as ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci which is valid beyond the one-loop approximation in diagrammatic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Besides, we analyse the loops-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' A series of recent single-molecule exper- iments have shown that the structural maintenance of chromosomes proteins, such as condensin and cohesin, when binding to DNA can exhibit ATP-dependent motor activity leading to progressive growth of DNA loops [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' These works provided long-awaited direct evidence of ac- tive loop extrusion – a hypothetical molecular mechanism previously introduced to explain a broad range of data on spatial organization of genome throughout the cell cycle [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Incorporation of loop extrusion mechanism into polymer models of chromatin folding has proven to be successful in explaining the experimental data on three dimensional genome organization in live cells available due to explosion of super-resolution imaging methods and sequencing-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In particular, the molecu- lar dynamics simulation of chromatin folding accounting for the motor units that randomly bind to chromatin fiber and extrude chromatin loops until stochastically dissoci- ating (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 1a) allows to reproduce the interphase domains observed in the population-averaged Hi-C maps [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Besides, computational models indicate that loop extrusion can explain condensin-mediated mitotic chromosome compaction and segregation [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Taken together, these results pave the way towards a better un- derstanding of how 3d chromatin architecture regulates the genome function [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The growing body of experimental data calls for de- velopment of analytical models that would give easily interpretable predictions concerning effect of loop extru- sion machinery on statistics of chromatin conformation avoiding the need to perform computationally intensive simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Recent theoretical work [21] has shown the promise of the fractal polymer model with quenched dis- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (a) A schematic of the loop extrusion model: over time a motor protein (depicted in red) binds chromatin, ex- trudes a loop, and unbinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (b) Variants of mutual arrange- ment of two neighboring cohesin-anchored loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' From left to right: two loops separated a gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' blocking configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' nested configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' two cohesins bypassing each other form a Z-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In sufficiently low cohesion concentration one can neglect the second and the third scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (c) Polymer chain with an array of sparse random loops as a model of interphase chromosome (loop bases are depicted in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' order of random loops for systematization of the experi- mentally available statistical information on the pairwise contacts in interphase genome of higher eukaryotes for genomic scales up to several megabases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Here we exploit the minimalistic version of this model where chromatin is treated as ideal chain with loops disorder to describe the expected footprints of cohesin-driven loop extrusion in the statistics of the physical distances between pairs of genomic loci in interphase chromosome, which can po- tentially be extracted via state-of-art microscopy-based techniques [22–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='03856v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='stat-mech] 10 Jan 2023 2 Model formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Let us list key assumptions under- lying our theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' First of all, based on esti- mates presented in previous studies [21, 33], we will as- sume that for interphase chromatin the fraction of nested, blocking and Z-like loop configurations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 1b) is relatively small, so that most of the cohesin-mediated loops are separated from each other by loops-free gaps as shown in Fig 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Since both cohesin-chromatin binding kinetics and ATP-consuming motor activity of cohesin are inherently stochastic, the array of cohesion-mediated loops should be characterized statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Given the previous assumption of a fairly low concentration of co- hesin, one can treat the lengths of loops and of inter- loops gaps as statistically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Assuming addi- tionally a constant extrusion speed, Poisson kinetics of cohesin binding/dissociation, uniform distribution of co- hesin binding sites and neglecting distinct loop extrusion barriers (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [13, 34]), we adopt the expo- nential probability densities for random lengths of loops and gaps with parameters λ and g denoting the mean loop length and mean gap length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The di- mensionless ratio λ/g is less than or of order of unity in interphase [21, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Next, simple estimates show that the characteristic time required for the cohesin complex to extrude a chromatin loop corresponding to a DNA region of ∼ 100 kbp, which corresponds to typical loop length in interphase estimated from in vivo Hi-C data, is long compared to the relaxation time of such a loop [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Given this argument, in our analytical calculations we will treat the loops disorder as frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Finally, com- pletely neglecting steric effects and affinity interactions, we will assume that chromatin is an ideal phantom chain with the Kuhn segment leff [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Summarizing the above assumptions, we arrive at a model of an equilibrium ideal chain with quenched dis- order of random loops, characterized by exponential probability densities of statistically independent con- tour lengths of loops and gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [21], the semi-analytical calculations and asymptotic one-loop analysis based on this model qualitatively reproduce spe- cific shape of experimental contact probability curves universal among mammalian cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Also, in the work [36] this model has been used to extract one-loop predic- tions regarding the scale-dependent conditional probabil- ities of triple contacts, which can be measured with the experimental techniques for detecting multiple contacts between more than two chromatin regions [24, 27, 37–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In this paper, we focus on the statistics of the physical distances between pairs of genome regions rather than on pairwise contact frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Overcoming the method- ological shortcomings of the semi-analytical and pertur- bative approaches used in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [15, 21, 36], here we present a method for exact summation of a diagrammatic series which allows us to derive an analytical answer for mean-squared distance between pair of loci and can po- tentially be generalized to the statistical moments of ar- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Four classes of diagrams contributing to the MSD between two points of the ideal chain with disorder of random loops: (a) both points reside beyond the cohesin-mediated loops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (b) one point resides at a loop, while another point is in inter-loop gap (t1 ≥ 0, 0 ≤ t2 ≤ s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (c) both points reside in the same loop (t1 ≥ 0, 0 ≤ s ≤ t2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (d) the points belong to two different loops (t1 ≥ 0, 0 ≤ t2 ≤ s, 0 ≤ τ ≤ s − t2, ˜T ≥ s − t2 − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note that the dash-dotted lines in diagrams (a), (b) and (d) may contain arbitrary number of random loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' bitrary order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In what follows, the key steps of derivation are outlined, whereas the technical details can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Outline of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Let us denote as ⃗R(s) the vec- tor between two points of the chain separated by the contour distance s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The main metric of interest for us is the mean-squared distance (MSD) defined as ⟨R2(s)⟩, where angular brackets denote averaging of the statis- tics of thermal noise and random loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Clearly, there are four scenarios for the relative arrangement of the se- lected points and bases of the cohesin-mediated loops, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Given this, the average physical separation can be represented as ⟨R2(s)⟩ = � α=a,b,c,d ⟨R2 α(s|{A}α)⟩loops, (1) where α enumerates the diagrams according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2, R2 α(s|{A}α) is the conditional MSD obtained by averag- ing of R2(s) over thermal noise at fixed pattern of ran- dom loops, {A}α represents the set of random variables parametrising the corresponding diagram, and ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⟩loops denotes averaging over variables {A}α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note that since the loop disorder is quenched by assumption, the averag- ing over thermal fluctuations precedes the averaging over the statistics of random loops in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To arrive at the MSD, one first needs to derive the con- ditional expressions R2 α(s|{A}α) associated with the dif- ferent diagrams, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' By virtue of the cen- tral limit theorem, the large-scale conformational statis- tics of the loop-free ideal chain is equivalent to that of the Brownian particle trajectory, with time measured in the units of the polymer contour length and diffusion coeffi- cient D = leff/6 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Thus, if λ, g ≫ leff and we are interested at scales s ≫ leff, then chromatin C3 conformation can be thought of as alternating free Brow- nian paths and Brownian bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In the absence of random loops, the MSD between two sites of an equilibrium Gaussian chain behaves as R2(s) = leffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As follows from analysis presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [15, 21], the conditional MSD R2 α(s|{A}α) associ- ated with fixed configuration of random loops obeys the same linear scaling law, but with an effective contour sep- aration ˜sα[s, {A}α] substituted for s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' More specifically, one obtains (see Appendix for details) R2 α(s|{A}α) = leff˜sα[s, {A}α], (2) where ˜sa[s, xs] = (1 − xs)s, (3) ˜sb[s, t1, t2, xs−t2] = (1 − xs−t2)(s − t2) + t1t2 t1+t2 , (4) ˜sc[s, t1, t2] = � 1 − s t1+t2 � s, (5) ˜sd[s, t1, t2, τ, xτ, ˜T] = t1t2 t1+t2 + (1 − xτ)τ + ˜t1˜t2 ˜t1+˜t2 , (6) and ˜t1 = ˜T + τ + t2 − s, ˜t2 = s − τ − t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Here t1, t2, ˜T and τ represent the contour lengths of the segments depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2a, b, and d, while xs, xs−t2 and xτ are the fractions of contour length occupied by loops in the segments depicted by dotted lines in diagrams (a), (b) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note, that this variables obey the constraint 0 ≤ xs, xs−t2, xτ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Next, we should average conditional MSD R2 α(s|{A}α) over the statistics of random variables {A}α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In order to derive the corresponding statistical weights, it is conve- nient to introduce a two-state Markov jump process in continuous time where time intervals are measured in the units of the polymer contour length and stochas- tic transitions between two states occur with the rates αl = λ−1 and αg = g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Clearly, the statistics of alter- nating loops and gaps in our original problem are equiv- alent to the statistics of time intervals that this auxiliary Markov process spends in different states in the course of its stochastic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [21] (see also Appendix), the exact analytical expressions for statisti- cal weights Wα({A}α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) can be derived from the basic properties of two-state Markov chain: Wa = pgπg→g(s)F(xs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (7) Wb = 2plα2 l e−αl(t1+t2)πg→g(s − t2)F(xs−t2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (8) Wc = plα2 l e−αl(t1+t2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (9) Wd = plα3 l e−αl(t1+t2+ ˜T )αgπg→g(τ)F(xτ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (10) where pg = αl αg+αl and pl = αg αg+αl give the probabili- ties that a starting point of the walker’s trajectory be- longs to a free Brownian path and loop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' πg→g(s) = 1 αg+αl (αl +αge−(αl+αg)s) is the probability to find Markov process in the gap state after time s given that initially it was in the same state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' and F(xs) repre- sents the probability density of xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Exact expression for F(xs) can be extracted from the Pendler’s work [47] on the occupation time statistics of two-state Markov pro- cess and is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (29) in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The loop-averaged conditional MSD ⟨R2 α(s|{A}α)⟩loops entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (1), is given by integration of R2 α(s|{A}α) with weight Wα({A}α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) over the variables {A}α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The main technical difficulties are associated with averaging over the random variables xs, which parametrizes the ex- pressions (3), (4) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' An exact probability density F(xs), while efficient for numerical analysis, is incon- venient for analytical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note, however, the conditional MSDs defined by expression (2) are linear with respect to the variable xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Exploiting properties of Markov bridge statistics one obtains (see Appendix) ⟨xs⟩ = 1 s � s 0 dtπg→l(t)πl→g(s − t) πg→g(s) , (11) where πg→l(s) = αg αg+αl (1 − e−(αg+αl)s) and πl→g(s) = αl αg+αl (1 − e−(αg+αl)s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (11) we can express conditional MSDs without usage of the cumbersome for- mula for F(xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note that such a trick does not work in more sophisticated case of contact probability calcu- lations where associated diagram contributions are non- linear in ⟨xs⟩ and should be analysed numerically [21, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Rather laborious calculation procedure finally leads us to surprisingly elegant analytical expression for the MSD ⟨R2(s)⟩ = leffs 1 + λ/g � 1 + λ g fMSD � s λ �� , (12) where fMSD(z) = 2 3(z−1(1 − e−z) + E3(z)) and En(z) = � +∞ 1 x−ne−zxdx is the exponential integral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Im- portantly, this result is non-perturbative in the sense that it takes into account all zoo of diagrams in our model and, thus, is formally valid for any value of the dimensionless ratio λ/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Let us pass to the analysis of the asymptotic behavior dictated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Since limz→0 fMSD(z) = 1, we see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) that the well-known ideal-chain scaling law, ⟨R2(s)⟩ = leffs, is recovered at s ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Clearly, this is because the sufficiently small segments of the chain are non-sensitive to the loops constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In the opposite limit one finds limz→∞ fMSD(z) = 0, so that ⟨R2(s)⟩ = leffs 1 + λ/g < leffs at s ≫ λ if λ/g ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This conclusion also has rather transparent explanation: the random loops compactify the large segments of ideal chain via effective shortening of contour distance between their end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As expected, the compactification degree is stronger for larger values of λ/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The double logarithmic scale graph of ⟨R2(s)⟩ is pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' We see that at λ ≲ g crossover between small- and large-s linear asymptotic regimes takes place at the scale s ∼ λ, whereas the mean inter-loop spacing 4 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (a) The MSD ⟨R2(s)⟩ (top panel), its log-log deriva- tive d log10⟨R2(s)⟩ d log10 s (middle panel) and d ds ⟨R2(s)⟩ s (bottom panel) in dependence on the contour separation s for different values of λ/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (b) Kurtosis coefficient K(s) as a function of contour separation s for the same set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (c) The minimum smin of the log-log derivative d log10⟨R2(s)⟩ d log10 s (top panel) and the maximum smax of the kurtosis coefficient K(s) (bottom panel) in their dependence on the dimensionless parameter λ/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' g affects only the magnitude of disorder-induced pertur- bation of the MSD profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This observation suggests how it would be possible to estimate the average length of the cohesin-anchored loops, having an experimentally measured profile of MSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Namely, analysis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) indicate that the minimum of expression s d ds[ ⟨R2(s)⟩ s ] in its dependence on the contour separation s, is determined by λ and is equal to s∗ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='14λ irrespectively of g, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Also, it may be informative to analyse the log- derivative d log10⟨R2(s)⟩ d log10 s which determines the slope of the MSD in the log-log scale plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 3c, the log-derivative exhibits local minimum whose position smin is of the order of λ and it changes by only 50% with a twenty-fold increase in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Beyond the MSD our model allows to explore how the functional form of the probability density of separation vector ⃗R(s) depends on the linear scale s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Qualitatively, one may expect that cohesin-mediated random loops do not destroy normality of statistics of sufficiently short chain segments of contour length s ≪ λ which are not affected by loops constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Also, Gaussianity must also restore at large scales, s ≫ λ, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Indeed, for each diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2 the conditional probability density of ⃗R(s) is Gaussian (see Appendix) with an effective contour separation whose fluctuations at s ≫ λ, g become small compared to the average value due to the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To quantify the possible deviations of scale-dependent two-point statistics from Gaussianity we calculate the kurtosis coefficient defined as K(s) = ⟨R4(s)⟩ ⟨R2(s)⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Clearly, the value 5/3 corresponds to the normal statistics of three-dimensional ideal chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The generalization of non- perturbative calculations presented above to the case of the fourth-order statistical moment ⟨R4(s)⟩, enter- ing the definition of the kurtosis, is possible in prin- ciple, but practically difficult to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' However, if λ/g ≪ 1 and s ≪ g, one can neglect the diagrams containing two or more cohesin-mediated loops due to their vanishing statistical weights, and analytical calcu- lations become feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Expanding statistical weights given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (32)-(40) in linear order upon small param- eters λ/g and s/g and using the relation R4 α(s|{A}α) = 5 3l2 eff˜s2 α[s, {A}α], which follows from the Gaussianity of conditional statistics of vector ⃗R for each diagram, we find the following asymptotic result (see Appendix) K(s) = 5 3 � α⟨˜s2 α[s, {A}α]⟩loops � α⟨˜sα[s, {A}α]⟩2 loops ≈ 5 3 + λ g fKurt � s λ � , (13) where fKurt(s) = 2 3s2 ((9 + 4s − 3s2)e−s − 9 + 5s + s2(5 + 3s)E3(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Equation (13) tells us that rare random loops produce a linear correction in small parameter λ/g ≪ 1 to the value 5/3 corresponding to normal statistics of three- dimensional ideal chain in the absence of loops disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The corresponding plot of the kurtosis coefficient K(s) as a function of s is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Data as- sociated with the regime λ/g ∼ 1 were generated via numerical integration of diagram contributions over ex- act statistical weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' We found that the one-loop pre- diction (dashed line) is rather accurate at λ/g ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='1, but underestimates K(s) when λ/g ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In agreement with the general arguments discussed above, the kurto- sis coefficient is close to 5/3 at s ≪ λ and s ≫ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' At intermediate scales of contour distances statistics of the separation vector ⃗R exhibits deviation from Gaussianity, and this effect is the more pronounced, the greater the dimensionless parameter λ/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Most importantly, the kur- tosis coefficient is peaked at the point s = smax, whose position is mainly determined by λ and changes by only 10% when g is changed by a factor of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Thus, we ex- pect that measurement of the scale-dependent kurtosis may provide an estimate for mean loop size λ along with the analysis of experimental MSD profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Note also that the loops-induced violation of normality predicted by our model cannot be reproduced in the framework of Hetero- geneous Loop Model [48–50] since it postulates normal statistics of chromatin at all genomic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The same applies to the modelling approach based on inference of the maximum entropy distribution of pair-wise distances with experimental mean-squared distances as constraints [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To the best to our knowledge, the exist- ing literature lacks the sufficient amount of relevant sta- tistical information characterised by high genomic and spatial resolution required to directly confront our pre- 5 dictions with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Nevertheless, we believe that the required data will become available in the coming years due to modern tools for high-throughput super- resolution imaging enabling direct visualization of the spatial positions of many genomic loci at the single-cell level [32, 52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Noteworthy, while modelling the chro- matin conformation by ideal chain seem to be reasonable for some types of data [21, 54, 56], quantitative agreement between theory and experiment in a wider range of sit- uations may require more complex polymer models that resist analytical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In particular, further (mostly numerical) work is required to establish how statistics of pairwise distances in the presence of loop extrusion is affected by excluded volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' thanks Leonid A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Mirny, Hugo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Brand˜ao and Kirill Polovnikov for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The work was supported by the Russian Science Foundation, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 22-72-10052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' APPENDIX The Appendix is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In the first section, we remind the basic statistical properties of free Brownian paths and Brownian bridges relevant for derivation of diagram contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Next, in the second section we discuss the basic properties of two-state Markov jump process required to construct exact statistical weights of the diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In sections III-VI, we derive the integral expressions for the loop-averaged contributions coming from each type of diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Finally, in the section VII, we provide details of one-loop calculations of the kurtosis coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Basic Statistical properties of Brownian paths In what follows we will heavily exploit the well-known analogy between a polymer and a random walk, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [35, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Within this analogy, the coordinate along the polymer plays a role of time and the polymer contour is thought of as the trajectory of a random walker, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Adopting this language, we, thus, obtain a random walk whose trajectory represents the alternating free Brownian paths, which correspond to the gap regions of the polymer, and the Brownian bridges corresponding to the cohesion-mediated loops in our original polymer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Let us recall the key statistical properties of Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The propagator of the free Brownian motion in three dimensions, Gfree(⃗r, t|⃗r0, 0) = 1 (4πDt)3/2 exp � −(⃗r − ⃗r0)2 4Dt � , (14) describes the probability to find the Brownian particle having diffusivity D in the point ⃗r after time t if it starts in ⃗r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In context of the polymer model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (14) represents the probability distribution of the separation vector ⃗r − ⃗r0 between two monomers inside a gap region of the polymer provided that their contour separation is t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The Brownian bridge is the Brownian trajectory subject to the condition that the particle must return to its starting position after a certain amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Propagator of a Brownian bridge of length T with a base in ⃗r0 is given by Gbridge(⃗r, t|⃗r0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⃗r0, T) = Gfree(⃗r, t|⃗r0, 0)Gfree(⃗r0, T|⃗r, t) Gfree(⃗r0, T|⃗r0, 0) = � T 4πDt(T − t) �3/2 exp � − T(⃗r − ⃗r0)2 4Dt(T − t) � , (15) where 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (15) describes the probability that the Brownian particle, which starts in ⃗r0 and returns to ⃗r0 after time T, will be in ⃗r at the moment of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Equivalently, this equation defines the probability distribution of the separation vector between the loop base and the monomer inside this loop given the contour separation t and the loop length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' More generally, the Brownian bridge pinned at two different points ⃗r1 and ⃗r2 at the moments of time t1 and t2, respectively, is characterised by the following probability distribution Gbridge(⃗r, t|⃗r1, t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⃗r2, t2) = � t2 − t1 4πD(t2 − t)(t − t1) �3/2 exp � − (⃗r2 − ⃗r)2 4D(t2 − t) − (⃗r − ⃗r1)2 4D(t − t1) + (⃗r2 − ⃗r1)2 4D(t2 − t1) � , (16) where t1 ≤ t ≤ t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In what follows the propagators determined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (14) and (15) play a role of building blocks of the diagram calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' But before proceeding to the corresponding calculations, we need to discuss the basic properties of the two-state Markov chain that will be required to derive the statistical weights of the diagrams depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2 in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (a) Based on the analogy between polymer conformation and random walk trajectory, we introduce a time axis with time intervals measured in the units of the polymer contour length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (b) The continuous time Markov jump process with two states, “Loop” and “Gap”, and transition rates αl = λ−1, αg = g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' By construction, statistics of time intervals that this auxiliary Markov process spends in different states coincides with the statistics of alternating loops and gaps in our polymer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Basic Statistical Properties of Two-State Markov Process Let us consider a Markov process with the transition rates αl = 1/λ and αg = 1/g between two states, “Gap” and “Loop”, which dictates the duration of random time intervals which the random walker introduced in previous section spends in the free and looped segments of its trajectory, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In other words, this auxiliary Markov process generates the random length of gaps and loops in the original polymer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The stochastic dynamics of the two-state continuous-time Markov jump process is described by the following pair of equations dπg ds = −αgπg + αlπl, (17) dπl ds = αgπg − αlπl, (18) where πg(s) and πl(s) represent the probabilities that the monomer having contour coordinate s lies on the gap or loop, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' It is straightforward to find the stationary solution of these equations pg = αl αg + αl , pl = αg αg + αl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (19) Clearly, pg (pl) gives the probability that a randomly chosen point of the polymer with disorder op loops belongs to a gap (loop) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The propagator πA→B(s) of the Markov process is defined as the probability to find the process in the state “B” after time s under the condition that it starts in the state “A”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' It is easy to find from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (17) and (18) that πg→g(s) = 1 αg + αl � αl + αge−(αl+αg)s� , (20) πg→l(s) = αg αg + αl � 1 − e−(αg+αl)s� , (21) πl→g(s) = αl αg + αl � 1 − e−(αg+αl)s� , (22) πl→l(s) = 1 αg + αl � αg + αle−(αg+αl)s� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (23) In the limit s → +∞, these expressions turn into statistically stationary probabilities pg and pl to find the process in given states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' lims→∞ πl→g(s) = lims→∞ πg→g(s) = pg, lims→∞ πl→l(s) = lims→∞ πg→l(s) = pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To perform averaging over the loop disorder (see below), we will also need to know the statistical moment ⟨xs⟩, where xs is the time spent in the “Loop” state during the time interval [0, s] under the condition that the Markov process occupies the “Gap” state at both ends of this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To calculate the expectation of xs, we introduce the stochastic variable ζ(t), which can take two values: ζ(t) = l if at the moment t the Markov jump process is in the “Loop” state, and ζ(t) = g if the process is currently in the “Gap” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Then the random variable xs can be represented as xs = 1 s � s 0 I[ζ(t) = l]dt, (24) O loop gap GAP LOOP7 where I[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='] is an indicator variable equal to one if the condition in its argument is true, and equal to zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Performing averaging one obtains ⟨xs⟩ = 1 s � s 0 ⟨I[ζ(t) = l]⟩dt = 1 s � s 0 Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g]dt, (25) where Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] is the probability of finding the Markov jump process in the “Loop” state at time t given that it was in the “Gap” state both at time 0 and at time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This probability can be easily calculated due to the lack of memory of the past in a Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Indeed, Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] = Pr[ζ(s) = g|ζ(t) = l]Pr[ζ(t) = l|ζ(0) = g] Pr[ζ(s) = g|ζ(0) = g] , (26) and since Pr[ζ(s) = g|ζ(t) = l] = πl→g(s − t), Pr[ζ(t) = l|ζ(0) = g] = πg→l(t) and Pr[ζ(s) = g|ζ(0) = g] = πg→g(s) we obtain Pr[ζ(t) = l|ζ(0) = g, ζ(s) = g] = πl→g(s − t)πg→l(t) πg→g(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (27) Substituting this result into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (25) yields ⟨xs⟩ = αgαl[2 + (αg + αl)s + e(αg+αl)s((αg + αl)s − 2)] s(αg + αl)2[αg + αle(αg+αl)s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (28) Beyond the mean value, the full statistics of the random variable xs can be extracted from the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Namely, the probability density F(xs) is given by F(xs) = e−αgsδ(xs) + � αgαl(1−xs)s2 xs I1 � 2 � αgαlxs(1 − xs)s2 � e−αg(1−xs)s−αlxss αl αg+αl + αg αg+αl e−(αg+αl)s , (29) where I1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=') denotes the modified Bessel function of the first kind [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Diagram A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (3) and (7) We wish to calculate the mean-squared displacement (MSD) of the random walker after time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Depending on the modes of the walker motion at the initial and final moments of time we should distinguish four cases represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' If the walker is in the free segments of its trajectory both initially and after time s, see the diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2a, then the probability density function of the walker’s displacement ⃗r is given by the Gaussian distribution Pa(⃗r|s, xs) = Gfree(⃗r, (1 − xs)s|⃗0, 0) = 1 (4πD˜sa[s, xs])3/2 exp � − r2 4D˜sa[s, xs] � , (30) with the effective contour separation ˜sa[s, xs] = (1 − xs)s, where xs denotes the fraction of time that walker spent performing Brownian bridges during the course of motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 0 ≤ xs < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' The intuition behind Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (30) is quite transparent: since the closed Brownian paths don’t produce the walker’s displacement, the overall effect of loops in diagram (a) is equivalent to reduction of the time allowed to the walker for exploration of the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' For the mean-squared displacement we, thus, obtain R2 a(s|xs) = � d3rr2Pa(⃗r|s, xs) = 6D˜sa[s, xs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (31) Next, using basic properties of two-state Markov jump process described in section II, we find that the diagram (a) is characterized by the following statistical weight Wa(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = pgπg→g(s)F(xs), (32) where pg = αl αg+αl gives the probability that a starting point of the walker’s trajectory belongs to a free Brownian path, πg→g(s) = 1 αg+αl (αl + αge−(αl+αg)s) is the probability to find the walker in the free segment of its trajectory 8 after time s under the condition that initially it is also in the free segment, and F(xs) is the probability distribution of the random variable xs determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' To average the contribution of the diagram (a) over the disorder of random loops, we should integrate the product R2 a(s|xs)Wa(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) over xs from 0 up to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' ⟨R2 a(s|xs)⟩loops = � 1 0 dxsR2 a(s|xs)Wa(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = 6Dspgπg→g(s)(1 − ⟨xs⟩), (33) where ⟨xs⟩ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Diagram B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (4) and (8) Next, let us assume that the walker starts in the loop and finds itself in the free segment of its trajectory after time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2b of the main text, the loop containing the starting point of the walker’s trajectory is parameterized by the time intervals t1 and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' After averaging over the position of the loop base ⃗r0, the probability density function of the walker’s displacement becomes Pb(⃗r|s, t1, t2, xs−t2) = � d3r0Gbridge(⃗0, 0|⃗r0, −t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⃗r0, t2)Gfree(⃗r, t2 + (1 − xs−t2)(s − t2)|⃗r0, t2) = = 1 (4πD˜sb[s, t1, t2, xs−t2])3/2 exp � − r2 4D˜sb[s, t1, t2, xs−t2] � , (34) where ˜sb[s, t1, t2, xs−t2] = (1 − xs−t2)(s − t2) + t1t2 t1+t2 , and 0 ≤ xs−t2 < 1, t1 ≥ 0, 0 ≤ t2 ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Now xs−t2 is the fraction of time the walker spend performing Brownian bridges during the time interval between t2 and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Therefore, the mean-squared displacement of the walker is given by R2 b(s|t1, t2, xs−t2) = � d3rr2Pb(⃗r|s, t1, t2, xs−t2) = 6D˜sb[s, t1, t2, xs−t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (35) Next, for the statistical weight of the diagram (b) we obtain Wb(t1, t2, xs−t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = 2plα2 l e−αl(t1+t2)πg→g(s − t2)F(xs−t2), (36) where pl = αg αg+αl gives the probability that a starting point of the walker’s trajectory belongs to a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Obviously, the case when the walker starts in the free segment and finishes in the closed segment is completely equivalent to the situation that we have just considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This explains the origin of factor 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (35) and (36) one obtains that the loops-averaged contribution of the diagram (b) is given by the following integral ⟨R2 b(s|t1, t2, xs−t2)⟩loops = � ∞ 0 dt1 � s 0 dt2 � 1 0 dxs−t2R2 b(s|t1, t2, xs−t2)Wb(t1, t2, xs−t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = = 12Dplα2 l � ∞ 0 dt1 � s 0 dt2 � (1 − ⟨xs−t2⟩)(s − t2) + t1t2 t1 + t2 � e−αl(t1+t2)πg→g(s − t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (37) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Diagram C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (5) and (9) Now let us consider the scenario when the starting and the final points of the walker’s trajectory belong to the same loop, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 2c in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Performing averaging over the position of the loop base we find the following result for the probability distribution of the walker’s displacement after time s Pc(⃗r|s, t1, t2) = � d3r0Gbridge(⃗0, 0|⃗r0, −t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⃗r0, t2)Gbridge(⃗r, s|⃗0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='⃗r0, t2) = = 1 (4πD˜sc[s, t1, t2])3/2 exp � − r2 4D˜sc[s, t1, t2] � , (38) 9 where ˜sc[s, t1, t2] = � 1 − s t1+t2 � s, and t1 ≥ 0, t2 ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (38) one obtains the mean-squared walker’s displacement R2 c(s|t1, t2) = � d3rr2Pc(⃗r|s, t1, t2) = 6D˜sc[s, t1, t2], (39) whereas for the the statistical weight of the trajectories described by the diagram (c) we find Wc(t1, t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = plα2 l e−αl(t1+t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (40) Thus, the loop-averaged contribution of the diagram (c) to mean-squared displacement is determined by the fol- lowing double integral ⟨R2 c(s|t1, t2)⟩loops = ∞ � 0 dt1 ∞ � s dt2R2 c(s|t1, t2)Wc(t1, t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = 6Dsplα2 l ∞ � 0 dt1 ∞ � s dt2 � 1 − s t1 + t2 � e−αl(t1+t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (41) VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Diagram D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (6) and (10) Finally, the probability distribution of the walker’s displacement in the situation when the initial and final point of its trajectory belong to different loops is given by Pd(⃗r|s, t1, t2, τ, xτ, T2) = 1 (4πD˜sd[s, t1, t2, τ, xτ, ˜T] exp � − r2 4D˜sd[s, t1, t2, τ, xτ, ˜T] � , (42) where ˜sd[s, t1, t2, τ, xτ, ˜T] = t1t2 t1+t2 + (1 − xτ)τ + ˜t1˜t2 ˜t1+˜t2 , and ˜t1 = ˜T + τ + t2 − s, ˜t2 = s − τ − t2, t1 ≥ 0, 0 ≤ t2 ≤ s, 0 ≤ τ ≤ s − t2, 0 ≤ xτ ≤ 1, ˜T ≥ s − t2 − τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In this case, xτ denotes the fraction of time the walker spend in ”Loop” state during the time interval between t2 and s − ˜t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (42) one obtains the mean-squared walker’s displacement R2 d(s|t1, t2, τ, xτ, ˜T) = � d3rr2Pc(⃗r|s, t1, t2) = 6D˜sd[s, t1, t2, τ, xτ, ˜T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (43) Clearly, the statistical weight of the trajectories described by the diagram (d) is given by Wd(t1, t2, τ, xτ, ˜T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = plα3 l e−αl(t1+t2+ ˜T )αgπg→g(τ)F(xτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (44) Thus, for the loops-averaged contribution of the diagram (d) we find ⟨R2 d(s|t1, t2, τ, xτ, ˜T)⟩loops = ∞ � 0 dt1 s � 0 dt2 s−t2 � 0 dτ ∞ � s−t2−τ d ˜T 1 � 0 dxτR2 d(s|t1, t2, τ, xτ, ˜T)Wd(t1, t2, τ, xτ, ˜T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' s) = 6Dplα3 l αg ∞ � 0 dt1 s � 0 dt2 s−t2 � 0 dτ ∞ � s−t2−τ d ˜T � (1 − ⟨xτ⟩)τ + t1t2 t1 + t2 + ( ˜T + t2 + τ − s)(s − t2 − τ) ˜T � e−αl(t1+t2+ ˜T )πg→g(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (45) Calculating the integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (33), (37), (41) and (45) and summing the resulting expressions, we arrive at the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' One-loop approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (13) The kurtosis coefficient of the random vector ⃗R(s) is defined as K(s) = ⟨R4(s)⟩ ⟨R2(s)⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (46) 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' For λ/g ≪ 1 and s/g ≪ 1 the two-point statistics of an ideal chain with a disorder of random loops can be computed in the one-loop approximation, leaving only those diagrams containing at most one cohesin-mediated loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In other words, diagram (d) can simply be ignored, and the dash-dotted line in diagrams (a) and (b) can be replaced by a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' We already know that the MSD ⟨R2(s)⟩ in our model is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As for the fourth order statistical moment ⟨R4(s)⟩, taking into account the Gaussian form of the conditional distribution functions (30), (34), (38), and (42), one readily obtains ⟨R4(s)⟩ = 60D2 � α=a,b,c,d ⟨˜s2 α[s, {A}α]⟩loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (47) Exact diagrammatic calculations accordingly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (47) are possible in principle, but practically difficult to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' However, analytical derivation of the fourth moment ⟨R4(s)⟩ becomes feasible in the rare loops limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' More specifically, if λ/g ≪ 1 and s/g ≪ 1, then one can neglect the realizations of diagrams where there is more than one loop, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Then, neglecting the diagram (d) and simplifying the formulas (3)-(5) from the main text, we obtain ⟨R4(s)⟩ ≈ 60D2 � α=a,b,c ⟨˜s2 α[s, {A}α]⟩one loop, (48) where ˜sa[s, xs] = (1 − xs)s, (49) ˜sb[s, t1, t2] = s − t2 + t1t2 t1+t2 , (50) ˜sc[s, t1, t2] = � 1 − s t1+t2 � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (51) When averaging over the disorder of the loops, it is convenient to pass to the new variables T and q defined as t1 = (1 − q)T, t2 = qT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (52) In terms of these variables, the diagrams (a), (b) and (c) depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' 5 are characterized by the following statistical weights Wa(xs|s) = pgπg→g(s)F(xs), for 0 ≤ xs < 1, (53) Wb(T, q, xs−qT |s) = 2pl˜ρl(T)πg→g(s − qT)F(xs−qT ), for 0 ≤ q ≤ min[1, s T ], T ≥ 0, 0 ≤ xs−qT < 1, (54) Wc(T, q|s) = pl˜ρl(T), for s T ≤ q ≤ 1, T ≥ s, (55) where ˜ρl(T) denote the probability density of the random loop length in the statistical experiment where loops are sampled by random choice of points along the polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Clearly, ˜ρl(T) = T λ ρl(T), where ρl(T) = 1 λ exp(− T λ ) is the actual loop length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Using the smallness of the dimensionless parameters λ/g ≪ 1 and s/g ≪ 1, we find from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (19), (20) and (29) pg = g g + λ ≈ 1 − λ g , pl = λ g + λ ≈ λ g , (56) 0() 0(1) 0() (011 and πg→g(s)F(xs) ≈ (1 − s g )δ(xs) + (1 − xs)s2 g ρl(xss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (57) By inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (56) and (57) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (53,54,55) and neglecting the terms nonlinear in the parameter λ/g one obtains Wa(xs|s) ≈ δ(xs) + λ g � −(1 + s λ)δ(xs) + (1−xs)s2 λ ρl(xss) � , for 0 ≤ xs < 1, (58) Wb(T, q|s) ≈ 2 λ g T λ ρl(T), for 0 ≤ q ≤ min[1, s T ], T ≥ 0, (59) Wc(T, q|s) ≈ λ g T λ ρl(T), for s T ≤ q ≤ 1, T ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (60) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' performing averaging over disorder of loops,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' we find that in the first order-approximation with respect to the ratio λ/g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' the fourth-order statistical moment of the random vector ⃗R(s) is given by ⟨R4(s)⟩ ≈ 60D2 � 1 0 dxs˜s2 a[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' xs]Wa(xs|s) + 60D2 � ∞ 0 dT � min[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='s/T ] 0 dq˜s2 b[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' q]Wb(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' q|s) + (61) +60D2 � ∞ s dT � 1 s/T dq˜s2 c[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' q]Wc(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' q|s) = (62) = 60(Ds)2 � 1 + λ g s2 λ �� 1 0 dxρl(xs)(−3 5x3 + 5 3x2 − 2x) + � +∞ 1 dxρl(xs)(− 3 5x2 + 5 3x − 2) �� = (63) = 60(Ds)2 � 1 + λ g f4( s λ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (64) where f4(s) = −54 − 96e−s − 10s(3s − 5) + 24(25 + 9s)E5(s) 15s2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (65) and En(s) = � +∞ 1 x−ne−sxdx is the exponential integral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) in the main text, the MSD in the same approximation is given by ⟨R2(s)⟩ ≈ 6Ds � 1 + λ d �2λ(1 − e− s λ ) 3s − 1 + 2 3E3( s λ) �� (66) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (61) and (66) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (46) finally yields K(s) ≈ 5 3 + λ g fKurt( s λ), (67) where fKurt(s) = 2 3s2 � (9 + 4s − 3s2)e−s − 9 + 5s + s2(5 + 3s)E3(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (68) This result matches equation (13) from the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' As noted above, the one-loop approximation relies on smallness of two dimensionless parameters: λ/g and s/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' However, by a happy coincidence the one-loop answer for MSD agrees with the exact result given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (12) in the main text for arbitrary large value of s/g provided λ/g ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' In other words, the large-scale behaviour of MSD obtained from one-loop calculations is accurate for any value of s, despite the one-loop approximation is justified only if s ≪ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' This fact allows us to conclude that since the statistics of zero-mean random vector ⃗R(s) is Gaussian at s ≫ g, λ, the one-loop prediction for kurtosis given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' (67) also remains valid for arbitrary s when λ/g ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' ∗ sergb27@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content='ru [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Ganji, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Shaltiel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Bisht, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Kim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Kalichava, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Haering, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Dekker, Science 360, 102 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE2T4oBgHgl3EQfYgfk/content/2301.03856v1.pdf'} +page_content=' Golfier, T.' metadata={'source': 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b/XtE3T4oBgHgl3EQfcAoE/content/tmp_files/2301.04520v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1175877689fd29b69409d6c8a0d66d6d5d84dd43 --- /dev/null +++ b/XtE3T4oBgHgl3EQfcAoE/content/tmp_files/2301.04520v1.pdf.txt @@ -0,0 +1,2958 @@ +Entangling spins using cubic nonlinear dynamics +Lingxia Wang,1 Yani Wang,1 Yujing Cheng,1 Zhiqi Yan,1 Lei Xie,1 Gang Liu,2 +Jinmin Fan,1 Di Wang,1 Yiling Song,1 Linli He,1, ∗ Wei Xiong,1, † and Mingfeng Wang1, ‡ +1Department of Physics, Wenzhou University, Zhejiang 325035, China +2School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China +Entangled states with a large number of N atomic spins are a key ingredient for quantum infor- +mation processing and quantum metrology. Nowadays, the preparation of such states has mainly +relied on the quadratic nonlinear dynamics. Here, we investigate the preparation of spin-spin mul- +tipartite entanglement, witnessed by quantum Fisher information, by using the cubic nonlinear +dynamics. We find that, in the regime of weak coupling, the cubic scheme can greatly speed up +the rate of entanglement generation as compared to the quadratic scheme (about N times faster). +In the strong coupling regime, the cubic nonlinear dynamics enables the periodic in time gener- +ation of a broad variety of new-type macroscopic superposition states, which allow us to realize +near-Heisenberg-limit phase sensitivity. In addition, we also reveal an interesting feature that the +amount of entanglement generated by the cubic scheme has a macroscopic sensitivity to the parity +of N, which has no counterpart in quadratic nonlinear dynamics and can be exploited for sensing +the parity of N at the single-spin level. We also propose a new approach for a fast and high-fidelity +generation of maximally entangled Greenberger-Horne-Zeilinger (GHZ) states. By using an alterna- +tive cubic-quadratic-admixture type of nonlinear interaction, we show that one may accelerate the +procedure of GHZ-state generation. The realization of the cubic nonlinear dynamics is also consid- +ered, showing that the cubic nonlinear dynamics can be realized by either repeatedly using linear- +and quadratic-nonlinear dynamics or utilizing light-mediated interactions in just one step. Finally, +by taking realistic imperfections into account, we find that the cubic scheme is sensitivity to the +single-spin decay in the strong coupling regime, while is robust against the collective dephasing. Our +proposed schemes offer potential possibilities for realizing high-sensitivity metrology in a variety of +platforms, including trapped ions and cold or warm atomic ensembles. +I. +INTRODUCTION +The generation of entanglement between a large num- +ber of spins is an extremely important subject in preci- +sion metrology and quantum science. In quantum metrol- +ogy [1], highly entangled spin states enable precision +metrology beyond the standard quantum limit (SQL) [2], +even approaching the Heisenberg limit (HL) [3, 4]. In the +field of quantum information [5, 6], entangled spin ensem- +bles are not only recognized as key resources for quantum +communication [7–9] but also considered as a promising +platform for quantum computation [10–13]. +To date, a variety of approaches have been developed +for producing entangled states of spin ensemble, which +can be classified into two main categories. One is based +on the projection measurement (such as quantum nonde- +molition measurement) [14–17]: first, entanglement is es- +tablished between the spin system and an auxiliary quan- +tum system (usually a light field), and then, a measure- +ment of the auxiliary quantum system will project the +spin state into a multipartite entangled state. Another +one has relied on unitary evolution of an initial product +spin state under a nonlinear spin-spin (NSS) interaction. +Among these NSS interactions, the most widely studied +one is possible the one-axis-twisting (OAT) interaction +∗ linlihe@wzu.edu.cn +† xiongweiphys@hotmail.com +‡ mfwang@wzu.edu.cn +[18–23], which, as shown by Ueda et al. [18], can pro- +duce pairwise spin-spin entanglement that is the origin +of spin squeezing [24, 25]. +Spin squeezing is probably +the most sought-after multipartite entangled resource in +the field of quantum metrology, as the phase estimation +based on spin squeezing is comparatively easy to imple- +ment in realistic experiments [25]. +Up till now, most +studies of NSS interaction have mainly been concentrated +on how to efficiently create highly squeezed spin states, +such as two-axis-twisting interaction [26–30], twist-and- +turn interaction [27, 31–33], and twisting-tensor interac- +tion [34]. +However, squeezed spin states are only one +category of multipartite entangled states that can bene- +fit the quantum metrology. Other categories of entangled +states, although having no spin-squeezing property, may +also be useful for quantum metrology and sensing, such +as the GHZ state enabling the phase sensitivity reaching +the HL [35]. In fact, apart from the spin squeezing, the +quantum Fisher information (QFI) provides a more gen- +eral and profound way to estimate whether a given spin +state is useful or not for quantum metrology [36–41]. The +larger the QFI of the entangled state, the more useful +the state might be. Therefore, it is of particulary neces- +sary to reconsider the entanglement generation induced +by the NSS interactions from the perspective of QFI. Ac- +cordingly, discovering and devising new NSS-interaction +schemes that can rapidly and efficiently generate large +QFI is of vital importance for realizing high-sensitivity +metrology. +In this paper, we propose to use cubic NSS interaction +arXiv:2301.04520v1 [quant-ph] 11 Jan 2023 + +2 +to entangle individual spins. Although exhibiting no spin +squeezing, the entangled state created by the cubic inter- +action have several advantages over the quadratic inter- +action from the perspective of QFI. First, in the weak- +coupling regime we find that the cubic scheme can pro- +duce QFI (and thus entanglement) much more rapidly +than the quadratic one. Quantitative analysis indicates +that the acceleration rate is proportional to the spin num- +ber of the system. The cubic scheme thus offers a great +advantage over the quadratic scheme in the case of large +spin systems. Second, the QFI of the cubic scheme in +the strong-coupling regime oscillates fast with coupling +strength, which, on average, is larger than the QFI pro- +duced by the quadratic scheme. Besides, the cubic NSS +interaction enables the production of a broad variety of +macroscopic superposition states that have large QFI, +which has no counterpart in the quadratic NSS dynamic. +We also analyze an interesting phenomena that has +not yet been discovered previously. +That is, the QFI +production of the cubic scheme is extremely sensitivity +to the parity of the total spin number N. We find that +the amount of QFI at a specific instant of time for even N +spins versus odd N + 1 spins change dramatically from +N (corresponding to no entanglement among spins) to +N 2 (maximal entanglement). This entanglement even- +odd effect is quite different from the one exhibited by +the OAT interaction [42], which, as we will show later, +is an orientation even-odd effect. We also show that this +entanglement even-odd effect enables us to design a new +type of sensing modality to detect the parity of the total +spin number of a spin system at the single-spin level. +Apart from the cubic NSS interaction, +we also +have studied a hybrid NSS interaction—cubic-quadratic- +admixture (CQA) interaction, which is a weighted sum +of the cubic and the quadratic interaction. We find that +the CQA interaction is an excellent tool for preparing the +GHZ states. High-fidelity GHZ states could be created +by simply applying the CQA evolution to the spin sys- +tem for a certain time interval. In contrast to the OAT +scheme [42], our hybrid scheme can greatly accelerate the +procedure of GHZ-state generation, which tremendously +eases experimental requirements. +To realize the cubic interactions in realistic spin sys- +tems, two approaches have been developed. One utilizes +the linear and quadratic interactions. +Unlike the har- +monic oscillator systems, where high-order interactions +can not be constructed from the quadratic interactions +(known as the Gaussian operations) [43, 44], we show +that the cubic interaction can be approximately con- +structed by repeatedly using linear and OAT interactions. +This method should be widely applicable to various spin +systems, as the OAT interactions have been experimen- +tally realized in a number of physical systems [20–22]. +Another one uses light-mediated interactions. The spin +system is placed inside an one-side optical cavity, forming +a spin-cavity system. We show that, by simply sending +an optical pulse, off-resonant with cavity mode, into the +spin-cavity system, the cubic NSS dynamics is realized +after the reflection of the pulse by the one-sided cavity. +Such method should be able to realize the cubic interac- +tion in just one step, which is rather attractive from the +perspective of experimental implementation. +Finally, we analyze the impact of spin damping, includ- +ing the single-spin decay and the collective-spin dephas- +ing. We reveal that, in the presence of damping, the cubic +scheme works much better than the quadratic one. That +is, in the weak-coupling regime, the cubic scheme can still +maintain is speed advantage in QFI production; besides, +the macroscopic superposition state created by the cu- +bic interaction is much more robust against decoherence +than the one created by the quadratic interaction. +The rest of the paper is organized as follows. In Sec. II +we introduce the multipartite entanglement of the collec- +tive spins and its correlations with QFI. In Sec. III we +first analytically derive the amount of achievable QFI in +the weak coupling regime. Then, we analysis the prop- +erties of the macroscopic superposition states created by +the cubic interaction. In Sec. IV we discuss the entangle- +ment even-odd effect. In Sec. V we describe how to speed +up the procedure of GHZ-state generation. In Sec. VI we +present two approaches to realize the cubic NSS dynam- +ics. In Sec. VII we analysis the impact of the decoherence +to the entanglement generation. Finally, we summarize +in Sec. VIII. +II. +MULTIPARTITE ENTANGLEMENT IN +QUANTUM SPIN SYSTEMS +We +consider +creating +multiparticle +entanglement +among spins in an ensemble consisting of N identical two- +level atoms with the excited state |↑⟩ and the ground +state |↓⟩. To describe the collective properties of such +system, we define the pseudo angular momentum oper- +ators Si = � +k σi +k/2(i = x, y, z) for atoms, which satisfy +the commutation relations [Si, Sj] = iεijkSk, with εijk +being the Levi-Civita symbol, where σi +k is a Pauli matrix +for the ith atom, e.g., σi +x = | ↑⟩i ⟨↓|i + |↓⟩i ⟨↑|i . Suppose +that all the elementary spins point in the same mean di- +rection (θ, φ), that is, each atom is prepared in the state +|θ, φ⟩i = cos θ +2| ↑⟩i+eiφ sin θ +2| ↓⟩i, forming the well-known +coherent spin state (CSS) [18] +|θ, φ⟩ = |θ, φ⟩⊗N +i += +2S +� +k=0 +� +(2S)! +(2S − k)!k! +× +� +sin θ +2 +�2S−k� +cos θ +2 +�k +eikφ |S, S − k⟩ , (1) +where the collective angular momentum states |S, m⟩ +(Dicke +states) +is +the +eigenstate +of +Sz, +satisfying +Sz |S, m⟩ = m |S, m⟩ with S = N/2. The CSSs are sepa- +rable (nonentangled), and a conventional way to entangle +the particles is to utilize the second order nonlinear pro- +cesses, e.g., OAT evolution UOAT = exp[−iχtS2 +x] [18], +where χ is the coupling constant. +To show how spin + +3 +entanglement is created by UOAT, assume that the col- +lective spin is polarized along the z direction, leading to +the initial state |ΨA⟩in = |↑⟩⊗N. +At short times, the +evolution of this state is found to be +|ΨA⟩out = UOAT|ΨA⟩in +≈ N +� +�|↑⟩⊗N − +2iα +N (1 − iα) +� +i̸=j +|↓i↓j⟩ |↑⟩⊗(N−2) +̸=i,j +� +� , +where N = (1−iα)/ +√ +1 + 3α2 is a normalization constant +with α = Nχt, and in deriving the last equality we have +kept terms up to first order in S2 +x and used the relations +σx +i |↓⟩i (|↑⟩i) = |↑⟩i (|↓⟩i). Obviously, the entanglement +between the initial and first coupled (double-spin-flipped) +states has been created. +Such pairwise entanglement +have garnered tremendous attention for many years [25], +as they are the origin of spin squeezing, which have im- +portant applications in quantum metrology as well as in +fundamental physics [18, 45]. In fact, irrespective of the +creation of spin squeezing, multiparticle entanglement +can also be produced by higher-order nonlinearity, such +as the three-order (cubic) evolution Ux = exp{−iχtS3 +x}. +For this evolution, one may also derive the time evolved +state at time t +|ΨA⟩out = Ux|ΨA⟩in +≈ N +� +�|↑⟩⊗N − 3iα +4N +� +i̸=j̸=k +|↓i↓j↓k⟩| ↑⟩⊗(N−3) +̸=i,j,k +� +� +with the normalization constant N = 1/ +� +1 + 3Nα2/32, +showing that the triple-wise entanglement among spins +is produced. Obviously, such a state exhibits no property +of spin squeezing [24], while a natural question arises: is +it useful for sub-shot-noise interferometry? +To answer this question we use the QFI to quantify +the degree of useful entanglement for quantum metrol- +ogy. The QFI is closely related to the multipartite en- +tanglement [37] and also gives the fundamental limit to +the precision achievable in an unknown-parameter esti- +mation protocol [36]. +Considering a scenario of phase +estimation, a probe spin state ρin is transformed into +ρβ = exp (−iβSn) ρin exp (iβSn) by the n-direction col- +lective spin generator Sn, where β denotes an unknown +phase shift to be estimated. The phase sensitivity is lim- +ited by the quantum Cram´er-Rao bound [46]: +∆β ≥ ∆βQCR = +1 +� +FQ [ρin, Sn] +, +(2) +where +FQ[ρ, Sn] = 2 +� +l,l′ +(λl − λl′)2 +λl + λl′ +|⟨l|J|l′⟩|2 +(3) +is the QFI, λl and |l⟩ are the eigenvalues and eigenvectors +of the probe state ρin, respectively. The QFI is a measure +of how susceptible of ρin to small influences induced by +Sn. The larger the value of QFI, the more precision the +estimation. In the case of pure state, ρin = |ψin⟩ ⟨ψin|, +Eq. (3) can be further simplified to [47] +FQ[ρin, Sn] = 4(∆Sn)2 +|ψin⟩, +(4) +where (∆A)2 +|ψ⟩ = ⟨ψ|A2|ψ⟩ − ⟨ψ|A|ψ⟩2 is the variance of +A in the state |ψ⟩. For a given probe state ρin, it is needed +to optimize the rotation direction, n → nop, to maximize +the variances of Sn (thus QFI) [35]. +If, for example, +the probe state is in the separable CSS |ψin⟩ = | π +2 , 0⟩, +one may choose nop = z to yield FQ = N, resulting +in a sensitivity ∆β = 1/ +√ +N, which is exactly the SQL +mentioned above. To overcome this limit, one should use +the entangled states, e.g., the GHZ states [48], |ψin⟩ = +1 +√ +2(| π +2 , 0⟩+| π +2 , π⟩), with which the QFI can be calculated +(by choosing nop = x) to give FQ = N 2, leading to the +HL sensitivity ∆β = 1/N. One thus can conclude that +any entangled states whose QFI satify N < FQ ≤ N 2 are +useful for sub-SQL sensitivity [36]. +III. +THE CUBIC INTERACTIONS +A. +Weak coupling regime +We now proceed with the derivation of the QFI of +the cubic-interaction-evolved states. For convenience, we +suppose that the spins are initially prepared in the CSS, +|ψ⟩ = | π +2 , 0⟩, which is subjected to the time evolution +Uz = exp +� +−iχtS3 +z +� +. +(5) +One thus obtains the probe state at time t +|ψin(t)⟩ = Uz |ψ⟩ = 1 +2S +2S +� +k=0 +� +(2S)! +(2S − k)!k! +×e−iχt(S−k)3 |S, S − k⟩ . +(6) +For this state, since [Sz, Uz] = 0, Sz is conserved during +evolution. Therefore, the uncertainties are redistributed +only in the x-y plane [see Fig. 1(c)], which predicts that +the optimal direction of the generator, nop, is in some +direction in the x-y plane. To see how the uncertainties +are redistributed, we next work in the Heisenberg picture. +The time evolution of the ladder operators S± = Sx±iSy +can be exactly evaluated to give [18]: +S−(t) = U † +zS−(0)Uz = e−iµ(Sz +2+Sz+ 1 +3)S−(0), +(7) +where µ ≡ 3χt. +The transverse components after the +cubic evolution are then given by +Sx(t) = 1 +2 +� +S+eiµ(S2 +z+Sz+ 1 +3) + e−iµ(S2 +z+Sz+ 1 +3)S− +� +, (8) +Sy(t) = 1 +2i +� +S+eiµ(S2 +z+Sz+ 1 +3) − e−iµ(S2 +z+Sz+ 1 +3)S− +� +.(9) + +4 +To find nop, we calculate the variance of an arbitrary +angular momentum operator along the φ direction, Sφ = +Sx(t) cos φ + Sy(t) sin φ, in the x-y plane, yielding +(∆Sφ)2 +|ψ⟩ = cos2φ(∆Sx)2 +|ψ⟩ + sin2φ(∆Sy)2 +|ψ⟩ ++ sin 2φ +�1 +2 ⟨{Sx, Sy}⟩ − ⟨Sx⟩⟨Sy⟩ +� +,(10) +where {., .} denotes the anticommutator of two observ- +ables. To calculate the first moments of the spin compo- +nents in Eq. (10), we turn to evaluate the mean of the +ladder operator +⟨S+ (t)⟩ = 2−2S +2S +� +k=0 +2S +� +l=0 +� +(2S)! +(2S − k)!k! +� +(2S)! +(2S − l)!l! +×⟨S, S − k|S+eiµ(S2 +z+Sz+1/3) |S, S − l⟩ += 2−2S +2S +� +l=1 +(2S)! +(2S − l)!(l − 1)!eiµ[(S−l)2+S−l+ 1 +3] += S +2S−1 +� +m=0 +P2S−1 (m) eiµ[(S−m)2−(S−m)+ 1 +3], (11) +where in the last equality we set m = l − 1 and the bi- +nomial distribution P2S−1(m) can be approximately con- +verted to the Gaussian distribution, +P2S−1 (m) = +(2S − 1)! +(2S − 1 − m)!m! +�1 +2 +�2S−1−m�1 +2 +�m +≃ +1 +√ +Sπ +e− (S−m) +S +2 +, +(12) +for large S (see Appendix A for details). We thus obtain +⟨S+ (t)⟩ = S +√π +1 +√ +S +√ +S +� +k=− +√ +S +e−(1−iµS)k2−iµ +√ +Sk+ 1 +3 iµ +≃ S +√π +� +∞ +−∞ +e−(1−iµS)k2−iµ +√ +Sk+ 1 +3 iµdk +≃ +S +√1 − iµS , +(13) +where k = (S − m)/ +√ +S, and, in the second equality, we +have transformed the sum to integral, which is valid only +when ∆k = 1/ +√ +S → 0 for, again, large S, and, in the last +equality, we also have used the approximation exp [iµ(4− +iµS)/12(1 − iµS)] ≈ 1 for µ ≪ 1. Along the same lines, +one may derive the quadratic expectation values ⟨S2 ++(t)⟩ +and ⟨S+(t)S−(t)⟩ (see Appendix B for more details), with +which we are able to calculate the means +⟨Sx⟩ = S +� +α1 (α1 + 1)/2, +⟨Sy⟩ = S +� +α1 (1 − α1)/2, +� +S2 +x +� += S +4 +� +(2S + 1) + (2S − 1) +� +α4 (α4 + 1)/2 +� +, +� +S2 +y +� += S +4 +� +(2S + 1) − (2S − 1) +� +α4 (1 − α4)/2 +� +, +⟨{Sx, Sy}⟩ = S +2 (2S − 1) +� +α4 (1 − α4)/2, +(14) +(a) +(b) +(c) +(i) +(ii) +(iii) +II1 +x +y +z +x +z +x +z +y +y +I1 +I2 +II2 II3 II4 +t +χ +FIG. 1. (Color online) QFI of the time-evolved states ver- +sus coupling strength (expressed in terms of either α or χt, +see text for clarification) for N = 200: exact numerical solu- +tion of the cubic scheme (solid orange curve) and the OAT +scheme (dash-dotted green curve). (a) The dashed blue curve +is the analytical result of Eq. (16). (b) The peaks marked +I1,2 are the maximum QFI that is achievable by the cubic +scheme, while the submaximal QFI are marked by II1−4. (c) +Quasiprobability distribution of different spin states: (i) is +a CSS, (ii) is a GHZ state, and (iii) is a four-components +Schr¨odinger cat state. +where we have defined the new parameters αk += +1/ +� +1 + kS2µ2 with k = 1, ..., 4. Substituting these val- +ues into Eq. (10) we finally arrive at +(∆Sφ)2 +|ψ⟩ = S +4 [2 (1 − α1) S + 1] ++ +� +A2 + B2 cos (2φ − 2δ) , +(15) +where +A = +S +4 +√ +2(2S − 1) +� +α4 (1 + α4) − S2 +2 α2 +1, +B = +S +4 +√ +2 (2S − 1) +� +α4 (1 − α4) − µS3 +2 α2 +1, +δ = 1 +2 arctan +� B +A +� +. +Eq. (15) is maximized when φ = δ, obtaining +FQ = 4 +� +A2 + B2 + S [2 (1 − α1) S + 1] . +(16) +For Sµ ≪ 1, Eq. (16) can be approximated as +FQ ≈ 2S + 9 +2S2α2 ≥ N, +(17) +which indicates that any nonzero α enables the sensi- +tivity to surpass the SQL. Therefore, the entanglement + +5 +created by Eq. +(5) is useful for quantum metrology. +For comparison, the QFI created by OAT interaction +in the weak coupling regime is also calculated to give: +FQOAT ≈ 2S + 2Sα2. Apparently, the QFI produced by +cubic interaction is about S times faster than OAT in- +teraction. This is quite a promising advantage, since the +ability to create entangled quantum resources rapidly is a +pursuit in quantum metrology. It should be emphasized +that the speed-up rate is closely connected to N (the +large the N, the faster the increase in QFI), which means +that the cubic scheme might be more suitable for atomic +systems with a large number of atoms [16, 17, 19, 49]. +In Fig. 1(a) we compare the analytical result of Eq. +(16) (dashed blue curve) and the exact numerical results +from Eq. (6) (solid orange curve). The two curves fit +pretty well in the weak coupling regime and gradually de- +viate when α increases. For large α, the numerical results +display various oscillating structures, which are lost by +analytical result due to the discrete-to-continuous conver- +sion in Eq. (13). In fact, each peak of QFI is related to a +macroscopic supposition of collective spin, as will be dis- +cussed below. Fig. 1(a) also confirms that the QFI of the +cubic scheme increases much more rapidly with α than +the OAT scheme (dash-dotted green curve). In Fig. 1(b) +we also plot the periodic evolution of QFI in time for both +schemes. It shows that the OAT scheme can saturate the +HL at t = π/2χ, which corresponds to the creation of +a GHZ state [42], as shown in Fig. 1(c)(ii). +The QFI +of the cubic scheme, however, has a quite complicated +structure. Although can not saturate the HL, there exist +two maximum peaks [labeled by I1,2 in Fig. 1(b)] that +are quite near the HL in a period of evolution, which +corresponds to a Schr¨odinger cat state with four super- +posed CSSs [see Fig. 1(c)(iii)]. Besides the two maximum +peaks, there also exist a number of lower peaks, e.g., four +secondary peaks [labeled by II1−4 in Fig. 1(b)]. Next, we +quantify the amount of QFI for these peaks and explore +the properties of these peak states . +B. +Strong coupling regime +For convenience, we first assume that N is even and +rewrite the state of Eq. (6) in the following form +|ψin(t)⟩ = 1 +2S +S +� +m=−S +� +(2S)! +(S + m)!(S − m)!e−iχtm3 |S, m⟩ +(18) +by setting k = S − m. +Considering the time evolved +state at special time t = π/nχ [50], where n is an integer, +the evolution factor exp(−iπm3/n) at this time has the +following periodic properties: +exp +� +−iπ +n (m + 2n)3 +� += exp +� +−iπ +n m3 +� +. +(19) +Such periodicity property enables us to expand the evo- +lution factor as a Fourier series [50] +exp +� +−iπ +n m3 +� += +2n−1 +� +q=0 +f e +q exp +� +−iπq +n m +� +, +(20) +where the coefficients f e +q are given by the inverse Fourier +transform +f e +q = 1 +2n +2n−1 +� +m=0 +exp +�iπq +n m +� +exp +� +−iπ +n m3 +� +. +(21) +Eq. (20) indicates that we have successfully converted an +exponentially cubic form into sums of exponentials linear +in m, which is a key step for the derivation. Inserting Eq. +(20) into Eq. (18), we obtain +����ψin +� π +nχ +�� += +2n−1 +� +q=0 +f e +q +���π +2 , πq +n +� +, +(22) +which shows that a Schr¨odinger-cat-like state (SCS) (a +superposition of the CSSs) can be produced by the cubic +evolution at the particular time t = π/nχ. The charac- +teristics of the SCSs are determined by the coefficients +f e +q . Specifically, by using Eqs. (21) and (22), one may +derive the form of SCS for n = 4, +����ψin +� π +4χ +�� += 1 +2 +����π +2 , 0 +� ++ +���π +2 , π +4 +� ++ +���π +2 , π +� +− +���� +π +2 , 5π +4 +�� +, +(23) +and for n = 12, +����ψin +� π +12χ +�� += 1 +2 +����π +2 , π +12 +� ++ +���π +2 , π +3 +� +− +���� +π +2 , 13π +12 +� ++ +���� +π +2 , 4π +3 +�� +. +(24) +Notably, the states (23) and (24) are just the two states +that create the two maximum QFI peaks I2 and I1 [see +Fig. 1(b)], respectively. +Next, we turn to derive the +amount of QFI of peak I2. +By using the state in Eq. +(23), one may directly calculate the means and variances +of the collective spin components, obtaining +⟨Sx⟩ = S +� +cos π +8 +�2S−1 +cos +�π +8 (2S + 1) +� +, +⟨Sy⟩ = S +� +cos π +8 +�2S−1 +sin +�π +8 (2S + 1) +� +, +� +S2 +x +� += 1 +8 +� +6S2 + S +� +, +� +S2 +y +� += 1 +8 +� +2S2 + 3S +� +, +⟨{Sx, Sy}⟩ = 1 +4 +� +2S2 − S +� +. +(25) +Substituting them into Eq. (10), after optimization of +(∆Sφ)2 +|ψin(π/4χ)⟩ we get +FQ = 2S2 +� +1 + 1 +√ +2 − +� +cos π +8 +�4S−2 +× +� +1 + cos +�πS +2 +��� ++ +� +1 − 1 +√ +2 +� +S +(26) + +6 +FIG. 2. (Color online) (a) QFI produced by the cubic scheme versus coupling strength χt for N = 200. The QFI peaks marked +I1, ..., V1 are produced by the states |ψin(π/12kχ)⟩ in Eq. (6) with k = 1, ..., 5, respectively. Insert: The QFI of the states +|ψin(π/12kχ)⟩ and |ψin(π/3(2k − 1)χ)⟩ vs k for N = 60 and N = 61, respectively. (b)–(e) The Fourier-coefficients distribution +[ given by Eq. (22)] (bottom) and the QDP (top) of the states of the peaks marked II1 − V1 in (a) for N = 1500. +TABLE I. The quantum state, QFI [calculated via Eq. (33)], and achievable sensitivity of each peak in Fig. 2(a). +Peaks +The quantum state of peaks (neglecting the normalization) +QFI +∆β +I1 +|ψin +� +π +12χ +� +⟩ = |GHZ− +π/12⟩ + |GHZ+ +π/3⟩ +0.85N 2 +1.08/N +II1 +|ψin +� +π +24χ +� +⟩ = |GHZ+ +π/6⟩ + +7 +10 |GHZ− +7π/24⟩ − +7 +10 |GHZ− +19π/24⟩ +0.75N 2 +1.15/N +III1 +|ψin +� +π +36χ +� +⟩ = |GHZ+ +0 ⟩ + |GHZ− +π/4⟩ + 1 +2 |GHZ+ +π/3⟩ − 1 +2 |GHZ+ +7π/12⟩ − 1 +3 |GHZ− +2π/3⟩ − 1 +3 |GHZ− +11π/12⟩ +0.70N 2 +1.20/N +IV1 +|ψin +� +π +48χ +� +⟩ = |GHZ+ +π/12⟩ + 9 +5 |GHZ− +7π/48⟩ + |GHZ+ +π/3⟩ + |GHZ+ +7π/12⟩ − 4 +5 |GHZ− +31π/48⟩ − |GHZ− +11π/6⟩ +0.67N 2 +1.22/N +V1 +|ψin +� +π +60χ +� +⟩ = 8 +5 |GHZ− +π/60⟩ + 3 +5 |GHZ+ +π/15⟩ + |GHZ− +13π/60⟩ + 8 +5 |GHZ+ +4π/15⟩ − |GHZ+ +7π/15⟩ +− |GHZ− +37π/60⟩ + 3 +5 |GHZ− +49π/60⟩ + |GHZ+ +13π/15⟩ +0.65N 2 +1.24/N +for φ = π/8. For large N, Eq. (26) is reduced down to +FQ ≈ 1 +2 +� +1 + 1 +√ +2 +� +N 2. +(27) +Eq. +(27) is the upper bound of QFI produced by the +cubic scheme in the case of even N. Inserting Eq. (27) +into Eq. (2) yields the best angular sensitivity achievable +by the cubic scheme, ∆β ≃ 1.08/N, which is very near +the HL. +In fact, the Heisenberg scaling of Eq. (27) originates +from the fact that the states of peaks I1,2 are in suppo- +sition of two GHZ states, i.e., +����ψin +� π +12χ +�� += +1 +√ +2 +� +|GHZ− +π/12⟩ + |GHZ+ +π/3⟩ +� +, (28) +where we have defined +��GHZ± +ϕ +� += +1 +√ +2 +����π +2 , ϕ +� +± +���π +2 , ϕ + π +�� +. +(29) +Obviously, the maximum-variance direction of the GHZ +states of Eq. (29) is ϕ, which from now on we call the +direction of a GHZ state. Corresponding to the states +(29), one may derive the variance of Sφ according to Eq. +(10), yielding +(∆Sφ)2 +|GHZ± +ϕ⟩ = 1 +4 +� +2S2 + S ++(2S2 − S) cos 2 (ϕ − φ) +� +. +(30) +This equation quantifies the amount of noise in the direc- +tion φ which deviates from the GHZ direction by an angle +ϕ − φ. It can maximized to (∆Sφ)2 +|GHZ± +ϕ ⟩ = S2 when the +two directions are exactly the same (φ = ϕ) and can be +minimized to (∆Sφ)2 +|GHZ± +ϕ ⟩ = S/2 when the two direc- +tions are orthogonal to each other (φ = ϕ − π/2). For +any 0 ≤ ϕ − φ ≤ π/2, we have S/2 ≤ (∆Sφ)2 +|GHZ± +ϕ ⟩ ≤ S2. +Therefore, Eq. (30) could also be regarded as the pro- +jection of the noise of a GHZ state in the GHZ direction +onto the φ direction. +Keeping this physical picture in mind, let us turn to +seek the maximum variance of the state (28). By pro- +jecting the noise of the two GHZ states, |GHZ− +π/12⟩ and + +3元/4 +FQ +/N2 +3元/4 +0元/2 +(a) +元/2 +0.8 +元/4 +π/4 +V0.4 +0.4 +0.6 +0.20.0 +0.05 +? +2 +5 +0 +2 +1 +3 +6 +1.0F +3 +4 +5 +6 +3元/4 +0.4 +3元/4 +0.9 +. N=61 +0 元/2 +元/2 +- N=60 +N +0.8F元/4 +.. +元/4 +III, +(c) +0.7 +V1 +0.2 +(e) +0.4 +0.4 +0.6 +. +0.5 +0.2 +0.20.0L! +.T/Xt +0.05 +0.05 +0.15 +0.20 +0.25 +0.30 0.0l +0.00 +0.10 +0 +? +1 +2 +4 +5 +6 +0 +1 +2 +3 +5 +6 +xt7 +|GHZ+ +π/3⟩, onto the φ direction, we get +(∆Sφ)2 +|ψin( +π +12χ)⟩ ≈ 1 +2 (∆Sφ)2���GHZ− +π/12 +� + 1 +2 (∆Sφ)2���GHZ+ +π/3 +� += 1 +4 +� +2S2 + S + 1 +2 +� +2S2 − S +� +× +� +cos 2 +� +φ − π +12 +� ++ cos 2 +� +φ − π +3 +��� += 1 +4 +� +2S2 + S + 1 +√ +2 +� +2S2 − S +� +×cos +� +2φ − 5π +12 +�� +, +(31) +where Sφ = Sx(0) cos φ + Sy(0) sin φ and, in the first +equality, we have neglected the nondiagonal terms, which +is reasonable when N is large (see Appendix C for more +details). Eq. (31) is maximized at φ = 5π/24 to give +(∆S5π/24)2 +|ψin(π/12χ)⟩ ≈ 1 +2 +� +1 + +1 +√ +2 +� +S2, which is exactly +the same as the maximum value of (∆Sφ)2 +|ψin(π/4χ)⟩. Such +result (that is, the QFI of I1,2 are equal) has also been +predicted by the numerical results in the previous section. +This confirms that the noise-projection method outlined +above provides a convenient way to derive the QFI of a +probe state in a superposition of arbitrary GHZ states, +that is, +|ψin⟩ = N +� +ϕ +Cϕ +��GHZ± +ϕ +� +, +(32) +where N is the normalization and Cϕ are the probability +amplitudes. Our task is to maximize the projected noises, +(∆Sφ)2 +|ψin⟩ ≈ +� +ϕ +|NCϕ|2 (∆Sφ)2 +|GHZ± +ϕ⟩ += 1 +4 +� +ϕ +|NCϕ|2 � +2S2 + S ++(2S2 − S) cos 2 (ϕ − φ) +� +, +(33) +over all values of φ. +We are now equipped to evaluate the QFI of the lower +peaks II1 − V1 [as shown in Fig. 2(a)]. In Figs. 2(b)-2(e) +we plot the Fourier coefficients distribution as well as +the quasiprobability distribution (QPD) [obtained from +the exact numerical evolution given by Eq. (18)] of each +peak state, showing that (i) the two results are consis- +tent with each other, and (ii) the appearance of large +QFI has always been accompanied with the generation +of macroscopic quantum-superposition state. Represent- +ing the peak states in terms of the GHZ states of Eq. +(29), we are able to show in table I the explicit forms for +each peak state, indicating that they have exactly the +same form as the states given in Eq. +(32). +One thus +can use Eq. (33) to approximately derive the amount of +QFI for each peak for the case of large N. As can be +seen from table I, the state of each peak could realize a +sensitivity near the HL. Interestingly, these peak states +appear regularly at a particular time tk = π/12kχ with +Ⅲ1 +Ⅰ1 +Ⅱ1 +Ⅳ1 +Ⅴ1 +(a) +(b) +x +y +z +probability +Ⅰ1 +Ι1 +≈ +  +12 +sin +S + +  +12 +sin +S + + +mx +FIG. 3. (Color online) (a) QFI produced by the cubic scheme +versus coupling strength χt for even (orange curve) and odd +(purple curve) number of spins. +Insert: +the QPD of the +state corresponds to the peak I1. (b) QFI produced by the +cubic scheme versus coupling strength χt. +The QFI peaks +labeled I1, ..., V1 are produced by |ψin(π/3(2k − 1)χ)⟩ with +k = 1, ..., 5, respectively. Insert: the Sx probability distribu- +tions of the peak GHZ state I1 (purple curve) and the initial +CSS state |π/2, 0⟩ (orange circle). We here take N = 201. +integer k = 1, ..., 5. In fact, the states of those not labeled +peaks on the left-hand side of the peak V1 [see Fig. 2(a)] +are also obtained at tk but with k ≥ 6. +It should be emphasized that the number of visible +peaks depends heavily on N. The larger the N, the more +QFI peaks it can be seen. This is because the number of +CSS components can be found in |ψin(π/12kχ)⟩ increases +with k. For large k but small N, these superposition CSS +components are overlapped with each other and become +indistinguishable. The distance between the CSS com- +ponents, however, can be increased by increasing N, as +their distance is proportional to N while the radius of a +CSS on the Bloch sphere is proportional to +√ +N. Once +all the CSS components of |ψin(π/12kχ)⟩ become distin- +guishable on the Bloch sphere, the kth QFI peak appears. + +T +T +儿 +儿N-60 +儿 +儿 +儿1.00 +0.03 +0.02 +0.01 +0.00 +-100 +-50 +0 +50 +1008 +In other words, the appearance of QFI peaks herald the +creation of macroscopic superposition states. As can be +seen from the insert in Fig. 2(a), QFI larger than 0.55N 2 +is still obtainable at t240 for N = 60. We emphasize that, +although the maximal QFI produced by the cubic scheme +is smaller than the maximal QFI of the OAT scheme (that +is, FQ = N 2 produced by the GHZ state at time π/2χ), +it has an advantage of short-preparation time, which, as +we will show below, makes the entanglement generation +robust against damping. +IV. +THE ENTANGLEMENT EVEN-ODD +EFFECT +An interesting feature of the cubic evolution of Eq. +(5) is an extreme sensitivity to the parity of the total +spin number N. As shown in Fig. 3(a), the evolutions +of QFI for N atoms vs N + 1 atoms are macroscopically +different: first, the maximum QFI of the cubic scheme +with odd-N spins can saturate to the HL [the peak I1 +in Fig. 3(a)], which is actually produced by the GHZ +state |GHZ− +7π/12⟩ at time t = 7π/12χ [see the inset of +Fig. 3(b)]; second, the state of each QFI peak [the peaks +I1 − V1 in Fig. 3(b)] for odd N appears at a different +but regular time tk = π/3(2k − 1)χ. These differences +originate from the fact that the evolution factor of the +odd-N spin system has a quite different periodic property, +exp +� +−iπ +n (m + 8n)3 +� += exp +� +−iπ +n m3 +� +. +(34) +Accordingly, the evolution factor can be reexpanded as +exp +� +−iπ +n m3 +� += +8n−1 +� +q=0 +f o +q exp +� +−iπq +n m +� +, +(35) +f o +q = 1 +8n +8n−1 +� +m=0 +exp +�iπq +n m +� +exp +� +−iπ +n m3 +� +. +(36) +Then, by using Eqs. (33) and (36) we are able to show in +the inset of Fig. 2(a) (the curve with circle) the amount +of QFI for each peak in case of large N. Despite the HL +QFI peak produced by the state |GHZ− +7π/12⟩, the spin +system with even N is superior over the one with odd N +in the production of multipartite entanglement. +A more striking feature is that the amount of QFI +at time t1 = π/3 changes dramatically with the par- +ity of N: for even N, the evolved spin state is a sep- +arable CSS with FQ = N, while, for odd N, the spin +state created is a maximally entangled GHZ state with +FQ = N 2. This entanglement even-odd effect is quite +different from the even-odd effect exhibited by the OAT +evolution UOAT = exp[−iχtS2 +z], which, at the instant t = +π/2χ, maps the initial CSS |π/2, 0⟩ to the N-dependent +GHZ state |ψin(π/2χ)⟩ = +1 +√ +2(eiπ/4 |π/2, −π(N − 1)/2⟩+ +e−iπ/4 |π/2, −π(N − 3)/2⟩) [42, 51], showing that the ori- +entation of the created GHZ state is sensitivity to the +parity of total spin number N. +FIG. 4. (Color online) (a) QFI produced by the interaction of +Eq. (37) versus coupling strength χt for N = 20. Insert (i): +schematic depiction of the evolution induced by the hybrid +dynamics on the Bloch sphere, with green flow lines denoting +S2 +z-dependent rotation of the QPD around the z axis caused +by the cubic part in Eq. (37) and orange flow lines repre- +senting twisting of the QPD induced by the quadratic part in +Eq. (37). Insert (ii): we plot the means ⟨Txy⟩ and ⟨Sz⟩ as a +function of χt. (b)-(e) are the QPD of the peak states marked +with α, ..., δ in (a). +The determination of the spin number in a realistic +quantum system is a critical first step toward the re- +alization of quantum metrology as well as quantum in- +formation processing [52–56]. Especially in the context +of spin-spin entanglement generation [57, 58], the spin +system will evolve into a highly entangled pure state or +a separable mixed state depending on the parity of N +[59–61]. It is thus of great importance to have the abil- +ity to determine the parity of N before performing the +protocols. Our entanglement even-odd effect might po- +tentially be used to detect the parity of the total spin +number N with a resolution at the single-spin level. The +parity detection proceeds as follows: the spin state is ini- +tially prepared in the CSS |π/2, 0⟩, and then is subjected +to the evolution described by Eq. (5) for a time duration +t = π/3χ. Finally, measuring Sx a particular outcome ˜Sx +is obtained. As can be seen from the inset of Fig. 3(b), +for even N, we have ˜Sx = S, while, for odd N, there will +be a large probability of finding ˜Sx around ±S sin(π/12) +(especially for the case of large N). Therefore, the mea- +surement of the angular momentum operator Sx provides +a direct way to estimate the parity of N. + +0 +(b) +(c) +03 +4 +0 +2. +3. +00.8 +(11) + π/2 the QPD is stretched along a direc- +tion determined by ϵ, in close analogy with the two-axis +twisting dynamics described by S2 +y −S2 +z [18]. As a result, +the QPD continuously spreads out and gradually arrives +at the position (π/2, π) on the Bloch sphere, forming a +Schr¨odinger-cat-like state α that has a quite large amount +of QFI, as shown in Fig. 4(a) and (b). This state, how- +ever, is an imperfect GHZ state as the two superposition +components are still overlap with each other. For this +state, it is not difficult to find that its expectation ⟨Sz⟩ +is negative, which evolves according to +d ⟨Sz⟩ /dt = −χ ⟨Txy⟩ , +(38) +indicating that the expectation ⟨Sz⟩ is determined +by +the +expectation +value +of +the +tensor +operator +Txy = SySx + SxSy. Once the QPD surrounds the Bloch +sphere, the sign of ⟨Txy⟩ is reversed at time t = t0, as can +be seen from the insert (ii) of Fig. 4(a). Then, ⟨Sz⟩ starts + +(b) +(a 4(0.9 +35 +V2Fo/N +0.8 +0.5 +Q +1.0 +30 +0.7 +0.0E.0.9 +N +0.6 +10 +20 +25 +30 +35 +40 +N25 +15 +0.8 +(c) 1.0b20 +0.7 +0.9 +0.615 +0.7 +15 +20 +25 +35 + 400.0 +25 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +20 +1.6 +15 +30 +35 +40 +N +E10 +to increase, which gradually eliminates the QPD between +the two superposition components, creating an unequally +weighted GHZ state β [see Fig. 4(c)]. Further evolution of +the dynamics of Eq. (37) will evenly distribute the QPD +to the antipodal CSSs [see Fig. 4(d)] and finally produce +a near-perfect GHZ state δ [see Fig. 4(e)]. It should be +stressed that, given an atom number N, the maximum +achievable QFI of the GHZ-like state δ depends heavily +on the relative coupling strength ϵ. Taking N = 20 as an +example, the maximum QFI FQ/N 2 = 0.99 is obtained +when ϵ = 0.29 at the fixed time tf = 0.65/χ. Obviously, +in contrast to the quadratic OAT interaction, the CQA +type interaction of Eq. (37) can speed up the procedure +of GHZ-state generation since the preparation time sat- +isfies tf < t π +2 . +In Fig. 5(a), we also plot the maximally achievable +QFI for different N as a function of the relative coupling +strength ϵ. We find that large QFI are (i) mainly concen- +trated within the regime of small ϵ and (ii) more easily +achieved when the atom number N is large. Fig. 5(b) +shows the maximal achievable QFI of the proposed pro- +tocol for different choices of N. QFI as large as 0.99N 2 +is obtainable, which yields an angular sensitivity ∆β ≃ +1.005/N. The insert of Fig. 5(b) indicates that the accel- +eration rate is outstanding in the small N regime, while +it starts to oscillate when N increases, which, however, +can be suppressed by slightly sacrificing the amount of +achievable QFI as shown in Fig. 5(c). Taking N = 30 as +an example, we have the preparation time tf = 0.28/χ +(≈ 0.18t π +2 ) while the achievable QFI is still as high as +0.97N 2, which corresponds to a GHZ state that has a +slight flaw as shown in the insert of Fig. 5(a). +VI. +IMPLEMENTATIONS +Next, we show how to implement the cubic interaction +in two-level atomic system. +A. +Cubic interaction induced by quadratic +interactions +In contrast to the cubic interaction, the quadratic +interactions are much easier to implement in realistic +atomic systems. +Among them, the most widely stud- +ied one is the OAT interaction, which, as mentioned +above, has been experimentally implemented in various +atomic systems [20, 21, 63–65]. +We next consider the +realization of cubic interaction by repeated application +of the OAT interactions. +Suppose that one is able to +freely apply the evolutions Uk,q(δ) = exp +� +iδSk +q +� +with +q ∈ {x, y, z} to atomic system, for k = 1, 2 and all δ ∈ R. +The evolutions U1,q(δ) denote a rotation of the collec- +tive spin around the q axis by a phase δ, which can be +easily realized by applying either a RF magnetic field +[66] or a circularly polarized optical pulse [67] to atoms +along the q direction. For the evolutions U2,q(δ), it is +only necessary to be able to perform the OAT evolu- +tion in a certain direction, e.g., U2,z(δ). The rest OAT +evolutions can be directly constructed from U1,k(δ) and +U2,z(δ), such as U2,x(δ) = U1,y(−π/2)U2,z(δ)U1,y(π/2). +With these evolutions, other quadratic evolutions can +then be constructed approximately by repeated applica- +tion of Uk,q(δ) to atoms, e.g., +E +� +S2 +x, Sy +� += U2,x(δ)U1,y(δ)U2,x(−δ)U1,y(−δ) += eiδS2 +xeiδSye−iδS2 +xe−iδSy += e[S2 +x,Sy]δ2 + O +� +δ3� +≈ eiδ2(SxSz+SzSx), +(39) +in the limit δ → 0. +Eq. +(39) is exactly the two-axis +twisting evolution presented in Ref. [18]. One can con- +clude from Eq. (39) that the result of the transformation +E(A, B) is the same as if we have applied the interaction +i[A, B] to the atomic system for time δ2 [43]. With this +in mind, let us consider the following transformations +UC = E +� +Sx, S2 +y +� +E +� +S2 +x, Sy +� +E +� +S2 +y, Sx +� +E +� +Sy, S2 +x +� +≈ eiδ2(SySz+SzSy)eiδ2(SxSz+SzSx) +×e−iδ2(SySz+SzSy)e−iδ2(SxSz+SzSx) += E (SySz + SzSy, SxSz + SzSx) +≈ e−i8δ4S3 +z+i(4S2+4S−1)Sz, +(40) +where the first term is the desired cubic term and the +second term linear in Sz generates a precession of the +collective spin around the z axis. To isolate the cubic +term it is convenient to apply a reverse-precession trans- +formation to atoms, resulting in the overall effect +U1,z +� +1 − 4S − 4S2� +UC ≈ e−i8δ4S3 +z, +(41) +which is exactly the cubic evolution Uz given in Eq. (5) +with χt = 8δ4. We thus have successfully developed a +general method, which should be widely applicable to a +variety of spin systems capable of performing OAT evo- +lution. Finally, it should be pointed out that such a re- +sult has no counterpart in bosonic system, where linear +together with quadratic Hamiltonians are unable to con- +struct a Hamiltonian of higher order [43]. +B. +Cubic interaction induced by atoms-light +interactions +The above cubic method we developed relies on multi- +step quantum operations, which might pose a technologi- +cal challenge to a realistic implementation. We next show +that it is possible to realize the cubic evolution in just one +step by utilizing properly designed light-mediated inter- +actions. We consider an ensemble of N two-level atoms +described above trapped inside a one-sided optical cavity +[see Fig. 6(a)]. The cavity field c couples the two states +|↓⟩ and |↑⟩ separated in energy by ℏωa with a detuning + +11 +x +0t +− +0 +0t +(a) +t +∆ +(b) +ω +κ +p +ω +Atoms-cavity system +y +z +c +FIG. 6. (Color online) Setup for realizing atomic cubic evolu- +tion. (a) Atoms with an excited state |↑⟩ and a ground state +|↓⟩ couple off-resonantly to an one-sided optical cavity. At +time t = −t0 a light pulse is sent to interact with the cavity +mode. After the light pulse is completely reflected by the cav- +ity at t = t0, a cubic-nonlinear transformation is successfully +applied to the atoms. (b) The central frequency ωp of the +incident optical pulse is detuned from cavity resonance, with +a detuning κ/2. +∆ in the Tavis-Cummings model [68], which can be de- +scribed by the Hamiltonian +Hcav = ωaSz + (ωa − ∆)c†c + gc†S− + H.c., +(42) +where g is the coupling constant. In the interaction pic- +ture with respect to ωaSz+(ωa − ∆)c†c, the Hamiltonian +of Eq. (42) can be expressed as: +˜Hcav = gc†S−e−i∆t + H.c.. +(43) +We now consider the case that ∆ is much larger com- +pared to g, to the linewidths of the cavity κ, and to the +linewidths of atom Γ. Besides, we assume the intracavity +photon number ⟨c†c⟩ is very small. As a result, the pop- +ulation of the excited state is small, which enables us to +adiabatically eliminate the state |↑⟩ to yield an effective +Hamiltonian +˜Heff +cav = Ω +� +2c†cSz + Sz − S2 +z +� +(44) +with Ω = g2/∆ [69], where the first term denotes the +ac Stark shift while the rest terms represent the cavity- +mode-induced backaction of atoms onto themselves. The +last two terms linear or quadratic in Sz will cause a pre- +cession or a shearing of the atomic pseudospin around +the z axis, while the first term generates entanglement +between the cavity mode and the collective spin, which +in the past is an unwelcome term and usually decoupled +to realize quadratic unitary transformation of atoms [70]. +Instead, we next show that how it can be engineered to +generate high-order nonlinear spin-spin dynamics. In the +following analysis, we closely follow the procedure out- +lined in Ref. [71]. +We assume that this cavity-atoms system is subjected +to interact with an external field, which can be conve- +niently described by the input-output formalism [72], re- +sulting in the effective Hamiltonian [71]: +H = (ωa + Ω) Sz + (ωa − ∆) c†c + +� +dωωb† +ωbω − ΩS2 +z ++2ΩSzc†c + i +� κ +2π +� +dω +� +b† +ωc − c†bω +� +, +(45) +where the third term denotes the energy of the exter- +nal field with the annihilation operators bω satisfying +[bω, b† +ω′] = δ(ω−ω′), and the last term represents the cou- +pling between the intracavity and external fields through +the partially transmissive input mirror of the cavity, and +κ stands for the cavity decay rate. Following the Fano’s +procedure [73], this Hamiltonian can be exactly diago- +nalized to give +H = (ωa + Ω) Sz − ΩS2 +z + +� +dωωa† +ωaω, +(46) +where the dressed annihilation operators are +aω = αωc + +� +dω′βω(ω′)bω′ +(47) +with +αω = i sin (∆ω) / +� +πκ/2, +(48) +βω(ω′) = 1 +π P sin (∆ω) +ω − ω′ − cos (∆ω) δ(ω − ω′), +(49) +∆ω = − arctan +κ/2 +ω + ∆ − ωa − 2ΩSz +, +(50) +where P denotes the principal part. The dressed oper- +ators {aω} obey the commutation relations [aω, a† +ω′] = +δ(ω −ω′) and describe a set of decoupled harmonic oscil- +lators. One can realize the photon excitations of aω by +increasing the field amplitudes of either the intracavity +mode or the outside continuum. A one-photon Fock state +of aω can then be created by acting a† +ω on the vacuum +states, yielding +|1ω⟩ = a† +ω |0c⟩ ⊗ |0bω⟩ += α∗ +ω |1c⟩ |0bω⟩ + +� +dω′β∗ +ω(ω′) +��1bω′ +� +|0c⟩ , (51) +which, for a given eigenstate |m⟩ of Sz, is the eigenstate of +H, that is, H |1ω⟩⊗|m⟩ = [(ωa+Ω)m−Ω2m2+ω] |1ω⟩ |m⟩. +Now, with the help of Eq. (46) we are able to describe +the interaction between a light pulse and the atoms- +cavity system. For simplicity, we consider the case that +a single-photon pulse is sent to the cavity at the initial +time t = −t0 that is far in the past, which has the form +|1b⟩ = +� +dωeiωt0B(ω)|1bω⟩, where B(ω) is the normalized +probability amplitude as a function of frequency, and the +cavity field starts in the vacuum, leading to the initial +state of the fields, |ψf⟩b = |1b⟩ ⊗ |0c⟩. +One thus can + +12 +express the initial state of the system as +|Ψ (−t0)⟩ = |ψa⟩ ⊗ |ψf⟩b += +S +� +m=−S +Cm +� +dω +� +dω′βω(ω′) +×eiω′t0B(ω′) |1ω⟩ |m⟩ += − +S +� +m=−S +Cm +� +dωei(ωt0+∆ω)B(ω) |1ω⟩ |m⟩, +(52) +where Cm is the normalized probability amplitude of an +arbitrary atomic state |ψa⟩ and in the second equality +we also have reexpressed the initial state |ψf⟩b in terms +of the eigenstates of the total field by using Eq. (51). +The phase ∆ω in the last equality can be understood as +a phase lag of the cavity field response to the drive of the +outside continuum photon at frequency ω [71]. The ini- +tial state (52), under the action of the unitary evolution +generated by Eq. (46) for time 2t0 that is long enough +to allow the light pulse to be completely reflected from +the cavity, transforms to +|Ψ (t0)⟩ = exp (−iH2t0) |Ψ (−t0)⟩ += +S +� +m=−S +Cme−i2t0[(ωa+Ω)m−Ωm2] +× +� +dωei(2∆ω−ωt0)B(ω) |1bω⟩ |m⟩, +(53) +where the field states have been transformed back into +|1bω⟩. +Obviously, the output pulse get entangled with +atoms as the phase factor ∆w in Eq. (53) contains in- +formation about the atoms. However, if the input light +pulse is near-monochromatic with frequency ωp and its +bandwidth is much less than the linewidth of cavity +[see Fig. 6(b)], then one may make the approximation +∆w → ∆wp [71], and the output state (53) becomes +|Ψ (t0)⟩ = +S +� +m=−S +Cme−2i[(ωa+Ω)t0m−Ωt0m2−∆ωp] |m⟩ +⊗ +� +dωe−iωt0B(ω) |1bω⟩, +(54) +which shows the output pulse is completely disentangled +from the atoms. +Although the interaction process has +nothing to do with the incident photon, it imposes an +additional m-dependent phase shift 2∆ωp to atoms, re- +sulting in the transformation of atomic state by the uni- +tary operator +U = e−2i[(ωa+Ω)t0Sz−Ωt0S2 +z−∆ωp], +(55) +indicating that the atomic subspace has experienced a +linear- and quadratic-Sz (OAT) interactions, while the +higher order interactions of interest are encoded in ∆ωp. +Next, we assume that the incident photon is off-resonance +with the cavity mode with frequency ωp = ωa − ∆ + κ/2 +and the cavity resonance is not shifted too much by atoms +[74], such that κ0 = Ω/κ ≪ 1/N, the operator ∆ωp can +then be expanded up to third order in the parameter κ0, +generating +U = e−2i{[(ωa+Ω)t0+2κ0]Sz+(4κ2 +0−Ωt0)S2 +z+ 16 +3 κ3 +0S3 +z} += e−2i{[(ωa+Ω)t0+2κ0]Sz+ 16 +3 κ3 +0S3 +z}, +(56) +where in the second equality we have set t0 = 4κ2 +0/Ω. +Analogously, the cubic term can be isolated by applying +a linear counter-rotating transformation around z axis to +atoms, finally arriving at +U1 = U1,z [2 (ωa + Ω) t0 + 4κ0] U = e−i 32 +3 κ3 +0S3 +z. (57) +Thus, we have successfully realized the cubic evolution +of the atomic state by simply injecting a single-photon +state into an atoms-cavity system. +A major concern of our proposed scheme might be the +coupling constant κ0, as it turns to be extremely weak +when N is large. One direct way to enhance the coupling +strength is to use more incident single photons. +Sup- +pose that we have n single-photon wave packets described +above, and they are sent one by one to interact with the +cavity mode at fixed interval 2t0. Each photon induces a +U1 transformation to atoms, then, after the nth interac- +tion, the atomic state evolves as if it has been applied a +unitary transformation +Un = e−iµnS3 +z +(58) +with µn = 32nκ3 +0/3 = n(ηΓ/∆)3/6, where η = 4g2/(κΓ) +is the single-atom cooperativity. Eq. (58) indicates that +the total coupling strength is now n times lager than the +coupling strength created by a single photon. Another +convenient way is to utilize a n-Fock state. If the incident +wave packet contains exactly n photons, its interaction +with cavity system would also lead to the cubic evolution +(58) [71]. As a specific example, for an atomic ensemble +with atom number N ∼ 103, if we take η = 0.04, Γ = 10g, +and ∆ = 150g, a coupling constant α ∼ 3 is obtainable +with the choice n ∼ 106, which enables the production of +QFI as high as FQ ∼ N 2/2 [according to Eq. (16)] with +the interaction time 2t0 ∼1 ns for g/(2π) ∼ 1 Mhz. +VII. +THE EFFECT OF DAMPING +Up till now we have only considered the perfect evo- +lution of the spin state. In realistic systems, however, +there are inevitable noise effect that will cause damping +of the spin state. Here, we mainly consider two types +of damping: one arises because of external field fluctua- +tions [75–77], which induces collective dephasing of the +atoms, and its influence on the evolution of the atomic +state ρ can be described by the dissipative superoperator +D[Sz]ρ, where D[O]ρ = 2OρO† − {O†O, ρ} is the stan- +dard Lindblad dissipative superoperator; another one is + +13 +FIG. 7. (Color online) Performance of the cubic scheme (solid curves) and the OAT scheme (dot-dashed curves) in the presence +of damping (in units of χ) for N = 20. QFI created by the two schemes versus interaction time (in units of 1/χ), in the presence +of either single-spin decay (a) or collective dephasing (b): no loss (diamond), 5% loss (circle), 10% loss (triangle). (c) QFI +created by the two schemes versus interaction time, in the presence of both decays: no loss (diamond), γ = Γ = 5% (circle), +γ = Γ = 10% (triangle). (d) QFI of the states of the peaks marked I1 and α in (c) as a function of γ(= Γ). Insert: QFI of the +GHZ state created by the cubic scheme (line with triangles) and the OAT scheme (line with diamonds) as a function of γ(= Γ) +for N = 21. +single-spin decay, which is normally caused by the spon- +taneous emission of photons by the individual atoms into +free space [78, 79] and can be described by the dissipative +superoperator � +k D[σk +−]ρ, where σk +− is the pseudo-spin +lowering operator for the kth atom. With these damp- +ing, the master equation for the atomic state ρ under the +interaction of H can be expressed as +˙ρ = −i[H, ρ] + ΓD[Sz]ˆρ + γ +� +k +D +� +σk +− +� +ρ, +(59) +where γ is the decay rate of the excited state |↑⟩ and Γ +is the collective dephasing rate. +The numerical solutions of Eq. +(59) are shown in +Fig. 7 for both H = χS2 +z and H = χS3 +z. In Fig. 7(a) +we plot the QFI evolves with time t (in units of 1/χ) +for Γ = 0 (in units of χ) and various values of γ (in +units of χ). In the weak coupling regime (region I), the +dampings have only small impact on the processes of en- +tanglement creation for both protocols. +Thus, in this +region our cubic scheme can still maintain its speed ad- +vantage in QFI creation. As the QFI increases (region +II), the cubic scheme is more susceptible to single-spin +decay. However, benefiting from the advantage of quan- +tity, the QFI of the cubic scheme before time t = π/12 are +still larger the QFI created by the ideal OAT evolution, +even when γ = 0.1. In the region labelled III, the noise +effect almost has the same impact on the two protocols +in the entanglement generation. +Figure 7(b) plots the +QFI in its dependence on t for γ = 0 and various values +of Γ, which indicates that our cubic method is consid- +erably more robust against the collective dephasing as +compared to the OAT method. In the presence of both +single- and collective-spin decay, we also plot the achiev- +able QFI versus t for both scheme in Fig. 7(c), showing +that the Heisenberg-limited QFI peak α (produced by +the OAT evolution at time t π +2 ) decreases rapidly with +the atomic decay. Although, in the ideal case, the peak +I1 (produced by the cubic evolution at time t π +12 ) has the + +(a) 0.8 +II +III0.40.2 +0.2 +用 +2 +12 +0.0 +0.0 +0. +0.2 +0.4 +0.5 +0. +0.2 +0.5t(units of 1/ x) +t (units of 1/ x) +(c) 1.0F +(d) 1.0F +0.8 +0.80:0 +0.6 +0.00....04. +.0.08...0.12 +0.6 +0.40.2 +0.2 +0.0 +0.0E0 +t(units of 1/ x) +0.1L +(=F)(units of x)14 +disadvantage of less QFI (in contrast to the peak α), it +is much more robust against decoherence [as also com- +pared in Fig. 7(d)], which makes it more attractive in +a realistic implementation. For odd N, a comparison of +the QFI of the GHZ state created by both protocols [see +the insert of Fig. 7(d)] indicates that the GHZ created +by our cubic scheme is less susceptible to decoherence. +VIII. +CONCLUSIONS +In this paper, we have proposed to entangle individ- +ual spins using the cubic nonlinear interaction. We find +that, although the multipartite entangled states created +by the cubic scheme have no spin squeezing, they are use- +ful for quantum metrology. In contrast to the traditional +OAT scheme, we have shown that the cubic scheme of- +fers several advantages. First, it can produce QFI much +more rapidly in the weak coupling regime. The larger +the total spin number N, the larger the acceleration rate, +which makes it particularly attractive for entanglement +generation in large-number spin system. Second, the cu- +bic scheme enables the preparation of a broad variety +of new-type macroscopic superposition states in a much +more short time. We showed that these states exhibit an +outstanding performance in the generation of large QFI, +which provide the possibility to realize near-HL phase +sensitivity. +Third, the cubic scheme is still capable of +producing much more spin-spin entanglement even in the +presence of large decays. +We also discovered a new even-odd effect of the cubic +evolution, that is, the entanglement created by the cubic +evolution is extremely macroscopic sensitive to the parity +of the total spin number N. We showed that such en- +tanglement even-odd effect might be exploited to design +new type of sensor modality, enabling the determination +of the parity of the spin number in a spin system at the +single-spin level. We also find a new mechanism to gen- +erate high-fidelity GHZ states. By using a hybrid NSS +interaction—CQA type of nonlinear interaction, one may +speed up the preparation of GHZ states as compared to +the methods based solely upon OAT interaction. +We also have presented two approaches to realize the +cubic evolution of the spin system. One is based on the +lower-order interactions. We showed that the cubic evo- +lution can be approximately constructed by repeatedly +using linear- and quadratic-nonlinear dynamics. +This +method is quite general and is widely applicable to a +variety of spin systems. Another one relies on the light- +mediated interactions. We found that, by suitably en- +gineering the light-mediated interactions, one is able to +realize the cubic NSS interaction among atoms in just +one step. +Our study provides a new angle in utilizing unitary +transformation to produce useful multipartite entangle- +ment among spins. Although the cubic (third-order) non- +linearity in a realistic spin system is normally weak, it +would greatly enrich the way of manipulating the spin +system, just like the Kerr nonlinearity in optical system +[80, 81]. We thus believe that the application of the cu- +bic interaction will not be restricted to the field of en- +tanglement generation. For instance, it has been shown +that the bosonized spin system is an excellent platform +for implementing continuous-variables quantum informa- +tion processing [10]. The cubic evolution (called the cu- +bic phase gate) then is a particularly convenient candi- +date for realizing non-Gaussian operations [82]. There- +fore, our proposed schemes would also benefit the field of +continuous-variables quantum computation. +ACKNOWLEDGMENTS +We thank Yanhong Xiao for helpful discussions. This +work was supported by the Natural Science Founda- +tion of China (Grants No. 22273067), the Natural Sci- +ence Foundation of Zhejiang province, China (Grant No. +LQ23A040001), and the Department of Education of +Zhejiang Province, China (Grant No. Y202146469). +Appendix A: Converting the binomial distribution +into the Gaussian Distribution +In this Appendix, we give the details of the derivation +of Eq. (12). +Defining M = 2S − 1, Eq. (12) can be +rewritten as +PM(m) = +M! +(M − m)!m!2−M. +(A1) +As shown in the main text, this binomial distribution +has the mean ⟨m⟩ = S and standard deviation ∆m ∼ +√ +S. We thus can use the Stirling’s approximation x! ≈ +√ +2πx(x/e)x for large S, obtaining +PM(m) = +√ +2πMM M2−M +� +2π (M − m)(M − m)M−m√ +2πmM m += +1 +√ +2Mπ +� +2 − 2 m +M +�−M� +m/M +1 − m/M +�−m +× +� m +M − m2 +M 2 +�−1/2 +. +(A2) +Taking the logarithm of Eq. (A2) leads to +ln PM(m) = ln +1 +√ +2Mπ +− M ln +� +2 − 2 m +M +� +−m ln +� +m/M +1 − m/M +� +− 1 +2 ln +� m +M − m2 +M 2 +� +. +(A3) + +15 +Using Taylor expansion around ⟨m⟩ = M/2, we arrive at +ln PM(m) ≃ ln +2 +√ +2Mπ ++ 2 +� +m − M +2 +� ++ 2 +M +� +m − M +2 +�2 +− 2 +� +m − M +2 +� +− 4 +M +� +m − M +2 +�2 ++ +2 +M 2 +� +m − M +2 +�2 +≃ ln +2 +√ +2Mπ +− 2 +M +� +m − M +2 +�2 +, +(A4) +where we have kept terms to second order and omitted +the last term in the first equality as it is much more +smaller when M is large. Finally, we exponentiate Eq. +(A4) to get +PM(m) = +2 +√ +2Mπ +exp +� +−2 +� +m − M +2 +�2 +M +� +≃ +1 +√ +Sπ +exp +� +−(m − S)2 +S +� +, +(A5) +where the last equality is valid for large S. +Appendix B: Calculation of the expectation values +of the evolved spin operators +To calculate the means and variances of Eqs. +(14), +one needs to calculate the expectation values ⟨S+ (t)⟩, +� +S2 ++ (t) +� +, and ⟨S+ (t) S− (t)⟩. Along the same line as cal- +culating ⟨S+ (t)⟩ given in the Eq. (11), one may derive +the quadratic expectation value +� +S2 ++ (t) +� += 2−2S +2S +� +k=0 +2S +� +l=0 +� +(2S)! +(2S − k)!k! +� +(2S)! +(2S − l)!l! +× ⟨S, S − k| +� +S+eiµ(S2 +z+Sz+1/3)�2 +|S, S − l⟩ += 2−2S +2S +� +l=2 +(2S)! +(2S − l)!(l − 2)!e2iµ[(S−l+1)2+ 1 +3] += S +� +S − 1 +2 +� 2S−2 +� +m=0 +P2S−2 (m) +× e2iµ[(S−m)2−2(S−m)+ 4 +3], +(B1) +where we set m = l −2. Again, the binomial distribution +P2S−2 (m) can be transformed into the Gaussian distri- +bution to give +� +S2 ++ (t) +� +≃ S (S − 1/2) +√ +πS +√ +S +� +k=− +√ +S +e−(1−2iµS)k2−4iµ +√ +Sk+ 8 +3 iµ +≃ S (S − 1/2) +√π +e +8 +3 iµ +× +� +∞ +−∞ +e−(1−2iµS)k2−4iµ +√ +Skdk +≃ S (S − 1/2) +√1 − 2iµS , +(B2) +and its complex conjugate +� +S2 +− (t) +� +≃ S (S − 1/2) +√1 + 2iµS . +(B3) +For the mean ⟨S+ (t) S− (t)⟩, it can be directly calculated +without approximation as +⟨S+ (t) S− (t)⟩ = ⟨S− (t) S+ (t)⟩ += 2−2S +2S +� +k=0 +2S +� +l=0 +� +(2S)! +(2S − k)!k! +� +(2S)! +(2S − l)!l! +× ⟨S, S − k| S+S− |S, S − l⟩ += 2−2S +2S +� +l=0 +(2S)! +(2S − l − 1)!l! (l + 1) += S2 + S +2 . +(B4) +Using Eqs. (B2)-(B4) and (13), we are able to calculate +the means and variances of Eqs. (14). Here, we derive +the mean ⟨Sx (t)⟩ as an example. According to Eq. (13) +we have +⟨Sx (t)⟩ = 1 +2 (⟨S+ (t)⟩ + ⟨S− (t)⟩) += S +2 +�√1 + iµS+√1 − iµS +� +� +1+µ2S2 +. +(B5) +Next, defining α0 = arccos(1/ +� +1+µ2S2), Eq. (B5) can +be reexpressed as +⟨Sx (t)⟩ = +S +4� +1+µ2S2 cos α0 +2 += S +� +1 + (1+µ2S2)1/2 +2(1+µ2S2) += S +� +α1 (α1 + 1) /2, +(B6) +where α1 = 1/ +� +1+µ2S2. +Appendix C: Calculation of the expectation values +of the nondiagonal elements +In this Appendix, we prove that it is reasonable to +ignore the nondiagonal terms in the calculation of Eq. + +16 +(31). +Suppose that | π +2 , φ1⟩ and | π +2 , φ2⟩ with φ1 ̸= φ2 +are two CSS components encoded in the spin states of +Eq. (32). Corresponding to this two CSSs, one needs +to evaluate the nondiagonal values ⟨ π +2 , φ1| Sφ | π +2 , φ2⟩ and +⟨ π +2 , φ1| S2 +φ | π +2 , φ2⟩ when deriving the projection noises of +Eq. (33). Let us first derive +�π +2 , φ1 +��� Sφ +���π +2 , φ2 +� += 1 +2e−iφ �π +2 , φ1 +��� S+ +���π +2 , φ2 +� ++ 1 +2eiφ �π +2 , φ1 +��� S− +���π +2 , φ2 +� += S +2 cos +�∆φ +2 +�2S−1 +cos +�∆φ +2 ++ φ1 − φ +� +eiS∆φ, +(C1) +where we have defined the new parameter ∆φ=φ2 − φ1. +Obviously, for a nonzero ∆φ, the value of Eq. (C1) decays +exponentially with S, resulting in ⟨ π +2 , φ1| Sφ | π +2 , φ2⟩ ≈ 0 +for the case of large S. +We now turn to evaluate the second moments +�π +2 , φ1 +��� S2 +φ +���π +2 , φ2 +� += 1 +4e−2iφ �π +2 , φ1 +��� S2 ++ +���π +2 , φ2 +� ++ 1 +4e2iφ �π +2 , φ1 +��� S2 +− +���π +2 , φ2 +� ++1 +4 +�π +2 , φ1 +��� S+S− +���π +2 , φ2 +� ++ 1 +4 +�π +2 , φ1 +��� S−S+ +���π +2 , φ2 +� +. +(C2) +To derive Eq. 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Wright, P. L. +McMahon, +and H. Mabuchi, Phys. Rev. Lett. 124, +240503 (2020). + diff --git a/XtE3T4oBgHgl3EQfcAoE/content/tmp_files/load_file.txt b/XtE3T4oBgHgl3EQfcAoE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..06e3447645e7f71e7ca37fc27f764912f99e5ba4 --- /dev/null +++ b/XtE3T4oBgHgl3EQfcAoE/content/tmp_files/load_file.txt @@ -0,0 +1,1500 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf,len=1499 +page_content='Entangling spins using cubic nonlinear dynamics Lingxia Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Yani Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Yujing Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Zhiqi Yan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Lei Xie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Gang Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='2 Jinmin Fan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Di Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Yiling Song,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1 Linli He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ∗ Wei Xiong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' † and Mingfeng Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ‡ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Wenzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Zhejiang 325035,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' China 2School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' China Entangled states with a large number of N atomic spins are a key ingredient for quantum infor- mation processing and quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Nowadays, the preparation of such states has mainly relied on the quadratic nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Here, we investigate the preparation of spin-spin mul- tipartite entanglement, witnessed by quantum Fisher information, by using the cubic nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We find that, in the regime of weak coupling, the cubic scheme can greatly speed up the rate of entanglement generation as compared to the quadratic scheme (about N times faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In the strong coupling regime, the cubic nonlinear dynamics enables the periodic in time gener- ation of a broad variety of new-type macroscopic superposition states, which allow us to realize near-Heisenberg-limit phase sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In addition, we also reveal an interesting feature that the amount of entanglement generated by the cubic scheme has a macroscopic sensitivity to the parity of N, which has no counterpart in quadratic nonlinear dynamics and can be exploited for sensing the parity of N at the single-spin level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We also propose a new approach for a fast and high-fidelity generation of maximally entangled Greenberger-Horne-Zeilinger (GHZ) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' By using an alterna- tive cubic-quadratic-admixture type of nonlinear interaction, we show that one may accelerate the procedure of GHZ-state generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The realization of the cubic nonlinear dynamics is also consid- ered, showing that the cubic nonlinear dynamics can be realized by either repeatedly using linear- and quadratic-nonlinear dynamics or utilizing light-mediated interactions in just one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Finally, by taking realistic imperfections into account, we find that the cubic scheme is sensitivity to the single-spin decay in the strong coupling regime, while is robust against the collective dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Our proposed schemes offer potential possibilities for realizing high-sensitivity metrology in a variety of platforms, including trapped ions and cold or warm atomic ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' INTRODUCTION The generation of entanglement between a large num- ber of spins is an extremely important subject in preci- sion metrology and quantum science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In quantum metrol- ogy [1], highly entangled spin states enable precision metrology beyond the standard quantum limit (SQL) [2], even approaching the Heisenberg limit (HL) [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In the field of quantum information [5, 6], entangled spin ensem- bles are not only recognized as key resources for quantum communication [7–9] but also considered as a promising platform for quantum computation [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To date, a variety of approaches have been developed for producing entangled states of spin ensemble, which can be classified into two main categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' One is based on the projection measurement (such as quantum nonde- molition measurement) [14–17]: first, entanglement is es- tablished between the spin system and an auxiliary quan- tum system (usually a light field), and then, a measure- ment of the auxiliary quantum system will project the spin state into a multipartite entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Another one has relied on unitary evolution of an initial product spin state under a nonlinear spin-spin (NSS) interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Among these NSS interactions, the most widely studied one is possible the one-axis-twisting (OAT) interaction ∗ linlihe@wzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='cn † xiongweiphys@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='com ‡ mfwang@wzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='cn [18–23], which, as shown by Ueda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' [18], can pro- duce pairwise spin-spin entanglement that is the origin of spin squeezing [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Spin squeezing is probably the most sought-after multipartite entangled resource in the field of quantum metrology, as the phase estimation based on spin squeezing is comparatively easy to imple- ment in realistic experiments [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Up till now, most studies of NSS interaction have mainly been concentrated on how to efficiently create highly squeezed spin states, such as two-axis-twisting interaction [26–30], twist-and- turn interaction [27, 31–33], and twisting-tensor interac- tion [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' However, squeezed spin states are only one category of multipartite entangled states that can bene- fit the quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Other categories of entangled states, although having no spin-squeezing property, may also be useful for quantum metrology and sensing, such as the GHZ state enabling the phase sensitivity reaching the HL [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In fact, apart from the spin squeezing, the quantum Fisher information (QFI) provides a more gen- eral and profound way to estimate whether a given spin state is useful or not for quantum metrology [36–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The larger the QFI of the entangled state, the more useful the state might be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Therefore, it is of particulary neces- sary to reconsider the entanglement generation induced by the NSS interactions from the perspective of QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Ac- cordingly, discovering and devising new NSS-interaction schemes that can rapidly and efficiently generate large QFI is of vital importance for realizing high-sensitivity metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In this paper, we propose to use cubic NSS interaction arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='04520v1 [quant-ph] 11 Jan 2023 2 to entangle individual spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Although exhibiting no spin squeezing, the entangled state created by the cubic inter- action have several advantages over the quadratic inter- action from the perspective of QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' First, in the weak- coupling regime we find that the cubic scheme can pro- duce QFI (and thus entanglement) much more rapidly than the quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Quantitative analysis indicates that the acceleration rate is proportional to the spin num- ber of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The cubic scheme thus offers a great advantage over the quadratic scheme in the case of large spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Second, the QFI of the cubic scheme in the strong-coupling regime oscillates fast with coupling strength, which, on average, is larger than the QFI pro- duced by the quadratic scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Besides, the cubic NSS interaction enables the production of a broad variety of macroscopic superposition states that have large QFI, which has no counterpart in the quadratic NSS dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We also analyze an interesting phenomena that has not yet been discovered previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' That is, the QFI production of the cubic scheme is extremely sensitivity to the parity of the total spin number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We find that the amount of QFI at a specific instant of time for even N spins versus odd N + 1 spins change dramatically from N (corresponding to no entanglement among spins) to N 2 (maximal entanglement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This entanglement even- odd effect is quite different from the one exhibited by the OAT interaction [42], which, as we will show later, is an orientation even-odd effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We also show that this entanglement even-odd effect enables us to design a new type of sensing modality to detect the parity of the total spin number of a spin system at the single-spin level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Apart from the cubic NSS interaction, we also have studied a hybrid NSS interaction—cubic-quadratic- admixture (CQA) interaction, which is a weighted sum of the cubic and the quadratic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We find that the CQA interaction is an excellent tool for preparing the GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' High-fidelity GHZ states could be created by simply applying the CQA evolution to the spin sys- tem for a certain time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In contrast to the OAT scheme [42], our hybrid scheme can greatly accelerate the procedure of GHZ-state generation, which tremendously eases experimental requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To realize the cubic interactions in realistic spin sys- tems, two approaches have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' One utilizes the linear and quadratic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Unlike the har- monic oscillator systems, where high-order interactions can not be constructed from the quadratic interactions (known as the Gaussian operations) [43, 44], we show that the cubic interaction can be approximately con- structed by repeatedly using linear and OAT interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This method should be widely applicable to various spin systems, as the OAT interactions have been experimen- tally realized in a number of physical systems [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Another one uses light-mediated interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The spin system is placed inside an one-side optical cavity, forming a spin-cavity system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We show that, by simply sending an optical pulse, off-resonant with cavity mode, into the spin-cavity system, the cubic NSS dynamics is realized after the reflection of the pulse by the one-sided cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Such method should be able to realize the cubic interac- tion in just one step, which is rather attractive from the perspective of experimental implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Finally, we analyze the impact of spin damping, includ- ing the single-spin decay and the collective-spin dephas- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We reveal that, in the presence of damping, the cubic scheme works much better than the quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' That is, in the weak-coupling regime, the cubic scheme can still maintain is speed advantage in QFI production;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' besides, the macroscopic superposition state created by the cu- bic interaction is much more robust against decoherence than the one created by the quadratic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' II we introduce the multipartite entanglement of the collec- tive spins and its correlations with QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' III we first analytically derive the amount of achievable QFI in the weak coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Then, we analysis the prop- erties of the macroscopic superposition states created by the cubic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' IV we discuss the entangle- ment even-odd effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' V we describe how to speed up the procedure of GHZ-state generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' VI we present two approaches to realize the cubic NSS dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' VII we analysis the impact of the decoherence to the entanglement generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Finally, we summarize in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' MULTIPARTITE ENTANGLEMENT IN QUANTUM SPIN SYSTEMS We consider creating multiparticle entanglement among spins in an ensemble consisting of N identical two- level atoms with the excited state |↑⟩ and the ground state |↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To describe the collective properties of such system, we define the pseudo angular momentum oper- ators Si = � k σi k/2(i = x, y, z) for atoms, which satisfy the commutation relations [Si, Sj] = iεijkSk, with εijk being the Levi-Civita symbol, where σi k is a Pauli matrix for the ith atom, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', σi x = | ↑⟩i ⟨↓|i + |↓⟩i ⟨↑|i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Suppose that all the elementary spins point in the same mean di- rection (θ, φ), that is, each atom is prepared in the state |θ, φ⟩i = cos θ 2| ↑⟩i+eiφ sin θ 2| ↓⟩i, forming the well-known coherent spin state (CSS) [18] |θ, φ⟩ = |θ, φ⟩⊗N i = 2S � k=0 � (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' × � sin θ 2 �2S−k� cos θ 2 �k eikφ |S, S − k⟩ , (1) where the collective angular momentum states |S, m⟩ (Dicke states) is the eigenstate of Sz, satisfying Sz |S, m⟩ = m |S, m⟩ with S = N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The CSSs are sepa- rable (nonentangled), and a conventional way to entangle the particles is to utilize the second order nonlinear pro- cesses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', OAT evolution UOAT = exp[−iχtS2 x] [18], where χ is the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To show how spin 3 entanglement is created by UOAT, assume that the col- lective spin is polarized along the z direction, leading to the initial state |ΨA⟩in = |↑⟩⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' At short times, the evolution of this state is found to be |ΨA⟩out = UOAT|ΨA⟩in ≈ N � �|↑⟩⊗N − 2iα N (1 − iα) � i̸=j |↓i↓j⟩ |↑⟩⊗(N−2) ̸=i,j � � , where N = (1−iα)/ √ 1 + 3α2 is a normalization constant with α = Nχt, and in deriving the last equality we have kept terms up to first order in S2 x and used the relations σx i |↓⟩i (|↑⟩i) = |↑⟩i (|↓⟩i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Obviously, the entanglement between the initial and first coupled (double-spin-flipped) states has been created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Such pairwise entanglement have garnered tremendous attention for many years [25], as they are the origin of spin squeezing, which have im- portant applications in quantum metrology as well as in fundamental physics [18, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In fact, irrespective of the creation of spin squeezing, multiparticle entanglement can also be produced by higher-order nonlinearity, such as the three-order (cubic) evolution Ux = exp{−iχtS3 x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For this evolution, one may also derive the time evolved state at time t |ΨA⟩out = Ux|ΨA⟩in ≈ N � �|↑⟩⊗N − 3iα 4N � i̸=j̸=k |↓i↓j↓k⟩| ↑⟩⊗(N−3) ̸=i,j,k � � with the normalization constant N = 1/ � 1 + 3Nα2/32, showing that the triple-wise entanglement among spins is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Obviously, such a state exhibits no property of spin squeezing [24], while a natural question arises: is it useful for sub-shot-noise interferometry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To answer this question we use the QFI to quantify the degree of useful entanglement for quantum metrol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The QFI is closely related to the multipartite en- tanglement [37] and also gives the fundamental limit to the precision achievable in an unknown-parameter esti- mation protocol [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Considering a scenario of phase estimation, a probe spin state ρin is transformed into ρβ = exp (−iβSn) ρin exp (iβSn) by the n-direction col- lective spin generator Sn, where β denotes an unknown phase shift to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The phase sensitivity is lim- ited by the quantum Cram´er-Rao bound [46]: ∆β ≥ ∆βQCR = 1 � FQ [ρin, Sn] , (2) where FQ[ρ, Sn] = 2 � l,l′ (λl − λl′)2 λl + λl′ |⟨l|J|l′⟩|2 (3) is the QFI, λl and |l⟩ are the eigenvalues and eigenvectors of the probe state ρin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The QFI is a measure of how susceptible of ρin to small influences induced by Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The larger the value of QFI, the more precision the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In the case of pure state, ρin = |ψin⟩ ⟨ψin|, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (3) can be further simplified to [47] FQ[ρin, Sn] = 4(∆Sn)2 |ψin⟩, (4) where (∆A)2 |ψ⟩ = ⟨ψ|A2|ψ⟩ − ⟨ψ|A|ψ⟩2 is the variance of A in the state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For a given probe state ρin, it is needed to optimize the rotation direction, n → nop, to maximize the variances of Sn (thus QFI) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' If, for example, the probe state is in the separable CSS |ψin⟩ = | π 2 , 0⟩, one may choose nop = z to yield FQ = N, resulting in a sensitivity ∆β = 1/ √ N, which is exactly the SQL mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To overcome this limit, one should use the entangled states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', the GHZ states [48], |ψin⟩ = 1 √ 2(| π 2 , 0⟩+| π 2 , π⟩), with which the QFI can be calculated (by choosing nop = x) to give FQ = N 2, leading to the HL sensitivity ∆β = 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' One thus can conclude that any entangled states whose QFI satify N < FQ ≤ N 2 are useful for sub-SQL sensitivity [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' THE CUBIC INTERACTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Weak coupling regime We now proceed with the derivation of the QFI of the cubic-interaction-evolved states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For convenience, we suppose that the spins are initially prepared in the CSS, |ψ⟩ = | π 2 , 0⟩, which is subjected to the time evolution Uz = exp � −iχtS3 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (5) One thus obtains the probe state at time t |ψin(t)⟩ = Uz |ψ⟩ = 1 2S 2S � k=0 � (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ×e−iχt(S−k)3 |S, S − k⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (6) For this state, since [Sz, Uz] = 0, Sz is conserved during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Therefore, the uncertainties are redistributed only in the x-y plane [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(c)], which predicts that the optimal direction of the generator, nop, is in some direction in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To see how the uncertainties are redistributed, we next work in the Heisenberg picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The time evolution of the ladder operators S± = Sx±iSy can be exactly evaluated to give [18]: S−(t) = U † zS−(0)Uz = e−iµ(Sz 2+Sz+ 1 3)S−(0), (7) where µ ≡ 3χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The transverse components after the cubic evolution are then given by Sx(t) = 1 2 � S+eiµ(S2 z+Sz+ 1 3) + e−iµ(S2 z+Sz+ 1 3)S− � , (8) Sy(t) = 1 2i � S+eiµ(S2 z+Sz+ 1 3) − e−iµ(S2 z+Sz+ 1 3)S− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (9) 4 To find nop, we calculate the variance of an arbitrary angular momentum operator along the φ direction, Sφ = Sx(t) cos φ + Sy(t) sin φ, in the x-y plane, yielding (∆Sφ)2 |ψ⟩ = cos2φ(∆Sx)2 |ψ⟩ + sin2φ(∆Sy)2 |ψ⟩ + sin 2φ �1 2 ⟨{Sx, Sy}⟩ − ⟨Sx⟩⟨Sy⟩ � ,(10) where {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='} denotes the anticommutator of two observ- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' To calculate the first moments of the spin compo- nents in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (10), we turn to evaluate the mean of the ladder operator ⟨S+ (t)⟩ = 2−2S 2S � k=0 2S � l=0 � (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' � (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ×⟨S, S − k|S+eiµ(S2 z+Sz+1/3) |S, S − l⟩ = 2−2S 2S � l=1 (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='eiµ[(S−l)2+S−l+ 1 3] = S 2S−1 � m=0 P2S−1 (m) eiµ[(S−m)2−(S−m)+ 1 3], (11) where in the last equality we set m = l − 1 and the bi- nomial distribution P2S−1(m) can be approximately con- verted to the Gaussian distribution, P2S−1 (m) = (2S − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2S − 1 − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' �1 2 �2S−1−m�1 2 �m ≃ 1 √ Sπ e− (S−m) S 2 , (12) for large S (see Appendix A for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We thus obtain ⟨S+ (t)⟩ = S √π 1 √ S √ S � k=− √ S e−(1−iµS)k2−iµ √ Sk+ 1 3 iµ ≃ S √π � +∞ −∞ e−(1−iµS)k2−iµ √ Sk+ 1 3 iµdk ≃ S √1 − iµS , (13) where k = (S − m)/ √ S, and, in the second equality, we have transformed the sum to integral, which is valid only when ∆k = 1/ √ S → 0 for, again, large S, and, in the last equality, we also have used the approximation exp [iµ(4− iµS)/12(1 − iµS)] ≈ 1 for µ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Along the same lines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' one may derive the quadratic expectation values ⟨S2 +(t)⟩ and ⟨S+(t)S−(t)⟩ (see Appendix B for more details),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' with which we are able to calculate the means ⟨Sx⟩ = S � α1 (α1 + 1)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ⟨Sy⟩ = S � α1 (1 − α1)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' � S2 x � = S 4 � (2S + 1) + (2S − 1) � α4 (α4 + 1)/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' � S2 y � = S 4 � (2S + 1) − (2S − 1) � α4 (1 − α4)/2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' ⟨{Sx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Sy}⟩ = S 2 (2S − 1) � α4 (1 − α4)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (14) (a) (b) (c) (i) (ii) (iii) II1 x y z x z x z y y I1 I2 II2 II3 II4 t χ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (Color online) QFI of the time-evolved states ver- sus coupling strength (expressed in terms of either α or χt, see text for clarification) for N = 200: exact numerical solu- tion of the cubic scheme (solid orange curve) and the OAT scheme (dash-dotted green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (a) The dashed blue curve is the analytical result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (b) The peaks marked I1,2 are the maximum QFI that is achievable by the cubic scheme, while the submaximal QFI are marked by II1−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (c) Quasiprobability distribution of different spin states: (i) is a CSS, (ii) is a GHZ state, and (iii) is a four-components Schr¨odinger cat state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' where we have defined the new parameters αk = 1/ � 1 + kS2µ2 with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Substituting these val- ues into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (10) we finally arrive at (∆Sφ)2 |ψ⟩ = S 4 [2 (1 − α1) S + 1] + � A2 + B2 cos (2φ − 2δ) , (15) where A = S 4 √ 2(2S − 1) � α4 (1 + α4) − S2 2 α2 1, B = S 4 √ 2 (2S − 1) � α4 (1 − α4) − µS3 2 α2 1, δ = 1 2 arctan � B A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (15) is maximized when φ = δ, obtaining FQ = 4 � A2 + B2 + S [2 (1 − α1) S + 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (16) For Sµ ≪ 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (16) can be approximated as FQ ≈ 2S + 9 2S2α2 ≥ N, (17) which indicates that any nonzero α enables the sensi- tivity to surpass the SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Therefore, the entanglement 5 created by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (5) is useful for quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For comparison, the QFI created by OAT interaction in the weak coupling regime is also calculated to give: FQOAT ≈ 2S + 2Sα2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Apparently, the QFI produced by cubic interaction is about S times faster than OAT in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This is quite a promising advantage, since the ability to create entangled quantum resources rapidly is a pursuit in quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' It should be emphasized that the speed-up rate is closely connected to N (the large the N, the faster the increase in QFI), which means that the cubic scheme might be more suitable for atomic systems with a large number of atoms [16, 17, 19, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(a) we compare the analytical result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (16) (dashed blue curve) and the exact numerical results from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (6) (solid orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The two curves fit pretty well in the weak coupling regime and gradually de- viate when α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For large α, the numerical results display various oscillating structures, which are lost by analytical result due to the discrete-to-continuous conver- sion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In fact, each peak of QFI is related to a macroscopic supposition of collective spin, as will be dis- cussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(a) also confirms that the QFI of the cubic scheme increases much more rapidly with α than the OAT scheme (dash-dotted green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(b) we also plot the periodic evolution of QFI in time for both schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' It shows that the OAT scheme can saturate the HL at t = π/2χ, which corresponds to the creation of a GHZ state [42], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(c)(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The QFI of the cubic scheme, however, has a quite complicated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Although can not saturate the HL, there exist two maximum peaks [labeled by I1,2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(b)] that are quite near the HL in a period of evolution, which corresponds to a Schr¨odinger cat state with four super- posed CSSs [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(c)(iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Besides the two maximum peaks, there also exist a number of lower peaks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', four secondary peaks [labeled by II1−4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Next, we quantify the amount of QFI for these peaks and explore the properties of these peak states .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Strong coupling regime For convenience, we first assume that N is even and rewrite the state of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (6) in the following form |ψin(t)⟩ = 1 2S S � m=−S � (2S)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (S + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (S − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='e−iχtm3 |S, m⟩ (18) by setting k = S − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Considering the time evolved state at special time t = π/nχ [50], where n is an integer, the evolution factor exp(−iπm3/n) at this time has the following periodic properties: exp � −iπ n (m + 2n)3 � = exp � −iπ n m3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (19) Such periodicity property enables us to expand the evo- lution factor as a Fourier series [50] exp � −iπ n m3 � = 2n−1 � q=0 f e q exp � −iπq n m � , (20) where the coefficients f e q are given by the inverse Fourier transform f e q = 1 2n 2n−1 � m=0 exp �iπq n m � exp � −iπ n m3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (21) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (20) indicates that we have successfully converted an exponentially cubic form into sums of exponentials linear in m, which is a key step for the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (20) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (18), we obtain ����ψin � π nχ �� = 2n−1 � q=0 f e q ���π 2 , πq n � , (22) which shows that a Schr¨odinger-cat-like state (SCS) (a superposition of the CSSs) can be produced by the cubic evolution at the particular time t = π/nχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The charac- teristics of the SCSs are determined by the coefficients f e q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Specifically, by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (21) and (22), one may derive the form of SCS for n = 4, ����ψin � π 4χ �� = 1 2 ����π 2 , 0 � + ���π 2 , π 4 � + ���π 2 , π � − ���� π 2 , 5π 4 �� , (23) and for n = 12, ����ψin � π 12χ �� = 1 2 ����π 2 , π 12 � + ���π 2 , π 3 � − ���� π 2 , 13π 12 � + ���� π 2 , 4π 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (24) Notably, the states (23) and (24) are just the two states that create the two maximum QFI peaks I2 and I1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1(b)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Next, we turn to derive the amount of QFI of peak I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' By using the state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (23), one may directly calculate the means and variances of the collective spin components, obtaining ⟨Sx⟩ = S � cos π 8 �2S−1 cos �π 8 (2S + 1) � , ⟨Sy⟩ = S � cos π 8 �2S−1 sin �π 8 (2S + 1) � , � S2 x � = 1 8 � 6S2 + S � , � S2 y � = 1 8 � 2S2 + 3S � , ⟨{Sx, Sy}⟩ = 1 4 � 2S2 − S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (25) Substituting them into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (10), after optimization of (∆Sφ)2 |ψin(π/4χ)⟩ we get FQ = 2S2 � 1 + 1 √ 2 − � cos π 8 �4S−2 × � 1 + cos �πS 2 ��� + � 1 − 1 √ 2 � S (26) 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (Color online) (a) QFI produced by the cubic scheme versus coupling strength χt for N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The QFI peaks marked I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', V1 are produced by the states |ψin(π/12kχ)⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (6) with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Insert: The QFI of the states |ψin(π/12kχ)⟩ and |ψin(π/3(2k − 1)χ)⟩ vs k for N = 60 and N = 61, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (b)–(e) The Fourier-coefficients distribution [ given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (22)] (bottom) and the QDP (top) of the states of the peaks marked II1 − V1 in (a) for N = 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The quantum state, QFI [calculated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (33)], and achievable sensitivity of each peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Peaks The quantum state of peaks (neglecting the normalization) QFI ∆β I1 |ψin � π 12χ � ⟩ = |GHZ− π/12⟩ + |GHZ+ π/3⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='85N 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='08/N II1 |ψin � π 24χ � ⟩ = |GHZ+ π/6⟩ + 7 10 |GHZ− 7π/24⟩ − 7 10 |GHZ− 19π/24⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='75N 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='15/N III1 |ψin � π 36χ � ⟩ = |GHZ+ 0 ⟩ + |GHZ− π/4⟩ + 1 2 |GHZ+ π/3⟩ − 1 2 |GHZ+ 7π/12⟩ − 1 3 |GHZ− 2π/3⟩ − 1 3 |GHZ− 11π/12⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='70N 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='20/N IV1 |ψin � π 48χ � ⟩ = |GHZ+ π/12⟩ + 9 5 |GHZ− 7π/48⟩ + |GHZ+ π/3⟩ + |GHZ+ 7π/12⟩ − 4 5 |GHZ− 31π/48⟩ − |GHZ− 11π/6⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='67N 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='22/N V1 |ψin � π 60χ � ⟩ = 8 5 |GHZ− π/60⟩ + 3 5 |GHZ+ π/15⟩ + |GHZ− 13π/60⟩ + 8 5 |GHZ+ 4π/15⟩ − |GHZ+ 7π/15⟩ − |GHZ− 37π/60⟩ + 3 5 |GHZ− 49π/60⟩ + |GHZ+ 13π/15⟩ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='65N 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='24/N for φ = π/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For large N, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (26) is reduced down to FQ ≈ 1 2 � 1 + 1 √ 2 � N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (27) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (27) is the upper bound of QFI produced by the cubic scheme in the case of even N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (27) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (2) yields the best angular sensitivity achievable by the cubic scheme, ∆β ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='08/N, which is very near the HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In fact, the Heisenberg scaling of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (27) originates from the fact that the states of peaks I1,2 are in suppo- sition of two GHZ states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', ����ψin � π 12χ �� = 1 √ 2 � |GHZ− π/12⟩ + |GHZ+ π/3⟩ � , (28) where we have defined ��GHZ± ϕ � = 1 √ 2 ����π 2 , ϕ � ± ���π 2 , ϕ + π �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (29) Obviously, the maximum-variance direction of the GHZ states of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (29) is ϕ, which from now on we call the direction of a GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Corresponding to the states (29), one may derive the variance of Sφ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (10), yielding (∆Sφ)2 |GHZ± ϕ⟩ = 1 4 � 2S2 + S +(2S2 − S) cos 2 (ϕ − φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (30) This equation quantifies the amount of noise in the direc- tion φ which deviates from the GHZ direction by an angle ϕ − φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' It can maximized to (∆Sφ)2 |GHZ± ϕ ⟩ = S2 when the two directions are exactly the same (φ = ϕ) and can be minimized to (∆Sφ)2 |GHZ± ϕ ⟩ = S/2 when the two direc- tions are orthogonal to each other (φ = ϕ − π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For any 0 ≤ ϕ − φ ≤ π/2, we have S/2 ≤ (∆Sφ)2 |GHZ± ϕ ⟩ ≤ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (30) could also be regarded as the pro- jection of the noise of a GHZ state in the GHZ direction onto the φ direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Keeping this physical picture in mind, let us turn to seek the maximum variance of the state (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' By pro- jecting the noise of the two GHZ states, |GHZ− π/12⟩ and 3元/4 FQ /N2 3元/4 0元/2 (a) 元/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='8 元/4 π/4 V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='05 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2 5 0 2 1 3 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='0F 3 4 5 6 3元/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='4 3元/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' N=61 0 元/2 元/2 N=60 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='8F元/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='. 元/4 III, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='7 V1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='2 (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='0L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='T/Xt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='0l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='10 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 1 2 4 5 6 0 1 2 3 5 6 xt7 |GHZ+ π/3⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' onto the φ direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' we get (∆Sφ)2 |ψin( π 12χ)⟩ ≈ 1 2 (∆Sφ)2���GHZ− π/12 � + 1 2 (∆Sφ)2���GHZ+ π/3 � = 1 4 � 2S2 + S + 1 2 � 2S2 − S � × � cos 2 � φ − π 12 � + cos 2 � φ − π 3 ��� = 1 4 � 2S2 + S + 1 √ 2 � 2S2 − S � ×cos � 2φ − 5π 12 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (31) where Sφ = Sx(0) cos φ + Sy(0) sin φ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' in the first equality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' we have neglected the nondiagonal terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' which is reasonable when N is large (see Appendix C for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (31) is maximized at φ = 5π/24 to give (∆S5π/24)2 |ψin(π/12χ)⟩ ≈ 1 2 � 1 + 1 √ 2 � S2, which is exactly the same as the maximum value of (∆Sφ)2 |ψin(π/4χ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Such result (that is, the QFI of I1,2 are equal) has also been predicted by the numerical results in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This confirms that the noise-projection method outlined above provides a convenient way to derive the QFI of a probe state in a superposition of arbitrary GHZ states, that is, |ψin⟩ = N � ϕ Cϕ ��GHZ± ϕ � , (32) where N is the normalization and Cϕ are the probability amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Our task is to maximize the projected noises, (∆Sφ)2 |ψin⟩ ≈ � ϕ |NCϕ|2 (∆Sφ)2 |GHZ± ϕ⟩ = 1 4 � ϕ |NCϕ|2 � 2S2 + S +(2S2 − S) cos 2 (ϕ − φ) � , (33) over all values of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We are now equipped to evaluate the QFI of the lower peaks II1 − V1 [as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(b)-2(e) we plot the Fourier coefficients distribution as well as the quasiprobability distribution (QPD) [obtained from the exact numerical evolution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (18)] of each peak state, showing that (i) the two results are consis- tent with each other, and (ii) the appearance of large QFI has always been accompanied with the generation of macroscopic quantum-superposition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Represent- ing the peak states in terms of the GHZ states of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (29), we are able to show in table I the explicit forms for each peak state, indicating that they have exactly the same form as the states given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' One thus can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (33) to approximately derive the amount of QFI for each peak for the case of large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' As can be seen from table I, the state of each peak could realize a sensitivity near the HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Interestingly, these peak states appear regularly at a particular time tk = π/12kχ with Ⅲ1 Ⅰ1 Ⅱ1 Ⅳ1 Ⅴ1 (a) (b) x y z probability Ⅰ1 Ι1 ≈ \uf028 \uf029 12 sin S \uf070 \uf028 \uf029 12 sin S \uf070 \uf02d mx FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (Color online) (a) QFI produced by the cubic scheme versus coupling strength χt for even (orange curve) and odd (purple curve) number of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Insert: the QPD of the state corresponds to the peak I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (b) QFI produced by the cubic scheme versus coupling strength χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The QFI peaks labeled I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', V1 are produced by |ψin(π/3(2k − 1)χ)⟩ with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Insert: the Sx probability distribu- tions of the peak GHZ state I1 (purple curve) and the initial CSS state |π/2, 0⟩ (orange circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We here take N = 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' integer k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' In fact, the states of those not labeled peaks on the left-hand side of the peak V1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(a)] are also obtained at tk but with k ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' It should be emphasized that the number of visible peaks depends heavily on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The larger the N, the more QFI peaks it can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This is because the number of CSS components can be found in |ψin(π/12kχ)⟩ increases with k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' For large k but small N, these superposition CSS components are overlapped with each other and become indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The distance between the CSS com- ponents, however, can be increased by increasing N, as their distance is proportional to N while the radius of a CSS on the Bloch sphere is proportional to √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Once all the CSS components of |ψin(π/12kχ)⟩ become distin- guishable on the Bloch sphere, the kth QFI peak appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' T T 儿 儿N-60 儿 儿 儿1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='00 100 50 0 50 1008 In other words, the appearance of QFI peaks herald the creation of macroscopic superposition states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' As can be seen from the insert in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(a), QFI larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='55N 2 is still obtainable at t240 for N = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' We emphasize that, although the maximal QFI produced by the cubic scheme is smaller than the maximal QFI of the OAT scheme (that is, FQ = N 2 produced by the GHZ state at time π/2χ), it has an advantage of short-preparation time, which, as we will show below, makes the entanglement generation robust against damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' THE ENTANGLEMENT EVEN-ODD EFFECT An interesting feature of the cubic evolution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (5) is an extreme sensitivity to the parity of the total spin number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3(a), the evolutions of QFI for N atoms vs N + 1 atoms are macroscopically different: first, the maximum QFI of the cubic scheme with odd-N spins can saturate to the HL [the peak I1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3(a)], which is actually produced by the GHZ state |GHZ− 7π/12⟩ at time t = 7π/12χ [see the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3(b)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' second, the state of each QFI peak [the peaks I1 − V1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3(b)] for odd N appears at a different but regular time tk = π/3(2k − 1)χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' These differences originate from the fact that the evolution factor of the odd-N spin system has a quite different periodic property, exp � −iπ n (m + 8n)3 � = exp � −iπ n m3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (34) Accordingly, the evolution factor can be reexpanded as exp � −iπ n m3 � = 8n−1 � q=0 f o q exp � −iπq n m � , (35) f o q = 1 8n 8n−1 � m=0 exp �iπq n m � exp � −iπ n m3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (36) Then, by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (33) and (36) we are able to show in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 2(a) (the curve with circle) the amount of QFI for each peak in case of large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Despite the HL QFI peak produced by the state |GHZ− 7π/12⟩, the spin system with even N is superior over the one with odd N in the production of multipartite entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' A more striking feature is that the amount of QFI at time t1 = π/3 changes dramatically with the par- ity of N: for even N, the evolved spin state is a sep- arable CSS with FQ = N, while, for odd N, the spin state created is a maximally entangled GHZ state with FQ = N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' This entanglement even-odd effect is quite different from the even-odd effect exhibited by the OAT evolution UOAT = exp[−iχtS2 z], which, at the instant t = π/2χ, maps the initial CSS |π/2, 0⟩ to the N-dependent GHZ state |ψin(π/2χ)⟩ = 1 √ 2(eiπ/4 |π/2, −π(N − 1)/2⟩+ e−iπ/4 |π/2, −π(N − 3)/2⟩) [42, 51], showing that the ori- entation of the created GHZ state is sensitivity to the parity of total spin number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (Color online) (a) QFI produced by the interaction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (37) versus coupling strength χt for N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Insert (i): schematic depiction of the evolution induced by the hybrid dynamics on the Bloch sphere, with green flow lines denoting S2 z-dependent rotation of the QPD around the z axis caused by the cubic part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (37) and orange flow lines repre- senting twisting of the QPD induced by the quadratic part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Insert (ii): we plot the means ⟨Txy⟩ and ⟨Sz⟩ as a function of χt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (b)-(e) are the QPD of the peak states marked with α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=', δ in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The determination of the spin number in a realistic quantum system is a critical first step toward the re- alization of quantum metrology as well as quantum in- formation processing [52–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Especially in the context of spin-spin entanglement generation [57, 58], the spin system will evolve into a highly entangled pure state or a separable mixed state depending on the parity of N [59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' It is thus of great importance to have the abil- ity to determine the parity of N before performing the protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Our entanglement even-odd effect might po- tentially be used to detect the parity of the total spin number N with a resolution at the single-spin level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' The parity detection proceeds as follows: the spin state is ini- tially prepared in the CSS |π/2, 0⟩, and then is subjected to the evolution described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' (5) for a time duration t = π/3χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Finally, measuring Sx a particular outcome ˜Sx is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' As can be seen from the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3(b), for even N, we have ˜Sx = S, while, for odd N, there will be a large probability of finding ˜Sx around ±S sin(π/12) (especially for the case of large N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' Therefore, the mea- surement of the angular momentum operator Sx provides a direct way to estimate the parity of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 0 (b) (c) 03 4 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content=' 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE3T4oBgHgl3EQfcAoE/content/2301.04520v1.pdf'} +page_content='8100 MeV +and integrated in different codes: COSMO (FORTRAN program) [58], +YIELDX (FORTRAN routine, including the latest updates of the equa- +tions) [57] and ACTIVIA (C++ computer package, using also experi- +mental data when available) [59]. +• The MC simulation of the interaction between nucleons or other pro- +jectiles and nuclei allows also computation of production cross sec- +tions. Many different models and codes have been developed and val- +idated considering the relevant processes (the formation and decay of +compound nuclei, the intranuclear cascade of nucleon interactions, de- +excitation processes like fission, fragmentation, spallation, or breakup) +[60]; some of these models have been implemented in general-purpose +codes like Geant4 [61] or FLUKA [62]. Evaluated libraries of produc- +tion cross sections have been elaborated, covering different types of re- +actions or projectiles and different energies, like TENDL (TALYS-based +Evaluated Nuclear Data Library)4 [63] (based on the TALYS code, for +protons and neutrons with energies up to 200 MeV); JENDL (Japanese +Evaluated Nuclear Data Library) [64] High Energy File5 (based on +the GNASH code, for protons and neutrons from 20 MeV to 3 GeV) +is an extension of the JENDL-4.0/HE library including results up to +200 MeV; HEAD-2009 (High Energy Activation Data) [65] (for protons +and neutrons with higher energies, from 150 MeV up to 1 GeV) uses a +3EXFOR: +http://www.nndc.bnl.gov/exfor/exfor.htm, +http://www- +nds.iaea.org/exfor/exfor.htm. +4https://tendl.web.psi.ch/tendl 2019/tendl2019.html +5JENDL +HE +library, +https://wwwndc.jaea.go.jp/ftpnd/jendl/jendl40he.html; +https://wwwndc.jaea.go.jp/jendl/jendl.html +13 + +selection of models and codes (CEM, CASCADE/INPE, MCNP, etc.) +dictated by an extensive comparison with EXFOR data. +4. Cosmogenic yields in Copper and Steel +As in many experiments, a significant amount of copper and stainless steel +are used in DarkSide detectors; both materials are known to become activated +and different specific studies on their activation are available [13, 14]. The +effect on DarkSide-20k of cosmogenic activity in the components made of +copper and stainless steel is analyzed here. +4.1. Production rates +The production rates of the radionuclides typically induced in these mate- +rials have been selected from measured and calculated results available in the +literature [13, 14]. Estimates using ACTIVIA, Geant4, and TALYS codes, +among others, have been made. Saturation activities have been measured +with sensitive germanium detectors in samples of copper [32, 38, 39] and +steel [38], exposed for long times to cosmic rays. In particular, in this work, +the production rates from dedicated measurements, using 125 kg of copper +provided by Norddeutsche Affinerie (now Aurubis) exposed for 270 days at +Gran Sasso and Nironit stainless steel exposed for 314 days, have been con- +sidered [38]; values are reproduced in Table 2. Among the different products +identified in copper, 60Co has the longest half-life and, unfortunately, there +is a significant disagreement on the production rates estimated for it [13, 14]; +the measured value in Ref. [38] is higher than most of all the other estimates +by a factor of up to a few times. No assessment of 60Co production in stainless +steel could be made in Ref. [38], as, being this isotope a typical contaminant +in steel, its cosmogenic activity is obscured by previous contamination at +similar level; then, the rate derived from Geant4 calculations [33] has been +used. Following the half-lives of the different cosmogenic isotopes identified +in copper and steel (also shown in Table 2), 54Mn, 57Co and 60Co could be +in principle the most relevant products. +4.2. Activity +To assess the possible effect of the cosmogenic isotopes in these mate- +rials for DarkSide-20k, activity A has been evaluated considering the se- +lected production rates at sea level, tcool =0 and extreme cases of exposure: +texp =1 month, texp =1 year and texp =10 years. It is worth noting that as +14 + +measured production rates have been taken into account, the deduced acti- +vation corresponds to all cosmic ray particles. The final expected activity is +obtained from the specific activities derived from the production rates (per +mass unit) using Eq. 1 and the mass of all the components used in the ex- +perimental set-up, which according to the present design of DarkSide-20k are +165.1 kg of copper (mainly from cables and PDMs electronic components) +and 225.655 tons of stainless steel (mainly from cryostat components) plus +12 tonnes from the inner detector. +Table 2 summarizes the total induced activity in copper and stainless +steel, respectively, for the relevant isotopes evaluated at the end of the differ- +ent exposure times; contribution from each individual component is propor- +tional to its mass (see Table 1). Following the decay mode of these nuclei, +γ emissions of the order of 1 MeV will be generated around the active vol- +ume by this cosmogenic activation. In the case of copper, even assuming +10 years of exposure, the total activity is at the level of 0.5 Bq. The deduced +activities can be compared with available measurements from radioassays; +for the copper from the Luvata company being considered in DarkSide-20k, +the measured activities using a HPGe detector in the Canfranc Underground +Laboratory are <0.30 mBq/kg of 60Co and <0.35 mBq/kg of 54Mn; then, this +upper limit set for 60Co would correspond to the exposure of a few years. For +all stainless steel components, some cosmogenic activities can be at the level +of a few hundreds of Bq, even for just 1 year of exposure; 54Mn is identified +as a potential relevant contributor to background. Comparing with avail- +able measurements from screening, the derived cosmogenic activity of 60Co +is much lower than for instance the one measured for the DUNE steel in the +Canfranc Underground Laboratory, finding (10.8±0.9) mBq/kg of 60Co and +(1.4±0.3) mBq/kg 54Mn; the measured activity of 54Mn would correspond to +an exposure of ∼1 year. +5. Cosmogenic yields in Titanium +Titanium is not part of the DarkSide-20k design. However, it was con- +sidered in previous designs and titanium is of interest in low-background +experiments generally, so we include it here. The possible impact of differ- +ent cosmogenic products has been analyzed and then, the production rate +and induced activity at sea level of the most relevant one, 46Sc, have been +quantified from available information and new calculations. To our knowl- +edge, no direct measurement of productions rates for activation is available +15 + +for titanium. Natural composition has been assumed. +5.1. Relevant isotopes +Titanium activation by cosmic rays, particularly during air transport, was +studied within the DarkSide Collaboration using a modified version of the +COSMO code; the induced activity was quantified under different exposure +and cooling conditions, including one with an exposure at sea level for a long +time (10,000 days). In Ref. [33], the cosmogenic activation at sea level of +several materials used in rare event search experiments, including titanium, +was quantified from Geant4 simulations (for neutrons, protons and muons +considering the Shielding physics list) and using the ACTIVIA code. In both +works, the yield of different products was evaluated: some of them (including +several Sc isotopes) are short-lived with half-lives of a few days at most; +others, like 40K and 50V, are very long-lived, so huge exposures comparable +to their half-lives would be required to produce a significant activity; among +the isotopes with intermediate half-lives, most of them produce emissions +which could not escape from titanium to reach the active volume, as they are +either pure or almost pure β− emitters (3H, 33P, 35S, 39Ar, 45Ca) or generate +X-rays or low energy (below ∼80 keV) γ rays (44Ti, 49V). Therefore, 46Sc +has been identified as the main product which could be relevant and the new +calculations performed here correspond just to this isotope. +16 + +Table 2: Estimates of induced activity in all copper and stainless steel components of DarkSide-20k at the end of the exposure +to cosmic rays. For each one of the products, the half-life [66], main γ emissions and corresponding probabilities are indicated +together with the production rates R at sea level assumed (from measurements in Ref. [38] except for 60Co in stainless steel, +taken from Ref. [33]) and the total activity A for the three exposure times considered (1 month, 1 year and 10 years). +7Be +46Sc +54Mn +59Fe +56Co +57Co +58Co +60Co +T1/2 (d) +53.22 +83.79 +312.19 +44.49 +77.24 +271.82 +70.85 +1923.95 +γ emissions (keV) +477.6 +889.3, 1120.5 +834.8 +1099.3, 1291.6 +846.8, 1238.3 +122.1 +810.8 +1173.2, 1332.5 +probability (%) +10.5 +99.98, 99.98 +99.98 +56.5, 43.2 +100, 67.6 +85.6 +99 +99.97, 99.99 +Copper +R (atoms/kg/day) +2.18±0.74 +8.85±0.86 +18.7±4.9 +9.5±1.2 +74±17 +67.9±3.7 +86.4±7.8 +A (1 m) (mBq) +0.92±0.31 +1.09±0.11 +13.3±3.5 +4.28±0.54 +10.4±2.4 +33.0±1.8 +1.77±0.16 +A (1 y) (mBq) +4.0±1.3 +9.39±0.91 +35.6±9.3 +17.5±2.2 +86±20 +126.1±6.9 +20.3±1.8 +A (10 y) (mBq) +4.2±1.4 +16.9±1.6 +35.7±9.4 +18.2±2.3 +141±32 +129.8±7.1 +121±11 +Stainless Steel +R (atoms/kg/day) +389±60 +19.0±3.5 +233±26 +20.7±3.5 +51.8±7.8 +6.27 +A (1 m) (Bq) +346±53 +11.5±2.1 +41.3±4.6 +13.4±2.3 +36.2±5.5 +0.19 +A (1 y) (Bq) +1061±164 +49.7±9.2 +356±40 +54.8±9.3 +138±21 +2.1 +A (10 y) (Bq) +1070±165 +52.3±9.6 +641±71 +56.9±9.6 +142±21 +13 +17 + +Table 3: Calculations of the production rate at sea level of 46Sc in titanium in this work +and from the literature. +Code +R (atoms/kg/day) +COSMO +289.4 +ACTIVIA +270.1 6 +[33] +Geant4 +275.5 +[33] +Estimated rate in this work +(271±68) +46Sc is a β− emitter with a half-life of 83.8 days and a transition energy of +2366.5 keV [66]; two γ rays of 889.3 keV and 1120.5 keV are produced with +almost 100% probability each. Its activity has been quantified in titanium +samples screened by different experiments: LUX-ZEPLIN analyzed many +items (finding values ranging from 0.2 to 23 mBq/kg, being most of them +around a few mBq/kg) and exposed in a controlled way a sample for 6 months +measuring afterwards an activity of (4.4±0.3) mBq/kg [67, 68]; XENON1T +also analyzed titanium of different grades, measuring activities from 1.0 to +2.7 mBq/kg [69]. In addition, the production rate at sea level of 46Sc was +computed using Geant4 and ACTIVIA [33] and can also be deduced from +COSMO calculations; Table 3 compares the different estimates, which point +to quite similar values. +5.2. Production rate +To evaluate the production rate of 46Sc at sea level using Eq. 2 and the cos- +mic neutron spectrum from Ref. [52], a selected description of the production +cross sections over the whole energy range from threshold up to 10 GeV, con- +sidering both neutrons and protons, has been defined. For libraries providing +individual reaction cross sections, the mechanisms indicated in Table 4 have +been considered. Figure 2 shows the full set of data on total production cross +sections taken into consideration from different libraries and a dedicated cal- +culation using YIELDX. Below 100 MeV, there are important discrepancies +between libraries and experimental data for protons, although this should +not be relevant for neutron activation if specific calculations for neutrons are +used; the only two available measurements on cross sections by neutrons are +in perfect agreement with TENDL-2019 results. Above 100 MeV, there is +a good agreement between different calculations and experimental data for +protons (except for one quite old series of data); it is worth noting than the +similarity between cross sections for neutrons and protons (usually assumed +18 + +Table 4: Production mechanisms for 46Sc in natural Ti isotopes by neutrons and protons. +Neutrons +Protons +46Ti (8.25%) +(n,p) +47Ti (7.44%) +(n,pn) +(p,2p) +48Ti (73.72%) +(n,p2n) +(p, 2pn) (p,pd) +49Ti (5.41%) +(n,p3n) +(p,2p2n) (p,2d) (p,pt) +50Ti (5.18%) +(n,p4n) +(p,2p3n) (p,2dn) (p,dt) (p,ptn) +in this range of higher energies) is fully confirmed by JENDL-HE, providing +independent results for them. +Taking into account all the available data, the following cross sections +σ(E) have been considered: +• Below 20 MeV, TENDL-2019, the only results that are available. +• From 20 to 200 MeV, production cross sections by neutrons from TENDL- +2019, JENDL-4.0 and JENDL-HE libraries. +• From 200 MeV to 1 GeV, results from JENDL-HE for neutrons and +HEAD-2009 library together with YIELDX calculations. +• From 1 to 10 GeV, YIELDX results and data from JENDL-HE for +neutrons (extrapolating the last available value at 3 GeV as constant +for all higher energies). +Figure 3 presents a closer view of the cross sections actually considered in +the calculations of the production rate for the low (top) and high (bottom) +energy regions. +The estimated contributions to the production rate of 46Sc for each energy +range and selected cross sections are summarized in Table 5. To sum the +contributions in the whole energy region, the calculations with the lowest +and highest rates in each region have been considered to get mean value and +uncertainty from the defined interval (from 202.9 to 338.9 atoms/kg/day); in +this way, the final result is (271±68) atoms/kg/day, in very good agreement +with all the previous estimates (see Table 3). +5.3. Activity +From the estimated production rate of 46Sc by neutrons at sea level, the +corresponding saturation activity according to Eq. 1 is (3.14±0.79) mBq/kg. +19 + +0.01 +0.1 +1 +10 +100 +1000 +1 +10 +100 +1000 +10000 +Production cross section (mb) +Energy (MeV) +TENDL n +TENDL p +JENDL n +JENDL p +JENDL-HE n +JENDL-HE p +HEAD2009 +YIELDX +Sisterson'2006 n +Greenwood'1987 n +Cervenak'2020 p +Parashari'2019 p +Garrido'2016 p +Hermanne'2014 p +Khandaker'2009 p +Jung'1991 p +Aleksandrov'1990 p +Fink'1990 p +Michel'1989 p +Michel'1980 p +Brodzinski'1971 p +Neumann'1999 p +Aleksandrov'1991 p +Leya'1997 p +Asano'1991 p +Asano,1983 p +Figure 2: Full compilation of production cross sections of 46Sc in natural Ti by nucleons +taken from different sources, including experimental data from the EXFOR database and +calculations following different approaches. +Table 5: Contributions to the production rate (in atoms/kg/day) of 46Sc in natural Ti by +cosmic neutrons at sea level estimated using Eq. 2, the neutron spectrum from Ref. [52] +and the different cross sections selected for each energy range. +TENDL +JENDL +YIELDX +HEAD2009 +JENDL-HE +(n) +(n) +(n) +<20 MeV +16.0 +20-200 MeV +146.6 +270.8 +187.7 +200-1000 MeV +37.9 +44.8 +49.5 +1-10 GeV +2.4 +2.6 +20 + +0.1 +1 +10 +100 +1000 +1 +10 +100 +Production cross section (mb) +Energy (MeV) +TENDL n +JENDL n +JENDL-HE n +HEAD2009 +YIELDX +Sisterson'2006 n +Greenwood'1987 n +1 +10 +100 +1000 +100 +1000 +10000 +Production cross section (mb) +Energy (MeV) +TENDL n +TENDL p +JENDL n +JENDL p +JENDL-HE n +JENDL-HE p +HEAD2009 +YIELDX +Aleksandrov'1990 p +Fink'1990 p +Michel'1989 p +Brodzinski'1971 p +Neumann'1999 p +Aleksandrov'1991 p +Leya'1997 p +Asano'1991 p +Asano,1983 p +Figure 3: Close view of production cross sections of 46Sc in natural Ti taken into consid- +eration in the estimate of the production rate in the low (top) and high (bottom) energy +regions. +21 + +Due to the half-life of 46Sc, this saturation value may be easily achieved for +usual exposure times; indeed, the measured activities of 46Sc in screened +samples are around this value. This number can be considered as a conser- +vative assumption for the induced activity at sea level; if exposure happens +at certain altitude, correction factors for the cosmic neutron flux should be +included. +The quantified activity has been obtained just for cosmic neu- +trons; according to the results based on Geant4 simulations in Ref. [33], the +neglected contribution from muons and protons would be just 1.7% of the +total. If the vessel of the inner detector was made of titanium, a total mass +of 9.5 tons would be used giving an overall activity of around 30 Bq of 46Sc +just when finishing the exposure to cosmic rays. According to the current +schedule for the installation of the detector, it will be underground more than +1.5 y before starting operation; then, after this cooling period, 46Sc activity +would have been reduced to 1.1% of the initial value. +6. Cosmogenic yields in Argon +Argon in the atmosphere contains stable 40Ar at 99.6%; cosmogenically +produced radioactive isotopes, mainly 39Ar but also 37Ar or 42Ar, can be a +significant background if argon obtained from air is used. The concentra- +tion of these three isotopes is much reduced in UAr, but the production of +cosmogenic radionuclides after extraction must be taken into consideration. +6.1. Relevant isotopes +39Ar is a β− emitter with a transition energy of 565 keV and half-life of +269 y [70]; it is mainly produced by the 40Ar(n,2n)39Ar reaction started by +cosmic neutrons [35]. The typical activity of 39Ar in AAr is at the level of +∼1 Bq/kg, as quantified by WARP [71], ArDM [72] and DEAP [73]. In UAr, +after a first study on argon from deep underground sources [74], the mea- +sured activity of 39Ar in the DarkSide-50 detector was (0.73 ± 0.11) mBq/kg +following a campaign of extracting and purifying argon from deep CO2 wells +in Colorado, US; as mentioned in Sec. 1, this means a reduction of a factor +(1.4±0.2)×103 relative to the AAr [8]. +Presence of cosmogenically produced 37Ar was also detected in the begin- +ning of the run of the DarkSide-50 detector with UAr [8]. It decays 100% +by electron capture to the ground state of the daughter nuclei with a half- +life of 35.02 days [66]; then, the binding energy of electrons from K-shell +(2.8 keV, at 90.21%) and L-shell (0.20-0.27 keV, at 8.72%) can be measured +22 + +as a distinctive signature. The main production channel is the 40Ar(n,4n)37Ar +reaction [35]. Production underground in UAr by thermal and epithermal +neutron capture is negligible, as for 39Ar, considering rates as in Ref. [35] +and neutron fluxes at LNGS. +42Ar is a pure β− emitter with a 32.9 y half-life and transition energy +of 599 keV, generating 42K, also a β− emitter with half-life of 12.36 h and +transition energy of 3525 keV [70]; this isotope can affect neutrinoless 2β +experiments using liquid argon as refrigerant and shielding, as shown by the +GERDA experiment [75]. There are two mechanisms for the production of +42Ar in AAr: a two-step neutron capture (requiring a high neutron flux be- +cause of the half-life of 41Ar, being of 1.8 h) and the (α,2p) reaction on 40Ar +[76]. The specific activity of 42Ar has been studied in the context of different +experiments using argon like ICARUS [77], DBA giving 92+22 +−46 µBq/kg [78] +and, more recently, DEAP, measuring 40.4 ± 5.9 µBq/kg [73]. The content +of 42Ar could not be quantified in DarkSide-50. For UAr, 42Ar should be +considered as a potential background for neutrinoless 2β decay searches (for +example, by doping LAr with 136Xe); but the threshold for the α reaction on +40Ar is much higher than the energy of α particles from natural radioactivity, +according to cross section values from TALYS and other sources [36]. This +is also the case for other processes which could produce 42Ar underground +from the rock, like 43Ca(n,2p)42Ar or 44Ca(n,n2p)42Ar, when considering the +typical energies of radiogenic neutrons from natural fission and (α,n) reac- +tions. The production rate of 42Ar in UAr at sea level from fast neutrons and +high energy muons and protons has been evaluated by Geant4 simulation as +5.8×10−3 atoms/kg/day in Ref. [36]; this rate would give a saturation activity +about three orders of magnitude lower than measured values in AAr. Taking +all this into account, the effect of 42Ar in DarkSide-20k will not be consid- +ered here although a specific study to quantify radiogenic and cosmogenic +production in the Earth’s crust is underway6. +3H in the detector medium of a dark matter experiment can be a very +relevant background source due to its decay properties: it is a pure β− emit- +ter with transition energy of 18.6 keV and a long half-life of 12.3 y [66]. The +quantification of its cosmogenic production is not easy, neither by calcula- +tions (3H can be generated by different reaction channels) nor experimentally +(the β emissions are hard to disentangle from other background contribu- +6https://indico.sanfordlab.org/event/29/contributions/487/ +23 + +tions). Estimates of the 3H production rate in several dark matter targets +were attempted in Ref. [79]; the rate has been measured for germanium from +EDELWEISS [19] and CDMSlite [21] data and for silicon and NaI(Tl) from +neutron irradiation [24, 28]. Possible presence of 3H has been observed also +in NaI(Tl) crystals by ANAIS [25, 80] and COSINE experiments [27, 81]. In +principle, purification systems for LAr may remove all non-noble radionu- +clides and 3H should not be a problem for DarkSide. This was also assumed +for liquid xenon, but 3H was considered as a possible explanation for the ex- +cess of electronic recoil events observed in the XENON1T experiment below +7 keV [82, 83], which has disappeared in XENONnT [5]. Activated 3H is +separated from argon with SAES Getters [84] and will be removed in situ +while the UAr recirculates. +Production of other radioisotopes with half-lives longer than 10 days in +argon was predicted by using the COSMO code, like 7Be and 22Na (giving γ +emissions) and 32,33P and 35S (being pure β− emitters); production rates at +sea level from fast neutrons, high energy muons and protons have been eval- +uated by Geant4 simulation in Ref. [36]. Assuming an efficient purification +of non-noble isotopes, they will not be considered in this study. +6.2. Production rates +The production rates of 37Ar and 39Ar from cosmic neutrons at sea level +were measured for the first time through controlled irradiation at Los Alamos +Neutron Science Center (LANSCE) with a neutron beam resembling the cos- +mic neutron spectrum and later direct counting with sensitive proportional +counters at Pacific Northwest National Laboratory (PNNL) [35]. Samples +of both AAr and UAr were irradiated. In addition, the study of other pro- +duction mechanisms due to muon capture, cosmic protons and high energy +γ rays at the Earth’s surface was made using available cross sections to com- +pute total production rates at sea level. The production rates obtained in +Ref. [35] for UAr are reproduced in Table 9 as they will be used to evalu- +ate the induced activity in DarkSide-20k. In addition, the production rates +of both 37Ar and 39Ar at sea level have been recently evaluated by Geant4 +simulation in Ref. [36] too. +The UAr to be used in DarkSide-20k is obtained in Colorado, which is +placed at a quite high altitude; then, the corresponding correction factors +f to the cosmic ray flux must be taken into consideration. +In Ref. [53], +high values of f are reported for neutrons at Colorado locations: 4.11 and +12.86 for Denver (at 5280 feet) and Leadville (at 10200 feet), respectively. +24 + +Table 6: Calculation of the correction factor f to be applied to the cosmic neutron flux at +sea level (in New York) for the location of the Urania facilities in Colorado. The relative +intensities I are derived from Eq. 3. The final factor for Urania is the average between +the deduced ones from Denver and Leadville data. +Location +H +A +f +Relative I +Deduced f +(ft) +(g/cm2) +from Ref. [53] +to Urania +for Urania +Denver +5280 +852.3 +4.11 +0.659 +6.24 +Leadville +10200 +705.2 +12.86 +1.942 +6.62 +Urania +7100 +795.5 +6.43 +These correction factors f have been adjusted to the altitude at the Urania +facilities (at 7100 feet), assuming that the ratio of f for different altitudes is +the same than the ratio of cosmic flux intensities. As described in Ref. [53], +the intensities I1 and I2 at two different altitudes A1 and A2 (converted to +g/cm2) are related as: +I2 = I1 exp[(A1 − A2)/L], +(3) +being L the absorption length for the cosmic ray particles. Calculations for +the cosmic neutron flux correction factor are summarized in Table 6, using +L =136 g/cm2; the final result for Urania is the average between the deduced +ones from Denver and Leadville data, f =6.43. +For cosmic protons and +muons, the correction factors have been obtained just from Eq. 3 considering +the corresponding absorption lengths (L = 110 g/cm2 for protons and L = +261 g/cm2 for muons [53]); the results are f = 8.67 for protons and f = 2.48 +for muons. +Following Eq. 2, a calculation of the production rates of relevant iso- +topes in argon (assuming 100% 40Ar) by cosmic neutrons from Ref. [52] has +been made considering a selection of excitation functions from libraries and +YIELDX calculations. Figure 4 shows the available information on produc- +tion cross sections of 3H, 37Ar and 39Ar by nucleons. For 39Ar, although no +experimental data at EXFOR was found for the total production cross sec- +tion, there are results for partial (n,2nγ) reactions in natural argon at 1-30 +MeV taken from Ref. [85]. For 3H, an irradiation experiment with neutrons +having an energy spectrum peaked at 22.5 MeV measured the corresponding +production cross section [86]. +The matching of the cross section data from different libraries, focused +on different energy ranges, is not good. +Several descriptions of the cross +25 + +0.1 +1 +10 +100 +1000 +10 +100 +1000 +10000 +Production cross section (mb) +Energy (MeV) +TENDL n +TENDL p +JENDL n +JENDL p +JENLD-HE n +JENDL-HE p +HEAD2009 +Qaim'78 Ar(n,X)t +1 +10 +100 +1000 +10 +100 +1000 +10000 +Production cross section (mb) +Energy (MeV) +TENDL p +JENDL n +JENDL p +JENDL-HE n +JENDL-HE p +HEAD2009 +YIELDX +1 +10 +100 +1000 +10 +100 +1000 +10000 +Production cross section (mb) +Energy (MeV) +TENDL n +TENDL p +JENDL n +JENDL p +JENDL-HE n +JENDL-HE p +HEAD2009 +YIELDX +(n,2ng), Eg=250 keV +(n,2ng), Eg=1267 keV +Figure 4: Production cross sections of 3H (top), 37Ar (medium) and 39Ar (bottom) in +40Ar by nucleons taken from different sources. +26 + +sections, even from different libraries below and above a particular energy cut, +have been considered to estimate the corresponding uncertainty; the obtained +maximum and minimum rates define an interval, whose central value and +half width have been considered as the final result and its uncertainty for the +evaluation of the production rates. Table 7 presents the obtained results for +37Ar and 39Ar, together with the measured production rate for fast neutrons +and different calculations from Refs. [35, 36]. The production rate of 39Ar +derived here is fully compatible with the measured value (and with several of +the calculations in Ref. [35]). The production rate of 37Ar is a factor 2 higher +than the measured one, but lower than the Geant4 estimate in Ref. [36]. +For calculating final activity yields of 37Ar and 39Ar, the values of the total +production rates obtained in Ref. [35] will be used; but this comparison can +be useful to assess the reliability of the production rates of isotopes estimated +only from calculations, like is the case of 3H in argon. +Evaluations of the production rate of 3H for several targets were applied +also for argon, using different codes like TALYS [16] and Geant4 and AC- +TIVIA [33]. It was also computed in Ref. [79] for cosmic neutrons, from +a selection of excitation functions considering the TENDL and HEAD2009 +libraries, following the same approach applied here; this study was cross- +checked against experimental data for NaI and germanium, reproducing prop- +erly measured production rates [19, 21, 28]. Now, new data for neutron cross +sections taken from the JENDL-HE library have been added in the analysis +in this work (giving a production rate of 221.6 atoms/kg/day) and then the +final production rate has been re-evaluated considering all the other previ- +ous estimates in Ref. [79] as (168±53) atoms/kg/day. It must be noted that +this value gives only production by neutrons; assuming equal flux and cross +sections of protons and neutrons above 1 GeV, it is estimated that protons +would increase the rate by 10% at most. Table 8 summarizes all the results +for 3H production in argon; an important dispersion of values is found. +27 + +Table 7: Calculations of the production rates R of 37Ar and 39Ar in Ar at sea level from this work considering different +descriptions of the excitation functions below (LE) and above (HE) a cut energy value; the final estimated rates are given by +the ranges defined between the maximum and minimum obtained rates (see text). Different calculations from the literature +(considering the same cosmic neutron spectrum from Ref. [52]) and the measured value for fast neutrons from Ref. [35] are +also shown for comparison. +37Ar +39Ar +This work +Cut +R +This work +Cut +R +LE+HE +(MeV) +(atoms/kg/day) +LE+HE +(MeV) +(atoms/kg/day) +TENDL(p)+HEAD2009 +150 +153.6 +TENDL+HEAD2009 +150 +726.4 +TENDL(p)+YIELDX +100 +93.5 +TENDL+YIELDX +100 +697.1 +TENDL(p)+YIELDX +200 +122.7 +TENDL+YIELDX +200 +646.0 +JENDL-HE(n) +30 +63.9 +TENDL+JENDL-HE(n) +20 +804.3 +Estimated rate in this work +109±45 +Estimated rate in this work +725±79 +Measurement [35] +51.0±7.4 +759±128 +ACTIVIA [35] +17.9±2.2 +200±25 +MENDL-2P [35] +155±19 +188±24 +TALYS [35] +76.8±9.6 +753±94 +INCL++ (ABLA07) [35] +79.3±9.9 +832±104 +TENDL-2015 [35] +726±91 +Geant4 [36] +176 +858 +28 + +Table 8: Production rate R of 3H in Ar at sea level from this work and from different +calculations from the literature. +R (atoms/kg/day) +Estimated rate in this work +168±53 +TENDL+HEAD2009 [79] +146± 31 +TALYS [16] +44.4 +Geant4 [33] +84.9 +ACTIVIA [33] +82.9 +6.3. Activity +The possible activity yields of relevant cosmogenic isotopes in Ar have +been analyzed for the DarkSide-20k detector considering Ar extraction, stor- +age and transportation and taking into account not only cosmogenic neutrons +but also other cosmic ray components. For 37Ar and 39Ar, the production +rates at sea level precisely determined with the LANSCE neutron beam and +the estimates for muons, protons and cosmic γ rays [35] have been considered, +while for 3H the production rate estimated in this work has been assumed. +It is planned to produce 120 t of UAr for DarkSide-20k, allocated as +follows: 100 t needed for filling the DarkSide-20k TPC, 4 t used during +conditioning and purging the cylinder skids, 4 t of argon left in Aria after +purification, and 12 t for contingency. The UAr extracted at the Urania +plant will be shipped firstly to the Aria facility for purification and then to +LNGS for storage and final operation. The current baseline design is to ship +the UAr in commercially available high-pressure (517 bar) gas cylinders that +are organized into skids capable of containing ∼2 t of UAr each. It is not +possible to predict accurately the final exposure conditions for the UAr, but +according to the present specifications of Urania and Aria, a baseline exposure +history with defined exposure times and places for the different steps of the +transportation process can be established and the main sources of uncertainty +in the process identified; then, activity yields have been computed for the +baseline exposure and the effect of uncertainties assessed. The following steps +are foreseen: +1. Storage of UAr at Urania: three skids will be filled before starting +transportation. Considering the time required to fill one, exposures of +8, 16 and 24 days have been assumed for each one of the three skids. +While at the Urania site, the UAr will always be on the surface while +29 + +being processed and once in the skids. The correction factors to the +sea level fluxes of cosmic neutrons, protons and muons evaluated for +Urania location in Colorado (see Sec. 6.2) have been included in this +step. +2. Trip from Urania to a shipping port: a container with the three skids +will transport the UAr from Urania to Houston by truck. An exposure +of 7 days has been considered. To take into account the different al- +titude across the trip, the average between the maximal (from Urania +altitude) and minimal (at sea level) expected activity has been calcu- +lated. +3. Trip overseas to Europe: 60 days of exposure at sea level have been +conservatively assumed for the trip by boat from Houston to Cagliari. +An additional exposure of 7 days is foreseen for custom issues and the +trip from Cagliari to the Aria location. +Steps 1 to 3 will be repeated over twenty times, running in parallel. +In total, 16 months are required for completing the extraction and +transportation of all the necessary UAr at Urania. +4. Processing and storage of UAr at Aria: once in Sardinia, the skids will +be stored near Aria and the UAr will be accumulated for processing. +At a purification rate of 1 ton per day, a minimal exposure of 120 t is +foreseen. Purified UAr will be stored locally in Sardinia until needed +for filling into DarkSide-20k. Underground storage at a depth of at +least some tens of m.w.e. would be recommended but, if not possible, +a virtually linear increase of 2.6 µBq/kg in the activity of 39Ar should +be considered per month of additional exposure at sea level. +5. Trip from Aria to LNGS: 10 days of exposure at sea level have been +considered for this trip by boat. It is expected to ship 12 t at a time +using six skids; then, this action should be repeated over ten times. +6. Storage at LNGS: skids will be stacked underground as they arrive. +All in all, under these assumptions, the total time from the beginning +of production at Urania to the end of processing at Aria is 614 days. +If +transport from Aria to LNGS starts once all the UAr has been processed, +100 additional days would be required to have all the UAr at LNGS. +Taking into account this exposure history, the induced activity by each +cosmic ray component has been computed for each one of the exposure steps +(at Urania, trip in US, overseas, at Aria and trip in Italy) from Eq. 1. Tables +9 and 10 show separately each contribution for 39Ar and 37Ar and for 3H, +30 + +respectively. The decay of the activities induced at each step during the rest +of the whole process is negligible for 39Ar and small for 3H, due to their long +half-lives, but extremely relevant for 37Ar; it is accounted for in the final +activities reported in Tables 9 and 10. +For both 39Ar and 37Ar, cosmogenic neutrons are responsible of the main +part of the induced activity. Under the assumed baseline conditions, the +relative contributions to the final 39Ar activity of each exposure step are the +following: Urania, 34.4%; US trip, 9.0%; overseas trip, 27.7%; at Aria, 24.8%; +and Italy trip, 4.1%. The exposure at Urania gives the largest contribution, +followed by that of the overseas trip and at Aria. For 37Ar, having a much +shorter half-life, the last exposure during the Italy trip is dominant, produc- +ing 55% of the final activity. Concerning 3H, the final activity in Table 10 +would apply if no purification procedure was considered; however, if a 100% +efficient removal of 3H was achieved in Aria, only the activity in the last step +for exposure in Italy would be produced. Table 11 summarizes the expected +activities once all the UAr is at LNGS. From values in Table 9, the final es- +timated activity of 39Ar is (20.7±1.5) µBq/kg; this equals 2.8% of measured +activity in DarkSide-50. For 37Ar, the effect of cooling is very important and +the expected activity when all the UAr is at LNGS is (103.0±8.6) µBq/kg. +From values in Table 10 for 3H, an activity of (2.97±0.94) µBq/kg is expected +at that time considering only activation after ideal purification in Aria; with +no purification, it would be around 25 times higher. +Uncertainties quoted for activities in Tables 9 and 10 come from those +of production rates. Concerning the correction factors of sea level cosmic +ray fluxes for exposure at Urania, it has been checked that considering a +description different to that applied in Sec. 6.2 produces very similar results; +correction factors computed from EXPACS spectra in the energy range rel- +evant for activation (1 MeV to 10 GeV) are f = 6.09 for neutrons, f = 7.60 +for protons and f = 1.61 for muons, giving a small decrease in the final activ- +ities: 1.0% for 39Ar, no change for 37Ar and 1.5% for 3H with no purification. +On the other hand, unexpected events can produce relevant deviations from +the baseline exposure conditions and their effect on the activation yields has +been assessed. +Doubling the exposure at Urania would increase the final +39Ar activity from (20.7±1.5) µBq/kg to (27.7±2.4) µBq/kg, which would +be 3.8% of the DarkSide-50 activity. Exposure at Aria has been evaluated +for the moment considering just the processing time, but activation produced +in the periods before and after the processing should be added if storage is +made above ground; to produce an additional 10% of the measured activity in +31 + +DarkSide-50 (which was determined with an uncertainty of 14%), 28 months +of additional exposure would be required, which is well above the period of +16 months needed for the extraction of the whole amount of UAr needed. +All in all, it can be concluded that there is enough contingency in the plan +for production, storage and shipping of the UAr so that cosmogenic 39Ar +activity does not endanger DarkSide-20k sensitivity. +7. Expected counting rates in DarkSide-20k +As described in Secs. 4.2 and 5.3, 54Mn in stainless steel and 46Sc in +titanium are identified as the most relevant cosmogenic products in these +materials. The effect of their contribution to the γ background of the ex- +periment has been evaluated finding for the former a negligible contribution +in comparison to the other sources of γ background. In a hypothetical de- +tector using a titanium vessel and considering the saturation activity when +going underground, 46Sc would add (0.41±0.10) Hz and (21.1±5.3) Hz, re- +spectively, to the estimated counting rates in the TPC and inner veto (see +Table 11). +The rates from the estimated cosmogenic activity of products in UAr, +under the assumed baseline exposure conditions, are also shown in Table 11. +Induced 39Ar due to the whole exposure from Urania to LNGS would add a +rate of (1.035±0.075) Hz for the TPC. The contribution of 3H to the TPC +counting rate is negligible (around 0.15 Hz) provided an efficient purification +at Aria is achieved while that of 37Ar (being (5.15±0.43) Hz if data taking +started just immediately after the arrival of all the UAr at LNGS) will de- +cay very quickly. Comparing these numbers with the total β and γ rates +presented in Sec. 2.2, it can be concluded that cosmogenic activity does not +produce a problematic increase of the TPC and Veto rates. +8. Conclusions +For DarkSide-20k, material cosmogenic activation is a source of β/γ back- +ground and it has been quantified for LAr and other massive components +from realistic exposure conditions in order to assess the contribution to the +counting rates and decide if additional exposure restrictions are necessary. +Main results are summarized in Table 11. +32 + +Table 9: Calculation of the expected induced activity in kg−1 d−1 of 39Ar and 37Ar in the UAr of the DarkSide-20k detector, +for the assumed production rates R and exposure times (see text). Different columns and rows show separate contributions by +cosmic ray components and exposure steps, respectively; relative contributions of each component to the total activity are also +quoted. Row labelled as ”Final” presents the sum of final activities from all exposure steps including properly their decays. +39Ar +Neutrons +Muons +Protons +γ rays +Total +R (atoms/kg/day) [35] +759±128 +172±26 +3.6±2.2 +112.8±20.9 +Urania +0.551±0.093 +0.0483±0.0073 +0.0035±0.0022 +0.0127±0.0024 +0.616±0.093 +US +0.139±0.024 +0.0148±0.0022 +0.0009±0.0005 +0.0056±0.0010 +0.161±0.024 +Overseas +0.359±0.061 +0.081±0.012 +0.0017±0.0010 +0.053±0.010 +0.495±0.063 +Aria +0.321±0.054 +0.073±0.011 +0.0015±0.0009 +0.048±0.0088 +0.444±0.056 +Italy +0.0536±0.0090 +0.0121±0.0018 +0.0003±0.0002 +0.0080±0.0015 +0.0739±0.0093 +Final +1.42±0.13 +0.229±0.018 +0.0078±0.0026 +0.127±0.014 +1.79±0.13 +(%) +79.6 +12.8 +0.4 +7.1 +37Ar +Neutrons +Thermal neutrons +Protons +γ rays +Total +R (atoms/kg/day) [35] +51±7.4 +0.9±0.3 +1.3±0.4 +3.5±0.7 +Urania +87±13 +2.99±0.92 +0.93±0.19 +0.239±0.080 +91±13 +US +24.5±3.6 +0.81±0.25 +0.453±0.091 +0.116±0.039 +25.9±3.6 +Overseas +37.5±5.4 +0.95±0.29 +2.57±0.51 +0.66±0.22 +41.7±5.5 +Aria +35.5±5.1 +0.90±0.28 +2.43±0.49 +0.63±0.21 +39.4±5.2 +Italy +9.2±1.3 +0.234±0.072 +0.63±0.13 +0.162±0.054 +10.2±1.3 +Final +8.03±0.74 +0.209±0.040 +0.524±0.070 +0.135±0.030 +8.90±0.75 +(%) +90.3 +2.3 +5.9 +1.5 +33 + +Table 10: Calculation of the expected induced activity in kg−1 d−1 of 3H by cosmic +neutrons in the UAr of the DarkSide-20k detector, for the production rate R estimated in +this work and the assumed exposure times (see text), considering no purification procedure. +Different rows show separate contributions by exposure steps. Row labelled as “Final” +presents the sum of final activities from all exposure steps including properly their decays +3H +Neutrons +R (atoms/kg/day) +168±53 +Urania +2.66±0.84 +US +0.67±0.21 +Overseas +1.73±0.54 +Aria +1.55±0.49 +Italy +0.259±0.082 +Final +6.5±1.1 +34 + +Table 11: Summary table of estimated activation in DarkSide-20k including isotope, material, main production channel, +calculation details, overall activity and counting rates in TPC and inner veto. All reported activity and rate values correspond +to the moment when the materials are brought underground. +For 3H, row (1) and (2) assume no purification and ideal +purification at Aria, respectively. +Isotope +Material +Main channel +Calculation +Activity +TPC rate +Veto rate +(µBq/kg) +(Hz) +(Hz) +39Ar +UAr +40Ar(n,2n)39Ar +Production rates from [35] +20.7±1.5 +1.035±0.075 +0.662±0.048 +37Ar +UAr +40Ar(n,4n)37Ar +Production rates from [35] +103.0±8.6 +5.15±0.43 +3.30±0.28 +3H (1) +UAr +40Ar(n,*)3H +σ(E) in Fig. 4+Gordon spectrum +76±12 +3.80±0.60 +2.43±0.38 +3H (2) +UAr +40Ar(n,*)3H +σ(E) in Fig. 4+Gordon spectrum +2.97±0.94 +0.148±0.047 +0.095±0.030 +35 + +For copper and stainless steel components, activation yields of isotopes +with relevant half-lives (like 54Mn, 57Co and 60Co) have been computed from +measured production rates at sea level at Ref. [38]. In copper, even for 10 y +of exposure to cosmic rays, estimated activities are below 0.5 Bq. In stainless +steel, hundreds of Bq are expected for some isotopes for just 1 y exposure; +the contribution to the counting rate of ER-like events in the TPC from +54Mn activity induced in steel components has been found to be negligible in +comparison to the estimated total rate from β/γ backgrounds. This allows +to relax additional limitations on the surface residency time. +For natural titanium, 46Sc has been identified as the main cosmogenic +product. Other radioisotopes induced are not considered as a potential rele- +vant background due to their half-lives or because their short-range emissions +are not expected to escape from titanium. The production rate at sea level +of 46Sc has been calculated from a selection of production cross sections +and considering the Gordon et al parametrization [52] for the cosmic neu- +tron spectrum, deriving a value of (271±68) atoms/kg/day, which is in very +good agreement with totally different estimates based on modified COSMO, +Geant4 simulation and the ACTIVIA code. The corresponding saturation +activity is (3.14±0.79) mBq/kg, in the range of most of the measurements +of 46Sc activity in samples found in the literature. Assuming exposure at sea +level for a long time, this saturation activity has been conservatively consid- +ered to quantify by MC simulation the possible effect of 46Sc emissions on +the ER background rate of DarkSide-20k if titanium was used, showing a +contribution to the TPC counting rate which is non-relevant, specially when +taking into account the cooling down underground before the start of the +data taking. +A total of 120 t of UAr depleted in 39Ar must be extracted and processed +for filling the TPC and inner veto of DarkSide-20k. The possible induced +activity on surface, from the extraction at Urania to the storage at LNGS, +has been analyzed not only for 39Ar but also for 37Ar and 3H. Production +rates from Ref. [35], based on a neutron irradiation experiment, have been +considered for the Ar isotopes while for 3H an estimate of the production rate +by cosmic neutrons made in this work obtaining (168±53) atoms/kg/day has +been used. The estimated cosmogenic activity of 39Ar when all the UAr ar- +rives to LNGS, (20.7±1.5) µBq/kg for the assumed baseline exposure history, +is considered acceptable as it is just 2.8% of the residual activity measured +in DarkSide-50 for UAr of the same source and would add ∼1 Hz to the +counting rate of the TPC. The quantified effect of some uncertain steps in +36 + +the procedure of UAr production shows that there is enough contingency. +Contributions from the induced activity of 37Ar and 3H are not problematic +thanks to short half-life and purification, respectively. The results of this +study of the cosmogenic activation of UAr will be useful to set exposure limi- +tations for the procurement of the large amounts of radiopure UAr necessary +in future LAr projects. +Acknowledgements +This report is based upon work supported by FSC 2014-2020 - Patto +per lo Sviluppo, Regione Sardegna, Italy, the U. S. National Science Foun- +dation (NSF) (Grants No. PHY-0919363, No. PHY-1004054, No. PHY- +1004072, No. PHY-1242585, No. PHY-1314483, No. PHY- 1314507, as- +sociated collaborative grants, No. PHY-1211308, No. PHY-1314501, and +No. PHY-1455351, as well as Major Research Instrumentation Grant No. +MRI-1429544), the Italian Istituto Nazionale di Fisica Nucleare (Grants +from Italian Ministero dell’Istruzione, Universit`a, e Ricerca Progetto Pre- +miale 2013 and Commissione Scientific Nazionale II), the Natural Sciences +and Engineering Research Council of Canada, SNOLAB, and the Arthur B. +McDonald Canadian Astroparticle Physics Research Institute. We acknowl- +edge the financial support by LabEx UnivEarthS (ANR-10-LABX-0023 and +ANR18-IDEX-0001), the S˜ao Paulo Research Foundation (Grant FAPESP- +2017/26238-4), Chinese Academy of Sciences (113111KYSB20210030) and +National Natural Science Foundation of China (12020101004). +The au- +thors were also supported by the Spanish Ministry of Science and Inno- +vation (MICINN) through the grant PID2019-109374GBI00, the “Atraccion +de Talento” Grant 2018-T2/ TIC-10494, the Polish NCN, Grant No. UMO- +2019/ 33/ B/ ST2/ 02884, the Polish Ministry of Science and Higher Ed- +ucation, MNi-SW, grant number 6811/IA/SP/2018, the International Re- +search Agenda Programme AstroCeNT, Grant No. MAB-/2018/7, funded by +the Foundation for Polish Science from the European Regional Development +Fund, the European Union’s Horizon 2020 research and innovation program +under grant agreement No 952480 (DarkWave), the Science and Technology +Facilities Council, part of the United Kingdom Research and Innovation, +and The Royal Society (United Kingdom), and IN2P3-COPIN consortium +(Grant No. 20-152). 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experiments investigating rare event phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' DarkSide-20k is a direct detection experiment for galactic dark matter particles, using a two-phase liquid argon time projection chamber filled with 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7 tonnes (active mass) of Underground Argon (UAr) depleted in 39Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Here, the cosmogenic activity of relevant long-lived radioisotopes induced in the argon and other massive components of the set-up has been estimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' production of 120 t of radiopure UAr is foreseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The expected exposure above ground and production rates, either measured or calculated, have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From the simulated counting rates in the detector due to cosmogenic isotopes, it is concluded that activation in copper and stainless steel is not problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activation of titanium, considered in early designs but not used in the final design, is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The activity of 39Ar induced during extraction, purification and transport on surface, in baseline conditions, is evaluated to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8% of the activity measured in UAr from the same source, and thus considered acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Other products in the UAr such as 37Ar and 3H are shown to not be relevant due to short half-life and assumed purification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Keywords: Cosmogenic activation, Argon, Dark matter, Rare events 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Introduction Great efforts have been devoted worldwide to unravel the nature of the dark matter [1] which could be pervading the galactic halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' One of the strate- gies followed is the search for Weakly Interacting Massive Particles (WIMPs) by direct detection via WIMP-nucleus elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This required mak- ing use of different kinds of very sensitive radiation detectors [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Noble elements like xenon and argon, being excellent scintillators and easily ionized, are ideal targets and massive experiments based on this detection technique presently have a leading role [4–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The expected counting rates from the interaction of WIMPs are extremely low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' therefore, dark matter detectors require ultra-low background condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Operating in deep underground locations, using active and passive shielding, carefully selecting radiopure materials, and developing background- 5 rejection methods in analysis are necessary for rare events experiments [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In this context, long-lived radioactive isotopes induced in the materi- als of the experiment by the exposure to cosmic rays at the earth’s surface (during fabrication, transport and storage) can be as relevant as residual contamination from primordial nuclides, and they may be very problematic, depending on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In principle, cosmogenic activation can be kept under control by minimizing exposure at surface and storing materials un- derground, avoiding flights, and even using shielding against the hadronic component of cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' But since these requirements usually complicate the preparation of experiments, it would be desirable to have reliable es- timates of activation yields to assess the real danger of exposing materials to cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Direct assay measurements of exposed materials, in very low background conditions, and calculations of production rates and yields, following different approaches, have been made for several materials in the context of dark matter, 2β decay, and neutrino experiments [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Results have been derived in the last years for detector media like germanium [15–23], silicon [24], NaI [20, 25–28], tellurium and TeO2 [29–31], xenon [32–34], ar- gon [20, 35, 36] and molybdate [37] as well as for materials commonly used in the set-ups like copper [18, 32, 33, 38, 39], lead [40] or stainless steel [33, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Liquid Argon (LAr) offers important advantages for radiation detection, like a high scintillation yield and easy purification from non-noble contam- inants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Radioactive backgrounds that produce electron recoils (ER) can be discriminated from nuclear recoil (NR) events, typically expected from WIMPs, as there is significant difference between the time distribution of their scintillation signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this provides an outstanding Pulse Shape Discrim- ination (PSD) power, as shown by the single-phase LAr detector DEAP-3600 with a rejection factor over 108 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Dual-phase Time Projection Chambers (TPCs) in which both the primary scintillation and the electroluminescence from electrons are detected have additional capabilities like better position resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this was the approach of the DarkSide-50 experiment carried out at the Laboratori Nazionali del Gran Sasso (LNGS) in Italy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' DarkSide-50 used Underground Argon (UAr), depleted of 39Ar by a factor 1400±200 with respect to the Atmospheric Argon (AAr) activity of ∼1 Bq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' DarkSide-50 demonstrated the dual-phase method also allows for a sensitive search for lighter WIMPs [9, 10] using the electroluminescence signal alone to obtain a lower energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Despite these excellent background discrimination capabilities, accep- tance losses via ER + NR pile-up in the TPC or accidental coincidence 6 between signals from the TPC and a veto detector that mimic the neutron capture signature can be produced by γ or β emitters in the set-up and therefore these background sources must be carefully considered too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The goal of this work is to quantify in particular the effect of cosmogenic activa- tion of detector materials on the expected counting rates of the DarkSide-20k detector, considering exposure on the Earth’s surface under realistic condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This allows requirements and procedures during the preparation and commissioning of the experiment to be set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The study has been carried out for the UAr acting as detector medium as well as for copper, and stainless steel, since the use of large quantities of these materials was foreseen in dif- ferent components, according to the design of DarkSide-20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The paper is structured as follows: the DarkSide-20k project is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the methodology applied to quantify cosmogenic activities is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3, showing the obtained results for different materials in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4, 5 and 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the counting rates expected from these activities are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 7, before summarizing conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The GADMC and the DarkSide-20k detector Following the success of several LAr dark matter experiments, the Global Argon Dark Matter Collaboration (GADMC) has been established to operate detectors pushing the sensitivity for WIMP detection down to the neutrino floor [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' As a first step, the DarkSide-20k experiment will be operated in the Hall C of LNGS with ∼20 t of UAr in the fiducial volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the data taking is intended to start in 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The ARGO detector, increasing this volume to ∼360 t, is foreseen towards the end of this decade with a target exposure of several thousand t·y at SNOLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' ARGO would also have excellent sensitivity to CNO neutrinos and galactic supernovae [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' GADMC One of the goals of GADMC is the procurement of large amounts of low-radioactivity UAr as detector target, which is essential to achieve the scientific goals as the content of 39Ar in AAr would be intolerable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' three projects are in development to ensure this: Extraction of argon with a naturally low concentration of radioactive 39Ar from an underground source (CO2 wells) will be carried out at the Urania plant, in Cortez, CO (US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This is the same source of UAr used for the DarkSide-50 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 7 UAr will be further chemically purified to detector-grade argon in the Aria facility, in Sardinia (Itay);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Aria will have a 350 m cryogenic dis- tillation column, called Seruci-I, currently being commissioned [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Assessing the ultra-low 39Ar content of the UAr is crucial for the GADMC projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This is the goal of the DArT detector [45], us- ing a small chamber placed at the centre of the ArDM detector in the Canfranc Underground Laboratory (LSC) in Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It aims to measure 39Ar below the mBq/kg level with 10% precision in one week of run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' There is a growing interest in the use of ultra-pure UAr also outside GADMC, as it has potential broader applications for measuring coherent neutrino scattering, environmental assay, neutrinoless 2β decay, and large DUNE-like detectors [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the challenges for its production and characteriza- tion are carefully addressed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' DarkSide-20k In DarkSide-20k the core of the apparatus is a dual-phase TPC, serving both as WIMP target and detector, filled and surrounded by low-radioactivity UAr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' a total of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 t of UAr is required, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 t inside the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The TPC has three dimensional space reconstruction capability that permits the defi- nition of a wall-less fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' SiPMs read the prompt scintillation in the liquid (S1) and delayed electroluminescence in the gas phase (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The TPC is contained in a gadolinium-loaded acrylic vessel (Gd-PMMA), which moderates and captures neutrons after they scatter in the TPC and produce a WIMP-like signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' γ-rays produced by the neutron capturing are detected in the UAr veto surrounding the Gd-PMMA vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The TPC is shaped as prism with octagonal base with a vertical drift length of 348 cm and an octag- onal inscribed circle diameter of 350 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The anode and cathode plates are realized by pure acrylic, PMMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' A total of 8448 and 1920 Photo-Detector Modules (PDMs) view the TPC volume and the inner veto, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The inner veto is housed within a vessel, made of stainless steel, immersed in a bath of 700 t of AAr acting as shield and outer veto detector for muons and associated products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The AAr is contained in a ProtoDUNE-like membrane cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' All the materials used to build the whole detector system are care- fully selected for low levels of radioactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Figure 1 shows cross views of the cryostat and the inner detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' A large amount of Gd-loaded PMMA, of the order of 11 t, is foreseen, but no hint of cosmogenic isotopes has been found 8 Figure 1: Cross sections of the cryostat of the outer veto (left) and of the vessel containing the inner veto and TPC (right) of the DarkSide-20k detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' OP stands for Optical Plane and PDU for Photo Detection Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' in the radiopurity measurements by γ spectroscopy performed for acrylic and Gd2O3 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Thus activation of this material has not been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Ta- ble 1 lists materials, masses and considered cosmogenic isotopes for the main components in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' DarkSide-20k is being designed to operate with <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 events over a 200 t·y exposure, thanks to the powerful PSD for ER background and the neutron veto capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For a fiducial mass of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 t, the projected sensitivities are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6×10−48 cm2 for the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' exclusion and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5×10−47 cm2 or the 5σ dis- covery of a 1 TeV/c2 WIMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In parallel, a much smaller detector specifically optimized for the investigation of low-mass dark matter, DarkSide-LowMass, is being considered [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Combined, DarkSide-20k, DarkSide-LowMass and ARGO will completely cover the spin-independent WIMP hypothesis param- eter space down to the irreducible neutrino background for WIMP masses from 1 GeV/c2 to several hundreds of TeV/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Planck-scale mass dark mat- ter, already investigated by DEAP-3600 [50], would also require large detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The achievement of an extremely low background rate requires thorough background studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' G4DS [51] is a Monte Carlo (MC) simulation frame- 9 Ti (or SS) vessel Top OP vPDU cm TPC Barre!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Calibration 350 cm Pipe Bottom Op Bottom OpTable 1: Materials and masses of the main components considered in the design of the DarkSide-20k detector shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmogenic isotopes considered for each material in this work are also indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Component Material Mass Induced isotopes Membrane cryostat Stainless steel 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 t See Table 2 Outer Veto: filling AAr 700 t 37Ar, 39Ar, 3H Inner Veto: vessel Stainless steel 12 t See Table 2 TPC: barrel Gd-loaded PMMA 11 t TPC: grids, frame, brackets Stainless steel 1055 kg See Table 2 TPC: cables Copper 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 kg See Table 2 Inner Veto+TPC: filling UAr 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 t 37Ar, 39Ar, 3H Electronic components Copper 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 kg See Table 2 work developed for DarkSide based on Geant4 providing accurate simulation of light production, propagation, and detection for background and signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It is designed with a modular architecture in order to include a full description of different detectors: DarkSide-50, DarkSide-20k and ARGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' G4DS has been extensively validated on DarkSide-50 data [51], demonstrat- ing the high accuracy required to optimize the geometry and establish the performance of the TPC and the neutron and muon vetoes for DarkSide- 20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For the TPC, it fully reproduces the responses of the detector in S1, S2, and time, the three primary variables on which the discrimination of β/γ background is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Great efforts have been devoted to the description of the physical properties of materials, especially the optical ones, exploited by G4DS to track each photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' A spatial event generator is implemented for each detector material in order to generate and track particles emitted by radioactive decays and to assess its impact on the DarkSide-20k background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In addition, an (α,n) event generator, based on the TALYS package1, is im- plemented to study the impact of the neutron background and the MC chain is completed by the electronics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' γ emissions from the full set of detector components have been simulated to estimate the corresponding background rates in the TPC and in the Veto from activities measured in an extensive material screening campaign based on the combination of different radioassay techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' discrimination techniques based on energy and posi- tion of the interactions are implemented to compute the rate in the fiducial 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='talys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='eu 10 volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Preliminary estimates of γ background rates point to values around 50 Hz in the TPC and 100 Hz in the Veto, with dominant contribution from PDMs and, to a lesser extent, from Gd-loaded acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The β contribution of 39Ar, considering the total active mass of UAr in the TPC (50 tonnes) and in the inner veto (32 tonnes) and the measured activity value in DarkSide-50, produces 36 Hz in the TPC and 26 Hz in the Veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Methodology One of the most relevant processes in the production of radioactive iso- topes in materials is the spallation of nuclei by high energy nucleons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' other reactions like fragmentation, induced fission or capture can be important for some nuclei too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' On Earth’s surface, as the proton to neutron ratio in cosmic rays decreases significantly at energies below the GeV scale because of the absorption of charged particles in the atmosphere, activation by neutrons is usually dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmogenic production of radionuclides underground can be considered in many cases negligible, as the flux of cosmic nucleons is suppressed by more than four orders of magnitude for depths of a few tens of meter water equivalent (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=') [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Radiogenic neutrons, with fluxes in deep underground facilities orders of magnitude lower than that of cos- mic neutrons on surface, have in addition energies (around a few MeV) too low to produce spallation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activation underground can be induced by muons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' muon spallation (virtual photon nuclear disintegration) and elec- tromagnetic and nuclear reactions from secondary particles are the relevant processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' As the muon energy spectra and fluxes depend on depth, under- ground activation can be very different for different sites and may impose a minimum required depth if on-site activation is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' To quantify the effect of material cosmogenic activation in a particular experiment, the first step is to know the production rates, R, of the relevant isotopes induced in the material targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Then, the produced activity, A, can be estimated according to the exposure history to cosmic rays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' for instance, considering just a time of exposure texp followed by a cooling time (time spent underground once shielded from cosmic rays) tcool, for an isotope with decay constant λ, the activity can be evaluated as: A = R[1 − exp(−λtexp)] exp(−λtcool).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' (1) Finally, the counting rate generated in the detector by this activity can be computed using G4DS [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 11 Some direct measurements of production rates at sea level have been carried out for a few materials from the saturation activity, obtained by sensitive screening of samples exposed in well-controlled conditions or by irradiating samples in high flux particle beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' However, in many cases, production rates must be evaluated from the flux of cosmic rays, φ, and the isotope production cross-section, σ, being both ingredients dependent on the particle energy E: R = Nt � σ(E)φ(E)dE, (2) with Nt the number of target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The spread for different calculations of productions rates is usually important, even within a factor 2 (see for instance Tables 5, 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In this work, measured production rates have been used whenever available and dedicated calculations have been performed otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmic ray flux An analytic expression for the cosmic neutron spectrum at sea level is presented by Gordon et al in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52], deduced by fitting data from a set of measurements for energies above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' with this parameterization, the integral flux from 10 MeV to 10 GeV is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6×10−3cm−2s−1 (for condi- tions of New York City).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [53], a similar parametrization is provided as well as correction factors, f, to the flux when considering exposure at different locations, as it depends on the altitude and geomagnetic rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For example, outside LNGS at an altitude of ∼1000 m, a correction factor f =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 was estimated [18] and should be considered in case of exposure to cosmic rays there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Alternatively, the EXPACS (“EXcel-based Program for calculating Atmospheric Cosmic-ray Spectrum”) program2 could be used to calculate fluxes of nucleons, muons, and other particles for different posi- tions and times in the Earth’s atmosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' in this way, possible temporal variations of the cosmic rays fluxes are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Although precise EXPACS calculations are being considered, results presented here are based on the parameterization from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52] and correction factor from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production cross sections Concerning the production cross sections, both measurements at fixed energies and calculations using different computational codes must be taken 2EXPACS: https://phits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jp/expacs/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 12 into account to choose the best description of the excitation functions σ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The following sources of data have been considered in this work: The Experimental Nuclear Reaction Data database (EXFOR, CSISRS in US) [54] provides nuclear reaction data and then measured produc- tion cross sections for a particular target, projectile, energy, or reaction whenever available3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The Silberberg and Tsao equations presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [55–57] are semiem- pirical formulae derived from proton-induced reactions for targets with mass number A ≥ 3, for products with A ≥ 6 and for energies >100 MeV and integrated in different codes: COSMO (FORTRAN program) [58], YIELDX (FORTRAN routine, including the latest updates of the equa- tions) [57] and ACTIVIA (C++ computer package, using also experi- mental data when available) [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The MC simulation of the interaction between nucleons or other pro- jectiles and nuclei allows also computation of production cross sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Many different models and codes have been developed and val- idated considering the relevant processes (the formation and decay of compound nuclei, the intranuclear cascade of nucleon interactions, de- excitation processes like fission, fragmentation, spallation, or breakup) [60];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' some of these models have been implemented in general-purpose codes like Geant4 [61] or FLUKA [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Evaluated libraries of produc- tion cross sections have been elaborated, covering different types of re- actions or projectiles and different energies, like TENDL (TALYS-based Evaluated Nuclear Data Library)4 [63] (based on the TALYS code, for protons and neutrons with energies up to 200 MeV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' JENDL (Japanese Evaluated Nuclear Data Library) [64] High Energy File5 (based on the GNASH code, for protons and neutrons from 20 MeV to 3 GeV) is an extension of the JENDL-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0/HE library including results up to 200 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' HEAD-2009 (High Energy Activation Data) [65] (for protons and neutrons with higher energies, from 150 MeV up to 1 GeV) uses a 3EXFOR: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='nndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='gov/exfor/exfor.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='ch/tendl 2019/tendl2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='html 5JENDL HE library, https://wwwndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jp/ftpnd/jendl/jendl40he.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='html;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' https://wwwndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='jp/jendl/jendl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='html 13 selection of models and codes (CEM, CASCADE/INPE, MCNP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=') dictated by an extensive comparison with EXFOR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmogenic yields in Copper and Steel As in many experiments, a significant amount of copper and stainless steel are used in DarkSide detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' both materials are known to become activated and different specific studies on their activation are available [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The effect on DarkSide-20k of cosmogenic activity in the components made of copper and stainless steel is analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production rates The production rates of the radionuclides typically induced in these mate- rials have been selected from measured and calculated results available in the literature [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Estimates using ACTIVIA, Geant4, and TALYS codes, among others, have been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Saturation activities have been measured with sensitive germanium detectors in samples of copper [32, 38, 39] and steel [38], exposed for long times to cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In particular, in this work, the production rates from dedicated measurements, using 125 kg of copper provided by Norddeutsche Affinerie (now Aurubis) exposed for 270 days at Gran Sasso and Nironit stainless steel exposed for 314 days, have been con- sidered [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' values are reproduced in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Among the different products identified in copper, 60Co has the longest half-life and, unfortunately, there is a significant disagreement on the production rates estimated for it [13, 14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the measured value in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [38] is higher than most of all the other estimates by a factor of up to a few times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' No assessment of 60Co production in stainless steel could be made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [38], as, being this isotope a typical contaminant in steel, its cosmogenic activity is obscured by previous contamination at similar level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, the rate derived from Geant4 calculations [33] has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Following the half-lives of the different cosmogenic isotopes identified in copper and steel (also shown in Table 2), 54Mn, 57Co and 60Co could be in principle the most relevant products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activity To assess the possible effect of the cosmogenic isotopes in these mate- rials for DarkSide-20k, activity A has been evaluated considering the se- lected production rates at sea level, tcool =0 and extreme cases of exposure: texp =1 month, texp =1 year and texp =10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It is worth noting that as 14 measured production rates have been taken into account, the deduced acti- vation corresponds to all cosmic ray particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The final expected activity is obtained from the specific activities derived from the production rates (per mass unit) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 1 and the mass of all the components used in the ex- perimental set-up, which according to the present design of DarkSide-20k are 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 kg of copper (mainly from cables and PDMs electronic components) and 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='655 tons of stainless steel (mainly from cryostat components) plus 12 tonnes from the inner detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 2 summarizes the total induced activity in copper and stainless steel, respectively, for the relevant isotopes evaluated at the end of the differ- ent exposure times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' contribution from each individual component is propor- tional to its mass (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Following the decay mode of these nuclei, γ emissions of the order of 1 MeV will be generated around the active vol- ume by this cosmogenic activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In the case of copper, even assuming 10 years of exposure, the total activity is at the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The deduced activities can be compared with available measurements from radioassays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' for the copper from the Luvata company being considered in DarkSide-20k, the measured activities using a HPGe detector in the Canfranc Underground Laboratory are <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='30 mBq/kg of 60Co and <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='35 mBq/kg of 54Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, this upper limit set for 60Co would correspond to the exposure of a few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For all stainless steel components, some cosmogenic activities can be at the level of a few hundreds of Bq, even for just 1 year of exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 54Mn is identified as a potential relevant contributor to background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Comparing with avail- able measurements from screening, the derived cosmogenic activity of 60Co is much lower than for instance the one measured for the DUNE steel in the Canfranc Underground Laboratory, finding (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9) mBq/kg of 60Co and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3) mBq/kg 54Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the measured activity of 54Mn would correspond to an exposure of ∼1 year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmogenic yields in Titanium Titanium is not part of the DarkSide-20k design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' However, it was con- sidered in previous designs and titanium is of interest in low-background experiments generally, so we include it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The possible impact of differ- ent cosmogenic products has been analyzed and then, the production rate and induced activity at sea level of the most relevant one, 46Sc, have been quantified from available information and new calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' To our knowl- edge, no direct measurement of productions rates for activation is available 15 for titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Natural composition has been assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Relevant isotopes Titanium activation by cosmic rays, particularly during air transport, was studied within the DarkSide Collaboration using a modified version of the COSMO code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the induced activity was quantified under different exposure and cooling conditions, including one with an exposure at sea level for a long time (10,000 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [33], the cosmogenic activation at sea level of several materials used in rare event search experiments, including titanium, was quantified from Geant4 simulations (for neutrons, protons and muons considering the Shielding physics list) and using the ACTIVIA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In both works, the yield of different products was evaluated: some of them (including several Sc isotopes) are short-lived with half-lives of a few days at most;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' others, like 40K and 50V, are very long-lived, so huge exposures comparable to their half-lives would be required to produce a significant activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' among the isotopes with intermediate half-lives, most of them produce emissions which could not escape from titanium to reach the active volume, as they are either pure or almost pure β− emitters (3H, 33P, 35S, 39Ar, 45Ca) or generate X-rays or low energy (below ∼80 keV) γ rays (44Ti, 49V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Therefore, 46Sc has been identified as the main product which could be relevant and the new calculations performed here correspond just to this isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 16 Table 2: Estimates of induced activity in all copper and stainless steel components of DarkSide-20k at the end of the exposure to cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For each one of the products, the half-life [66], main γ emissions and corresponding probabilities are indicated together with the production rates R at sea level assumed (from measurements in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [38] except for 60Co in stainless steel, taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [33]) and the total activity A for the three exposure times considered (1 month, 1 year and 10 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 7Be 46Sc 54Mn 59Fe 56Co 57Co 58Co 60Co T1/2 (d) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='79 312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='19 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='49 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='24 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='82 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='85 1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='95 γ emissions (keV) 477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3, 1120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 1099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3, 1291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8, 1238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 1173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2, 1332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 probability (%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='98, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='98 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5, 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 100, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 99 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='97, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='99 Copper R (atoms/kg/day) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='74 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='86 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 74±17 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 A (1 m) (mBq) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='11 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='54 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='16 A (1 y) (mBq) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='91 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 86±20 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 A (10 y) (mBq) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 141±32 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 121±11 Stainless Steel R (atoms/kg/day) 389±60 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 233±26 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='27 A (1 m) (Bq) 346±53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='19 A (1 y) (Bq) 1061±164 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 356±40 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 138±21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 A (10 y) (Bq) 1070±165 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 641±71 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 142±21 13 17 Table 3: Calculations of the production rate at sea level of 46Sc in titanium in this work and from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Code R (atoms/kg/day) COSMO 289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 ACTIVIA 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 6 [33] Geant4 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 [33] Estimated rate in this work (271±68) 46Sc is a β− emitter with a half-life of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 days and a transition energy of 2366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 keV [66];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' two γ rays of 889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 keV and 1120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 keV are produced with almost 100% probability each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Its activity has been quantified in titanium samples screened by different experiments: LUX-ZEPLIN analyzed many items (finding values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 to 23 mBq/kg, being most of them around a few mBq/kg) and exposed in a controlled way a sample for 6 months measuring afterwards an activity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3) mBq/kg [67, 68];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' XENON1T also analyzed titanium of different grades, measuring activities from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7 mBq/kg [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In addition, the production rate at sea level of 46Sc was computed using Geant4 and ACTIVIA [33] and can also be deduced from COSMO calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 3 compares the different estimates, which point to quite similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production rate To evaluate the production rate of 46Sc at sea level using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2 and the cos- mic neutron spectrum from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52], a selected description of the production cross sections over the whole energy range from threshold up to 10 GeV, con- sidering both neutrons and protons, has been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For libraries providing individual reaction cross sections, the mechanisms indicated in Table 4 have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Figure 2 shows the full set of data on total production cross sections taken into consideration from different libraries and a dedicated cal- culation using YIELDX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Below 100 MeV, there are important discrepancies between libraries and experimental data for protons, although this should not be relevant for neutron activation if specific calculations for neutrons are used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the only two available measurements on cross sections by neutrons are in perfect agreement with TENDL-2019 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Above 100 MeV, there is a good agreement between different calculations and experimental data for protons (except for one quite old series of data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' it is worth noting than the similarity between cross sections for neutrons and protons (usually assumed 18 Table 4: Production mechanisms for 46Sc in natural Ti isotopes by neutrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Neutrons Protons 46Ti (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='25%) (n,p) 47Ti (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='44%) (n,pn) (p,2p) 48Ti (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='72%) (n,p2n) (p, 2pn) (p,pd) 49Ti (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='41%) (n,p3n) (p,2p2n) (p,2d) (p,pt) 50Ti (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='18%) (n,p4n) (p,2p3n) (p,2dn) (p,dt) (p,ptn) in this range of higher energies) is fully confirmed by JENDL-HE, providing independent results for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Taking into account all the available data, the following cross sections σ(E) have been considered: Below 20 MeV, TENDL-2019, the only results that are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From 20 to 200 MeV, production cross sections by neutrons from TENDL- 2019, JENDL-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0 and JENDL-HE libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From 200 MeV to 1 GeV, results from JENDL-HE for neutrons and HEAD-2009 library together with YIELDX calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From 1 to 10 GeV, YIELDX results and data from JENDL-HE for neutrons (extrapolating the last available value at 3 GeV as constant for all higher energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Figure 3 presents a closer view of the cross sections actually considered in the calculations of the production rate for the low (top) and high (bottom) energy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The estimated contributions to the production rate of 46Sc for each energy range and selected cross sections are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' To sum the contributions in the whole energy region, the calculations with the lowest and highest rates in each region have been considered to get mean value and uncertainty from the defined interval (from 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 to 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 atoms/kg/day);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' in this way, the final result is (271±68) atoms/kg/day, in very good agreement with all the previous estimates (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activity From the estimated production rate of 46Sc by neutrons at sea level, the corresponding saturation activity according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 1 is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='79) mBq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content="1 1 10 100 1000 1 10 100 1000 10000 Production cross section (mb) Energy (MeV) TENDL n TENDL p JENDL n JENDL p JENDL-HE n JENDL-HE p HEAD2009 YIELDX Sisterson'2006 n Greenwood'1987 n Cervenak'2020 p Parashari'2019 p Garrido'2016 p Hermanne'2014 p Khandaker'2009 p Jung'1991 p Aleksandrov'1990 p Fink'1990 p Michel'1989 p Michel'1980 p Brodzinski'1971 p Neumann'1999 p Aleksandrov'1991 p Leya'1997 p Asano'1991 p Asano," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1983 p Figure 2: Full compilation of production cross sections of 46Sc in natural Ti by nucleons taken from different sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' including experimental data from the EXFOR database and calculations following different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 5: Contributions to the production rate (in atoms/kg/day) of 46Sc in natural Ti by cosmic neutrons at sea level estimated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2, the neutron spectrum from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52] and the different cross sections selected for each energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' TENDL JENDL YIELDX HEAD2009 JENDL-HE (n) (n) (n) <20 MeV 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0 20-200 MeV 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7 200-1000 MeV 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 1-10 GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content="1 1 10 100 1000 1 10 100 Production cross section (mb) Energy (MeV) TENDL n JENDL n JENDL-HE n HEAD2009 YIELDX Sisterson'2006 n Greenwood'1987 n 1 10 100 1000 100 1000 10000 Production cross section (mb) Energy (MeV) TENDL n TENDL p JENDL n JENDL p JENDL-HE n JENDL-HE p HEAD2009 YIELDX Aleksandrov'1990 p Fink'1990 p Michel'1989 p Brodzinski'1971 p Neumann'1999 p Aleksandrov'1991 p Leya'1997 p Asano'1991 p Asano," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1983 p Figure 3: Close view of production cross sections of 46Sc in natural Ti taken into consid- eration in the estimate of the production rate in the low (top) and high (bottom) energy regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 21 Due to the half-life of 46Sc, this saturation value may be easily achieved for usual exposure times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' indeed, the measured activities of 46Sc in screened samples are around this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This number can be considered as a conser- vative assumption for the induced activity at sea level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' if exposure happens at certain altitude, correction factors for the cosmic neutron flux should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The quantified activity has been obtained just for cosmic neu- trons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' according to the results based on Geant4 simulations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [33], the neglected contribution from muons and protons would be just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7% of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' If the vessel of the inner detector was made of titanium, a total mass of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 tons would be used giving an overall activity of around 30 Bq of 46Sc just when finishing the exposure to cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' According to the current schedule for the installation of the detector, it will be underground more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 y before starting operation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, after this cooling period, 46Sc activity would have been reduced to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1% of the initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Cosmogenic yields in Argon Argon in the atmosphere contains stable 40Ar at 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' cosmogenically produced radioactive isotopes, mainly 39Ar but also 37Ar or 42Ar, can be a significant background if argon obtained from air is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The concentra- tion of these three isotopes is much reduced in UAr, but the production of cosmogenic radionuclides after extraction must be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Relevant isotopes 39Ar is a β− emitter with a transition energy of 565 keV and half-life of 269 y [70];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' it is mainly produced by the 40Ar(n,2n)39Ar reaction started by cosmic neutrons [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The typical activity of 39Ar in AAr is at the level of ∼1 Bq/kg, as quantified by WARP [71], ArDM [72] and DEAP [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In UAr, after a first study on argon from deep underground sources [74], the mea- sured activity of 39Ar in the DarkSide-50 detector was (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='11) mBq/kg following a campaign of extracting and purifying argon from deep CO2 wells in Colorado, US;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 1, this means a reduction of a factor (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2)×103 relative to the AAr [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Presence of cosmogenically produced 37Ar was also detected in the begin- ning of the run of the DarkSide-50 detector with UAr [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It decays 100% by electron capture to the ground state of the daughter nuclei with a half- life of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='02 days [66];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, the binding energy of electrons from K-shell (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 keV, at 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='21%) and L-shell (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='20-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='27 keV, at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='72%) can be measured 22 as a distinctive signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The main production channel is the 40Ar(n,4n)37Ar reaction [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production underground in UAr by thermal and epithermal neutron capture is negligible, as for 39Ar, considering rates as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35] and neutron fluxes at LNGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 42Ar is a pure β− emitter with a 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 y half-life and transition energy of 599 keV, generating 42K, also a β− emitter with half-life of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='36 h and transition energy of 3525 keV [70];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this isotope can affect neutrinoless 2β experiments using liquid argon as refrigerant and shielding, as shown by the GERDA experiment [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' There are two mechanisms for the production of 42Ar in AAr: a two-step neutron capture (requiring a high neutron flux be- cause of the half-life of 41Ar, being of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8 h) and the (α,2p) reaction on 40Ar [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The specific activity of 42Ar has been studied in the context of different experiments using argon like ICARUS [77], DBA giving 92+22 −46 µBq/kg [78] and, more recently, DEAP, measuring 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 µBq/kg [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The content of 42Ar could not be quantified in DarkSide-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For UAr, 42Ar should be considered as a potential background for neutrinoless 2β decay searches (for example, by doping LAr with 136Xe);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' but the threshold for the α reaction on 40Ar is much higher than the energy of α particles from natural radioactivity, according to cross section values from TALYS and other sources [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This is also the case for other processes which could produce 42Ar underground from the rock, like 43Ca(n,2p)42Ar or 44Ca(n,n2p)42Ar, when considering the typical energies of radiogenic neutrons from natural fission and (α,n) reac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The production rate of 42Ar in UAr at sea level from fast neutrons and high energy muons and protons has been evaluated by Geant4 simulation as 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8×10−3 atoms/kg/day in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this rate would give a saturation activity about three orders of magnitude lower than measured values in AAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Taking all this into account, the effect of 42Ar in DarkSide-20k will not be consid- ered here although a specific study to quantify radiogenic and cosmogenic production in the Earth’s crust is underway6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3H in the detector medium of a dark matter experiment can be a very relevant background source due to its decay properties: it is a pure β− emit- ter with transition energy of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 keV and a long half-life of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 y [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The quantification of its cosmogenic production is not easy, neither by calcula- tions (3H can be generated by different reaction channels) nor experimentally (the β emissions are hard to disentangle from other background contribu- 6https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='sanfordlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='org/event/29/contributions/487/ 23 tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Estimates of the 3H production rate in several dark matter targets were attempted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [79];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the rate has been measured for germanium from EDELWEISS [19] and CDMSlite [21] data and for silicon and NaI(Tl) from neutron irradiation [24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Possible presence of 3H has been observed also in NaI(Tl) crystals by ANAIS [25, 80] and COSINE experiments [27, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In principle, purification systems for LAr may remove all non-noble radionu- clides and 3H should not be a problem for DarkSide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This was also assumed for liquid xenon, but 3H was considered as a possible explanation for the ex- cess of electronic recoil events observed in the XENON1T experiment below 7 keV [82, 83], which has disappeared in XENONnT [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activated 3H is separated from argon with SAES Getters [84] and will be removed in situ while the UAr recirculates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production of other radioisotopes with half-lives longer than 10 days in argon was predicted by using the COSMO code, like 7Be and 22Na (giving γ emissions) and 32,33P and 35S (being pure β− emitters);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' production rates at sea level from fast neutrons, high energy muons and protons have been eval- uated by Geant4 simulation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Assuming an efficient purification of non-noble isotopes, they will not be considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production rates The production rates of 37Ar and 39Ar from cosmic neutrons at sea level were measured for the first time through controlled irradiation at Los Alamos Neutron Science Center (LANSCE) with a neutron beam resembling the cos- mic neutron spectrum and later direct counting with sensitive proportional counters at Pacific Northwest National Laboratory (PNNL) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Samples of both AAr and UAr were irradiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In addition, the study of other pro- duction mechanisms due to muon capture, cosmic protons and high energy γ rays at the Earth’s surface was made using available cross sections to com- pute total production rates at sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The production rates obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35] for UAr are reproduced in Table 9 as they will be used to evalu- ate the induced activity in DarkSide-20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In addition, the production rates of both 37Ar and 39Ar at sea level have been recently evaluated by Geant4 simulation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [36] too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The UAr to be used in DarkSide-20k is obtained in Colorado, which is placed at a quite high altitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, the corresponding correction factors f to the cosmic ray flux must be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [53], high values of f are reported for neutrons at Colorado locations: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='11 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='86 for Denver (at 5280 feet) and Leadville (at 10200 feet), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 24 Table 6: Calculation of the correction factor f to be applied to the cosmic neutron flux at sea level (in New York) for the location of the Urania facilities in Colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The relative intensities I are derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The final factor for Urania is the average between the deduced ones from Denver and Leadville data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Location H A f Relative I Deduced f (ft) (g/cm2) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [53] to Urania for Urania Denver 5280 852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='659 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='24 Leadville 10200 705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='942 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='62 Urania 7100 795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='43 These correction factors f have been adjusted to the altitude at the Urania facilities (at 7100 feet), assuming that the ratio of f for different altitudes is the same than the ratio of cosmic flux intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' As described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [53], the intensities I1 and I2 at two different altitudes A1 and A2 (converted to g/cm2) are related as: I2 = I1 exp[(A1 − A2)/L], (3) being L the absorption length for the cosmic ray particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Calculations for the cosmic neutron flux correction factor are summarized in Table 6, using L =136 g/cm2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the final result for Urania is the average between the deduced ones from Denver and Leadville data, f =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For cosmic protons and muons, the correction factors have been obtained just from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3 considering the corresponding absorption lengths (L = 110 g/cm2 for protons and L = 261 g/cm2 for muons [53]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the results are f = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='67 for protons and f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='48 for muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2, a calculation of the production rates of relevant iso- topes in argon (assuming 100% 40Ar) by cosmic neutrons from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52] has been made considering a selection of excitation functions from libraries and YIELDX calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Figure 4 shows the available information on produc- tion cross sections of 3H, 37Ar and 39Ar by nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 39Ar, although no experimental data at EXFOR was found for the total production cross sec- tion, there are results for partial (n,2nγ) reactions in natural argon at 1-30 MeV taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 3H, an irradiation experiment with neutrons having an energy spectrum peaked at 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 MeV measured the corresponding production cross section [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The matching of the cross section data from different libraries, focused on different energy ranges, is not good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Several descriptions of the cross 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content="1 1 10 100 1000 10 100 1000 10000 Production cross section (mb) Energy (MeV) TENDL n TENDL p JENDL n JENDL p JENLD-HE n JENDL-HE p HEAD2009 Qaim'78 Ar(n," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='X)t 1 10 100 1000 10 100 1000 10000 Production cross section (mb) Energy (MeV) TENDL p JENDL n JENDL p JENDL-HE n JENDL-HE p HEAD2009 YIELDX 1 10 100 1000 10 100 1000 10000 Production cross section (mb) Energy (MeV) TENDL n TENDL p JENDL n JENDL p JENDL-HE n JENDL-HE p HEAD2009 YIELDX (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2ng),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Eg=250 keV (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2ng),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Eg=1267 keV Figure 4: Production cross sections of 3H (top),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 37Ar (medium) and 39Ar (bottom) in 40Ar by nucleons taken from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 26 sections, even from different libraries below and above a particular energy cut, have been considered to estimate the corresponding uncertainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the obtained maximum and minimum rates define an interval, whose central value and half width have been considered as the final result and its uncertainty for the evaluation of the production rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 7 presents the obtained results for 37Ar and 39Ar, together with the measured production rate for fast neutrons and different calculations from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The production rate of 39Ar derived here is fully compatible with the measured value (and with several of the calculations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The production rate of 37Ar is a factor 2 higher than the measured one, but lower than the Geant4 estimate in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For calculating final activity yields of 37Ar and 39Ar, the values of the total production rates obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35] will be used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' but this comparison can be useful to assess the reliability of the production rates of isotopes estimated only from calculations, like is the case of 3H in argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Evaluations of the production rate of 3H for several targets were applied also for argon, using different codes like TALYS [16] and Geant4 and AC- TIVIA [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It was also computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [79] for cosmic neutrons, from a selection of excitation functions considering the TENDL and HEAD2009 libraries, following the same approach applied here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this study was cross- checked against experimental data for NaI and germanium, reproducing prop- erly measured production rates [19, 21, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Now, new data for neutron cross sections taken from the JENDL-HE library have been added in the analysis in this work (giving a production rate of 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 atoms/kg/day) and then the final production rate has been re-evaluated considering all the other previ- ous estimates in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [79] as (168±53) atoms/kg/day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It must be noted that this value gives only production by neutrons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' assuming equal flux and cross sections of protons and neutrons above 1 GeV, it is estimated that protons would increase the rate by 10% at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 8 summarizes all the results for 3H production in argon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' an important dispersion of values is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 27 Table 7: Calculations of the production rates R of 37Ar and 39Ar in Ar at sea level from this work considering different descriptions of the excitation functions below (LE) and above (HE) a cut energy value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the final estimated rates are given by the ranges defined between the maximum and minimum obtained rates (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Different calculations from the literature (considering the same cosmic neutron spectrum from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [52]) and the measured value for fast neutrons from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35] are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 37Ar 39Ar This work Cut R This work Cut R LE+HE (MeV) (atoms/kg/day) LE+HE (MeV) (atoms/kg/day) TENDL(p)+HEAD2009 150 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 TENDL+HEAD2009 150 726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 TENDL(p)+YIELDX 100 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 TENDL+YIELDX 100 697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 TENDL(p)+YIELDX 200 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7 TENDL+YIELDX 200 646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0 JENDL-HE(n) 30 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 TENDL+JENDL-HE(n) 20 804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 Estimated rate in this work 109±45 Estimated rate in this work 725±79 Measurement [35] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 759±128 ACTIVIA [35] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 200±25 MENDL-2P [35] 155±19 188±24 TALYS [35] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 753±94 INCL++ (ABLA07) [35] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 832±104 TENDL-2015 [35] 726±91 Geant4 [36] 176 858 28 Table 8: Production rate R of 3H in Ar at sea level from this work and from different calculations from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' R (atoms/kg/day) Estimated rate in this work 168±53 TENDL+HEAD2009 [79] 146± 31 TALYS [16] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4 Geant4 [33] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 ACTIVIA [33] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Activity The possible activity yields of relevant cosmogenic isotopes in Ar have been analyzed for the DarkSide-20k detector considering Ar extraction, stor- age and transportation and taking into account not only cosmogenic neutrons but also other cosmic ray components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 37Ar and 39Ar, the production rates at sea level precisely determined with the LANSCE neutron beam and the estimates for muons, protons and cosmic γ rays [35] have been considered, while for 3H the production rate estimated in this work has been assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It is planned to produce 120 t of UAr for DarkSide-20k, allocated as follows: 100 t needed for filling the DarkSide-20k TPC, 4 t used during conditioning and purging the cylinder skids, 4 t of argon left in Aria after purification, and 12 t for contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The UAr extracted at the Urania plant will be shipped firstly to the Aria facility for purification and then to LNGS for storage and final operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The current baseline design is to ship the UAr in commercially available high-pressure (517 bar) gas cylinders that are organized into skids capable of containing ∼2 t of UAr each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It is not possible to predict accurately the final exposure conditions for the UAr, but according to the present specifications of Urania and Aria, a baseline exposure history with defined exposure times and places for the different steps of the transportation process can be established and the main sources of uncertainty in the process identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, activity yields have been computed for the baseline exposure and the effect of uncertainties assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The following steps are foreseen: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Storage of UAr at Urania: three skids will be filled before starting transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Considering the time required to fill one, exposures of 8, 16 and 24 days have been assumed for each one of the three skids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' While at the Urania site, the UAr will always be on the surface while 29 being processed and once in the skids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The correction factors to the sea level fluxes of cosmic neutrons, protons and muons evaluated for Urania location in Colorado (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2) have been included in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Trip from Urania to a shipping port: a container with the three skids will transport the UAr from Urania to Houston by truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' An exposure of 7 days has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' To take into account the different al- titude across the trip, the average between the maximal (from Urania altitude) and minimal (at sea level) expected activity has been calcu- lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Trip overseas to Europe: 60 days of exposure at sea level have been conservatively assumed for the trip by boat from Houston to Cagliari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' An additional exposure of 7 days is foreseen for custom issues and the trip from Cagliari to the Aria location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Steps 1 to 3 will be repeated over twenty times, running in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In total, 16 months are required for completing the extraction and transportation of all the necessary UAr at Urania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Processing and storage of UAr at Aria: once in Sardinia, the skids will be stored near Aria and the UAr will be accumulated for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' At a purification rate of 1 ton per day, a minimal exposure of 120 t is foreseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Purified UAr will be stored locally in Sardinia until needed for filling into DarkSide-20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Underground storage at a depth of at least some tens of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' would be recommended but, if not possible, a virtually linear increase of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 µBq/kg in the activity of 39Ar should be considered per month of additional exposure at sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Trip from Aria to LNGS: 10 days of exposure at sea level have been considered for this trip by boat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' It is expected to ship 12 t at a time using six skids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' then, this action should be repeated over ten times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Storage at LNGS: skids will be stacked underground as they arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' All in all, under these assumptions, the total time from the beginning of production at Urania to the end of processing at Aria is 614 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' If transport from Aria to LNGS starts once all the UAr has been processed, 100 additional days would be required to have all the UAr at LNGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Taking into account this exposure history, the induced activity by each cosmic ray component has been computed for each one of the exposure steps (at Urania, trip in US, overseas, at Aria and trip in Italy) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Tables 9 and 10 show separately each contribution for 39Ar and 37Ar and for 3H, 30 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The decay of the activities induced at each step during the rest of the whole process is negligible for 39Ar and small for 3H, due to their long half-lives, but extremely relevant for 37Ar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' it is accounted for in the final activities reported in Tables 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For both 39Ar and 37Ar, cosmogenic neutrons are responsible of the main part of the induced activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Under the assumed baseline conditions, the relative contributions to the final 39Ar activity of each exposure step are the following: Urania, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' US trip, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' overseas trip, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' at Aria, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' and Italy trip, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The exposure at Urania gives the largest contribution, followed by that of the overseas trip and at Aria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 37Ar, having a much shorter half-life, the last exposure during the Italy trip is dominant, produc- ing 55% of the final activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Concerning 3H, the final activity in Table 10 would apply if no purification procedure was considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' however, if a 100% efficient removal of 3H was achieved in Aria, only the activity in the last step for exposure in Italy would be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Table 11 summarizes the expected activities once all the UAr is at LNGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From values in Table 9, the final es- timated activity of 39Ar is (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5) µBq/kg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' this equals 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8% of measured activity in DarkSide-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 37Ar, the effect of cooling is very important and the expected activity when all the UAr is at LNGS is (103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6) µBq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' From values in Table 10 for 3H, an activity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='94) µBq/kg is expected at that time considering only activation after ideal purification in Aria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' with no purification, it would be around 25 times higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Uncertainties quoted for activities in Tables 9 and 10 come from those of production rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Concerning the correction factors of sea level cosmic ray fluxes for exposure at Urania, it has been checked that considering a description different to that applied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 produces very similar results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' correction factors computed from EXPACS spectra in the energy range rel- evant for activation (1 MeV to 10 GeV) are f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='09 for neutrons, f = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='60 for protons and f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='61 for muons, giving a small decrease in the final activ- ities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0% for 39Ar, no change for 37Ar and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5% for 3H with no purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' On the other hand, unexpected events can produce relevant deviations from the baseline exposure conditions and their effect on the activation yields has been assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Doubling the exposure at Urania would increase the final 39Ar activity from (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5) µBq/kg to (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='4) µBq/kg, which would be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8% of the DarkSide-50 activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Exposure at Aria has been evaluated for the moment considering just the processing time, but activation produced in the periods before and after the processing should be added if storage is made above ground;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' to produce an additional 10% of the measured activity in 31 DarkSide-50 (which was determined with an uncertainty of 14%), 28 months of additional exposure would be required, which is well above the period of 16 months needed for the extraction of the whole amount of UAr needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' All in all, it can be concluded that there is enough contingency in the plan for production, storage and shipping of the UAr so that cosmogenic 39Ar activity does not endanger DarkSide-20k sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Expected counting rates in DarkSide-20k As described in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3, 54Mn in stainless steel and 46Sc in titanium are identified as the most relevant cosmogenic products in these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The effect of their contribution to the γ background of the ex- periment has been evaluated finding for the former a negligible contribution in comparison to the other sources of γ background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In a hypothetical de- tector using a titanium vessel and considering the saturation activity when going underground, 46Sc would add (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='10) Hz and (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3) Hz, re- spectively, to the estimated counting rates in the TPC and inner veto (see Table 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The rates from the estimated cosmogenic activity of products in UAr, under the assumed baseline exposure conditions, are also shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Induced 39Ar due to the whole exposure from Urania to LNGS would add a rate of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='035±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='075) Hz for the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The contribution of 3H to the TPC counting rate is negligible (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='15 Hz) provided an efficient purification at Aria is achieved while that of 37Ar (being (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='43) Hz if data taking started just immediately after the arrival of all the UAr at LNGS) will de- cay very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Comparing these numbers with the total β and γ rates presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2, it can be concluded that cosmogenic activity does not produce a problematic increase of the TPC and Veto rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Conclusions For DarkSide-20k, material cosmogenic activation is a source of β/γ back- ground and it has been quantified for LAr and other massive components from realistic exposure conditions in order to assess the contribution to the counting rates and decide if additional exposure restrictions are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Main results are summarized in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 32 Table 9: Calculation of the expected induced activity in kg−1 d−1 of 39Ar and 37Ar in the UAr of the DarkSide-20k detector, for the assumed production rates R and exposure times (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Different columns and rows show separate contributions by cosmic ray components and exposure steps, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' relative contributions of each component to the total activity are also quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Row labelled as ”Final” presents the sum of final activities from all exposure steps including properly their decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 39Ar Neutrons Muons Protons γ rays Total R (atoms/kg/day) [35] 759±128 172±26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='2 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8±20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='75 (%) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 33 Table 10: Calculation of the expected induced activity in kg−1 d−1 of 3H by cosmic neutrons in the UAr of the DarkSide-20k detector, for the production rate R estimated in this work and the assumed exposure times (see text), considering no purification procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Different rows show separate contributions by exposure steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Row labelled as “Final” presents the sum of final activities from all exposure steps including properly their decays 3H Neutrons R (atoms/kg/day) 168±53 Urania 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='66±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='84 US 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='21 Overseas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='54 Aria 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='49 Italy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='259±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='082 Final 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='1 34 Table 11: Summary table of estimated activation in DarkSide-20k including isotope, material, main production channel, calculation details, overall activity and counting rates in TPC and inner veto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' All reported activity and rate values correspond to the moment when the materials are brought underground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For 3H, row (1) and (2) assume no purification and ideal purification at Aria, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Isotope Material Main channel Calculation Activity TPC rate Veto rate (µBq/kg) (Hz) (Hz) 39Ar UAr 40Ar(n,2n)39Ar Production rates from [35] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='035±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='662±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='048 37Ar UAr 40Ar(n,4n)37Ar Production rates from [35] 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='0±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='28 3H (1) UAr 40Ar(n,*)3H σ(E) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4+Gordon spectrum 76±12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='38 3H (2) UAr 40Ar(n,*)3H σ(E) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 4+Gordon spectrum 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='148±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='095±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='030 35 For copper and stainless steel components, activation yields of isotopes with relevant half-lives (like 54Mn, 57Co and 60Co) have been computed from measured production rates at sea level at Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In copper, even for 10 y of exposure to cosmic rays, estimated activities are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5 Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' In stainless steel, hundreds of Bq are expected for some isotopes for just 1 y exposure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' the contribution to the counting rate of ER-like events in the TPC from 54Mn activity induced in steel components has been found to be negligible in comparison to the estimated total rate from β/γ backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' This allows to relax additional limitations on the surface residency time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' For natural titanium, 46Sc has been identified as the main cosmogenic product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Other radioisotopes induced are not considered as a potential rele- vant background due to their half-lives or because their short-range emissions are not expected to escape from titanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The production rate at sea level of 46Sc has been calculated from a selection of production cross sections and considering the Gordon et al parametrization [52] for the cosmic neu- tron spectrum, deriving a value of (271±68) atoms/kg/day, which is in very good agreement with totally different estimates based on modified COSMO, Geant4 simulation and the ACTIVIA code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The corresponding saturation activity is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='79) mBq/kg, in the range of most of the measurements of 46Sc activity in samples found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Assuming exposure at sea level for a long time, this saturation activity has been conservatively consid- ered to quantify by MC simulation the possible effect of 46Sc emissions on the ER background rate of DarkSide-20k if titanium was used, showing a contribution to the TPC counting rate which is non-relevant, specially when taking into account the cooling down underground before the start of the data taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' A total of 120 t of UAr depleted in 39Ar must be extracted and processed for filling the TPC and inner veto of DarkSide-20k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The possible induced activity on surface, from the extraction at Urania to the storage at LNGS, has been analyzed not only for 39Ar but also for 37Ar and 3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Production rates from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' [35], based on a neutron irradiation experiment, have been considered for the Ar isotopes while for 3H an estimate of the production rate by cosmic neutrons made in this work obtaining (168±53) atoms/kg/day has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The estimated cosmogenic activity of 39Ar when all the UAr ar- rives to LNGS, (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='5) µBq/kg for the assumed baseline exposure history, is considered acceptable as it is just 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='8% of the residual activity measured in DarkSide-50 for UAr of the same source and would add ∼1 Hz to the counting rate of the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The quantified effect of some uncertain steps in 36 the procedure of UAr production shows that there is enough contingency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Contributions from the induced activity of 37Ar and 3H are not problematic thanks to short half-life and purification, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The results of this study of the cosmogenic activation of UAr will be useful to set exposure limi- tations for the procurement of the large amounts of radiopure UAr necessary in future LAr projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' Acknowledgements This report is based upon work supported by FSC 2014-2020 - Patto per lo Sviluppo, Regione Sardegna, Italy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' National Science Foun- dation (NSF) (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-0919363, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1004054, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY- 1004072, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1242585, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1314483, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY- 1314507, as- sociated collaborative grants, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1211308, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1314501, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' PHY-1455351, as well as Major Research Instrumentation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' MRI-1429544), the Italian Istituto Nazionale di Fisica Nucleare (Grants from Italian Ministero dell’Istruzione, Universit`a, e Ricerca Progetto Pre- miale 2013 and Commissione Scientific Nazionale II), the Natural Sciences and Engineering Research Council of Canada, SNOLAB, and the Arthur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' McDonald Canadian Astroparticle Physics Research Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' We acknowl- edge the financial support by LabEx UnivEarthS (ANR-10-LABX-0023 and ANR18-IDEX-0001), the S˜ao Paulo Research Foundation (Grant FAPESP- 2017/26238-4), Chinese Academy of Sciences (113111KYSB20210030) and National Natural Science Foundation of China (12020101004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' The au- thors were also supported by the Spanish Ministry of Science and Inno- vation (MICINN) through the grant PID2019-109374GBI00, the “Atraccion de Talento” Grant 2018-T2/ TIC-10494, the Polish NCN, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' UMO- 2019/ 33/ B/ ST2/ 02884, the Polish Ministry of Science and Higher Ed- ucation, MNi-SW, grant number 6811/IA/SP/2018, the International Re- search Agenda Programme AstroCeNT, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' MAB-/2018/7, funded by the Foundation for Polish Science from the European Regional Development Fund, the European Union’s Horizon 2020 research and innovation program under grant agreement No 952480 (DarkWave), the Science and Technology Facilities Council, part of the United Kingdom Research and Innovation, and The Royal Society (United Kingdom), and IN2P3-COPIN consortium (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' 20-152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content='A is supported in part by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} +page_content=' We also wish to ac- knowledge the support from Pacific Northwest National Laboratory, which is 37 operated by Battelle for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFOT4oBgHgl3EQf9jQ2/content/2301.12970v1.pdf'} 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b/YdE1T4oBgHgl3EQfcQQe/content/tmp_files/2301.03181v1.pdf.txt @@ -0,0 +1,2273 @@ +arXiv:2301.03181v1 [math.QA] 9 Jan 2023 +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC +PAIRS +MICHAEL EHRIG AND KAIXUAN GAN +Abstract. A way to construct the natural representation of the quantized affine algebra +Uv(ˆslℓ) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl +modules for Uq(slN) were q is a root of unity to the action of Uv(ˆslℓ) as N tends towards +infinity. In this paper we investigate the situation outside of type A. In classical types, +we construct embeddings of the Grothendieck group of finite dimensional Uq(g)-modules +into Fock spaces of different charges and define an action of an affine quantum symmetric +pair that plays the role of the quantized affine algebra. +We describe how the action is +related to the linkage principal for quantum groups at a root of unity and tensor product +multiplicities. +Contents +1. +Introduction +1 +2. +Preliminaries +3 +3. +Fock spaces, affine Weyl groups, and quantum symmetric pairs +7 +4. +Type C +12 +5. +Type B +19 +6. +Type D and beyond +28 +References +30 +1. Introduction +The Fock space (of charge zero) F0 arises in mathematical physics. In the context of +representation theory it gives a particularly nice realisation of a representation for an affine +Kac-Moody algebras (e.g. [Kac90, Chapter 14]). With a basis labelled by all partitions, the +basis elements can naturally be interpreted as the classes of irreducible finite dimensional +highest weight modules L(λ) for glN(C) with dominant polynomial highest weights when N +tends towards infinity. One major step is to consider F0 = F0 ⊗Q Q(v) the Q(v)-deformation +of F0. Here and for the rest of the paper v is an indeterminate. In this case Misra and Miwa +[MM90], following the work of Hayahsi [Hay90] defined an action of the quantized universal +enveloping algebra Uv(ˆslℓ) on F. The action is of a particularly nice form in the sense that +the image of a partition under a Chevalley generators has, as coefficients, monomials in v +that are combinatorially easy to describe. +In turn F0 can be realized via the affine Hecke algebra, in this way it obtains the struc- +ture of a KL-module depending on a parameter ℓ. This leads to a second natural basis, +the Kazhdan-Lusztig basis. One now adopts the point of view that the standard basis of +partitions corresponds to classes of Weyl modules ∆q(λ) for the quantum group Uq(glN) spe- +cialized at a 2ℓ’s root of unity and N tending towards infinity. The Kazhdan-Lusztig basis +corresponds to the classes of simple modules in the root of unity case and the transition +matrix between the two basis evaluated at 1 gives the multiplicities for the modules in terms +of evaluations of parabolic affine Kazhdan-Lusztig modules, see [LT00]. From a Lie theo- +retic point of view the coefficients for the action of Chevalley generators can be connected +1 + +2 +M. EHRIG AND K. GAN +to Shapovalov determinants as shown in [RT10]. Note that here and in the rest of the paper +q is a root of unity such that q2 has order ℓ. +The combinatorial Fock space was defined in [LRS19]. It is isomorphic, after extension +of scalars, to the Grothendieck group of finite dimensional representations for Uq(g), where +g is an arbitrary finite dimensional semi-simple complex Lie algebra. This comes naturally +equipped with a basis corresponding to dominant weights or via the isomorphism to the +classes of Weyl modules. Depending on the order ℓ, [LRS19] establishes the structure of a +KL-module on this space. This in turn yields a Kazhdan-Lusztig type basis that corresponds +to the classes of simple modules with the transition matrix being given by parabolic affine +Kazhdan-Lusztig polynomials. In type AN these Fock space models can be embedded in +each other for growing N and in the direct limit give the space F0 from above. +This paper is a step to generalize the setup in the following sense. Outside of type A, +we embed the combinatorial Fock space into a traditional Fock space, this plays the role of +F0. This is done via the combinatorics of certain sequences that are a natural generalisation +of Young tableaux in type A. We then define an action of an affine quantum symmetric +pair related to Uv(ˆslℓ) on this Fock space. Such algebras were studied and classified in the +non-affine situation (see [Let02] and [Let03]) and in the Kac-Moody setting (see [Kol14]). +This action is compatible with the linkage principal for the quantum group at a root of unity +and describes certain tensor product multiplicities for Weyl modules in a similar way as the +action of the quantum affine algebra does in type A. Roughly said the action of generators +describes, in the Grothendieck group, the image of a Weyl module under the translation +functor given by taking the tensor product with the “natural” representation for a fixed +rank of g. +The structure of the paper is as follows. In Section 2 we recall the necessary definitions for +weight combinatorics for quantum groups, the definition of the combinatorial Fock space from +[LRS19], and some facts about quantum groups at a root of unity and their representation +theory. +In Section 3 we introduce the combinatorial definition of the analogue of F0 in +the type A setting as a space of certain sequences as well as most of the combinatorics for +sequences that are needed later on. We briefly recall the type A situation translated into these +combinatorics. We introduce some notions for affine Weyl groups and alcove combinatorics +needed later on, as well as the affine quantum symmetric pair that acts outside of type A. +In Section 4 we investigate the type C case. This is the easiest case with the least amount of +complications. We first specialize the combinatorics to this case, then relate linear operators +on the Fock space to the linkage principal in type C and then define the action of the affine +quantum symmetric pair (Definition 4.8). +In Theorem 4.12 we describe the relationship +between the action and tensor product multiplicities and in Proposition 4.14 the analogue +of letting N tend to infinity here. Section 5 has then the same structure but for type B. +Note that here one has to distinguish between the case of ℓ even and ℓ odd. The definition +of the action can be found in Definitions 5.9 and 5.22. The relationship with tensor product +multiplicities is divided into Theorems 5.14, 5.15, 5.24, and 5.25. The situation of letting +N tend towards infinity is described in and before Proposition 5.16 for ℓ odd. The same +results for ℓ even can be derived in a similar way. Finally in Section 6 we give a sketch of +how to apply this construction to type D and what the corresponding results look like. We +also elaborate on why the construction is not fully detailed in the paper. +Acknowledgements. We thank Daniel Tubbenhauer for comments and remarks. +This +work was supported by the National Natural Science Foundation of China under Grant No. +12050410261. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +3 +2. Preliminaries +In this part we review the fundamental definitions for the three main objects involved. +The weight combinatorics of semi-simple Lie algebras, the combinatorial Fock space, and the +quantum group at a root of unity corresponding to a non-exceptional semi-simple complex +Lie algebra. Throughout the paper we use an integer ℓ that is the order of a root of unity +in Section 2.3. In general we assume +ℓ > 3. +This is done to avoid special cases of quantum symmetric pairs in case of small ℓ. +2.1. Lie algebras and weight combinatorics. We fix g a finite dimensional semi-simple +complex Lie algebra. In the following we are only interested in the non-exceptional cases. +Hence from now on we make the assumption that g is of type AN, BN, CN or DN. We denote +this type by XN in what follows. To not having to deal with the various isomorphisms in +small ranks we assume N > 1 in type BN, N > 2 in type CN, and N > 3 in type DN. +Remark 2.1. All constructions outside of Sections 4 and 5 can be made for exceptional Lie +algebras. We go into more details why we are restricting to the classical cases in Section 6. +Fix a Cartan subalgebra h and a Borel subalgebra b containing h. With the choice of h we +denote by Φ the root system of g and by Φ+ the positive roots with respect to b. By W we +denote the Weyl group corresponding to g. Fix a non-degenerate W-invariant bilinear form +(−, −) : h∗ × h∗ → C. For α ∈ Φ+, the corresponding coroot is defined by α∨ = 2α/(α, α). +We label the simple roots in Φ+ by α1, . . . , αN. The lattice of integral weights is denoted by +X = {λ ∈ h∗ | (λ, α∨) ∈ Z, for α ∈ Φ+}. +The elements ωi ∈ X such that (ωi, α∨ +j ) = δij are called the fundamental weights. Inside the +weight lattice we fix the set of dominant weights +X+ = {λ ∈ X | (λ, α∨) ≥ 0, for α ∈ Φ+}. +On X we have the action of the Weyl group W with the reflection sα given by sα(λ) = +λ − (λ, α∨)α for α ∈ Φ+. +2.2. The combinatorial Fock space. Fix the element ρ = 1 +2 +� +α∈Φ+ α and the set of ρ- +shifted dominant weights X+ +ρ = X+ + ρ. We denote by Q(v) the rational functions over +Q. +Following [LRS19, Section 1.1], the combinatorial Fock space F(XN) of type XN is the +Q(v)-vector space with basis {λ = λ + ρ | λ ∈ X+}. +Remark 2.2. In [LRS19] the Fock space is originally defined to be generated as a Z[v, v−1]- +module by elements indexed by X modulo relations, but [LRS19, Theorem 1.1] shows that +the elements indexed by X+ form a basis as a free Z[v, v−1]-module. We simple shift the +indexing set of the basis elements by ρ and extend scalars to the field Q(v). +We do not make use of the description, but it should be noted that one of the main and +most intricate results of [LRS19] is to endow the combinatorial Fock space F(XN) with the +structure of a KL-module. For this it is alternatively constructed by starting from the affine +Hecke algebra. For this purpose an action of an affine Weyl group has to be introduced and +this action depends on an integer ℓ. Thus in contrast to [LRS19] we drop the label ℓ in +the notation for the Fock space, as we do not make use of the description via affine Hecke +algebras. + +4 +M. EHRIG AND K. GAN +2.3. Quantum group at a root of unity. Starting with a fixed semi-simple complex +Lie algebra of type XN we define the corresponding quantized enveloping algebra, following +[Lus10]. For 1 ≤ i, j ≤ N we set aij = (αj, α∨ +i ) ∈ Z. For 1 ≤ i ≤ N we fix di = 1 if αi is +short root and di = 2 if αi is a long root. Let vi = vdi ∈ Q(v). This is extended to all roots, +by setting dα = di if α and αi are in the same Weyl group orbit. +Remark 2.3. Note that in type AN and type DN all simple roots are short. For the non +simply-laced cases we have the Dynkin diagrams +BN : +and +CN : +. +Hence in type BN there is precisely one short simple root αN corresponding to the right +most vertex in the diagram above, while in type CN there is precisely one long simple root, +again corresponding to the right-most vertex in the diagram above. For (d1, . . . , dN), with di +corresponding to the i-th node from the left in the diagrams above, we have in type BN the +vector (2, 2, . . . , 2, 1) and in type CN the vector (1, 1, . . . , 1, 2). +To fix the notation for the quantum integers we define for n ∈ Z≥0 and k ∈ Z the following +elements in Z[v, v−1] +[n]v = vn − v−n +v − v−1 , [n]v! = +n +� +m=1 +vm − v−m +v − v−1 , and +� +k +n +� +v += +n +� +m=1 +vk+1−m − v−k−1+m +vm − v−m +. +We use the same notation if we substitute v for vi or an element of a field. +The quantized enveloping algebra Uv(g) is the associative algebra over Q(v) generated by +elements {Ei, Fi, K±1 +i +| 1 ≤ i ≤ N} subject to the following relations for all 1 ≤ i, j ≤ N +(1) KiK−1 +i += 1 = K−1 +i +Ki, KiKj = KjKi, +(2) KiEj = vaij +i +EjKi, KiFj = v−aij +i +FjKi, +(3) EiFj − FjEi = δij +Ki−K−1 +i +vi−v−1 +i +(commutator relation), +(4) �1−aij +k=0 (−1)kE(1−aij−k) +i +EjE(k) +i += 0, for i ̸= j (quantum Serre relations), +(5) �1−aij +k=0 (−1)kF (1−aij−k) +i +FjF (k) +i += 0, for i ̸= j (quantum Serre relations), +where E(k) +i += Ek +i /[k]vi! and F (k) +i += F k +i /[k]vi! are the divided powers. +For a Uv(g)-module M and λ ∈ X we fix the λ-weight space as +Mλ = +� +m ∈ M | Kim = v(λ,α∨ +i ) +i +m +� +. +We denote by Uv(g)-mod the full subcategory of finite dimensional Uv(g)-modules consisting +of modules M of type 1, i.e. those finite dimensional modules satisfying M = � +λ∈X Mλ. +We denote by [Uv(g)-mod] the Grothendieck group of Uv(g)-mod and to avoid unnecessary +clutter of notations, we assume that the scalars of the Grothendieck group are extended from +Z to Q(v). +Following [Lus90], we fix the subring A = Z[v, v−1] ⊂ Q(v) and denote by UA(g) the +A-form of Uv(g), i.e. the A subalgebra of Uv(g) generated by divided powers E(k) +i +, F (k) +i +, +K±1 +i +and the elements +� +Ki; c +k +� += +k +� +s=1 +Kivc+1−s +i +− K−1 +i +vs−1−c +i +vs +i − v−s +i +for c ∈ Z, k ∈ Z>0. +Note that for k = 1 and c = 0 this is exactly equal to [Ei, Fi], while other elements of this +form appear in generalised versions of commutator relation for divided powers. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +5 +Fix q ∈ C with q2 is primitive ℓ-th root of unity and consider Uq = UA(g) ⊗A C, where +v ∈ A acts by multiplication with q in C. For a Uq-module M and λ ∈ X we set +Mλ = +� +m ∈ M | Kim = qdi(λ,α∨ +i )m, +� +Ki; 0 +k +� +m = +� +(λ, α∨ +i ) +k +� +qdi +m +� +We refer to Uq simply as the quantum group at a root of unity. +In analogy to the generic case, we denote by Uq-mod the full subcategory of finite dimen- +sional Uq-modules consisting of modules M such that M = � +λ∈X Mλ. We denote again by +[Uq-mod] the Grothendieck group and assume that the scalars are extended from Z to Q(v). +In both cases of Uv(g)-mod and Uq-mod we have the character of a module given as +ch(M) = � +λ∈X dim(Mλ)eλ for formal symbols eλ and the dimension taken over the re- +spective fields Q(v) and C. Replacing the dimension by the rank over A, one obtains the +character of a UA(g)-module that is free as an A-module. +In the generic case, for λ ∈ X+, we denote by ∆v(λ) = Lv(λ) the irreducible highest +weight module of highest weight λ and we fix xλ ∈ Lv(λ) a highest weight vector. Following +[Tan04, Section 7], this can be lifted to the A-form by defining ∆A(λ) to be the UA(g)- +submodule of ∆v(λ) generated by xλ. Alternatively, [APW91] construct the module as a +quotient of the Verma module defined for UA(g). +Then ∆A(λ) is a free A-module with +∆v(λ) = ∆A(λ) ⊗A Q(v). Especially one obtains, see [APW91, Proposition 1.22], +ch(∆v(λ)) = ch(∆A(λ)), +both of them given by Weyl’s character formula. As noted in [APW91, Remark 1.25] this +does not follow the usual tradition from algebraic groups to construct Weyl modules. In +those setting one would induce from the opposite Borel to obtain the dual Weyl module +and then use the duality to obtain the Weyl module. +This more involved approach has +many advantages, but since we are only interested in the classes of Weyl modules in the +Grothendieck group we use this simpler construction here. +Since ∆A(λ) is free over A, we can set ∆q(λ) = ∆A(λ) ⊗A C and naturally +ch(∆q(λ)) = ch(∆A(λ)) = ch(∆v(λ)). +(1) +The module ∆q(λ) is called the Weyl module with highest weight λ. By construction it is +a quotient of the corresponding Verma module Mq(λ) of highest weight λ and ∆q(λ) has a +unique irreducible quotient Lq(λ). Note that every simple object in Uq-mod can be obtained +in this way. +Remark 2.4. Since ∆q(λ) has Lq(λ) as its head and by highest weight theory all other +composition factors are of the form Lq(µ) for λ − µ a positive, non-zero, sum of positive +roots, the classes [∆q(λ)] for λ ∈ X+ form a basis of the Grothendieck group [Uq-mod]. +The Weyl modules play the key role for our situation here. The following observation is +well-known to experts, but since it is central to our arguments we formulate it here. +Proposition 2.5. Let λ, µ ∈ X+ and consider the tensor product decomposition +∆v(λ) ⊗Q(v) ∆v(µ) ∼= +r +� +i=1 +∆v(νi) +in Uv(g)-mod for some dominant weights ν1, . . . , νr. Then in the Grothendieck group [Uq-mod] +it holds +[∆q(λ) ⊗C ∆q(µ)] = +r +� +i=1 +[∆q(νi)]. + +6 +M. EHRIG AND K. GAN +Proof. Consider the tensor product ∆A(λ) ⊗A ∆A(µ). Then, by (1), it holds +ch(∆q(λ) ⊗ ∆q(µ)) = ch(∆A(λ) ⊗A ∆A(µ)) = ch(∆v(λ) ⊗Q(v) ∆A(µ)) +Since the classes of Weyl modules form a basis of [Uq-mod] and (1), we thus obtain that +the decomposition of the character in the generic case, which is just the tensor product +decomposition, gives the decomposition in the Grothendieck group in the root of unity +case. +□ +We only need the statement about the Grothendieck group from Proposition 2.5, but a +stronger statement in the category also holds. +Remark 2.6. With the same notations as in Proposition 2.5, ∆q(λ)⊗C∆q(µ) has a filtration +with subquotient isomorphic to Weyl modules of the form ∆q(νi) for ν1, . . . , νr some dominant +weights. +The existence of a filtration with Weyl modules as subquotient can be derived from the +literature in different ways. In [AST18] (see arXiv-Appendix, Claim 3.10.1) this is worked +out in the simply-laced case and for dual Weyl modules (hence one needs to apply a duality). +In [Par94, Theorem 3.3] this can be found more general under the name of good filtrations, +which then dualizes to a filtration by Weyl modules. +To determine which Weyl modules +appear in the filtration one uses the same argument about characters as in Proposition 2.5 +above. +In contrast to Uv(g)-mod which is a semi-simple category, Uq-mod is not semi-simple. For +a criterion to decide when two simple modules respectively two Weyl modules can be in the +same block, one introduces an action of an affine reflection group on X. Namely for α ∈ Φ+ +define ℓα = ℓ/gcd(ℓ, dα). Then denote by Wℓ the group generated by the affine reflections of +the form +sα,k · λ = sα · λ + kℓαα, +for α ∈ Φ+, k ∈ Z and w · λ = w(λ + ρ) − ρ for w ∈ W the dot action of W on X. +Remark 2.7. Note that in our situation dα is either 1 or 2, hence in case that ℓ is odd +ℓα = ℓ for all α. In which case Wℓ is the affine Weyl group attached to W, except that the +action is scaled by a factor ℓ after the shift by ρ. In case that ℓ is even ℓα = ℓ/2 for a long +root α. In this case the group Wℓ is acting as the affine Weyl group for the dual root system, +shifted by ρ and scaled by a factor ℓ. +We call two weights λ, µ ∈ X+ linked, if there exists an element w ∈ Wℓ such that λ = w·µ. +This correlates to extensions between simple modules as follows. +Theorem 2.8. [And03, Theorem 4.3] Let λ, µ ∈ X+. If Ext1(Lq(λ), Lq(µ)) ̸= 0 then λ and +µ are linked and not equal. +Since ∆q(λ) for λ ∈ X+ is indecomposable, we thus get that if ∆q(λ) and ∆q(µ) are in +the same block, then λ and µ are linked. +As mentioned, the Lq(λ) for λ ∈ X+ form a complete set of irreducible modules in +Uq-mod. We fix the following identification from now on to view classes of Weyl modules as +basis elements of the corresponding combinatorial Fock space. +Lemma 2.9. There is a Q(v)-vector space isomorphism between F(XN) and [Uq-mod] ⊗Z +Q(v), mapping the basis vector λ = λ + ρ to [∆q(λ)]. +Note that the interpretation of F(XN) as a KL-module in [LRS19] allows to identify the +KL-basis of F(XN) with the basis given by the classes of irreducible modules in [Uq-mod]⊗Z +Q(v). + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +7 +3. Fock spaces, affine Weyl groups, and quantum symmetric pairs +In this section we introduce the necessary combinatorics of sequences to define the spaces +that the F(XN) are embedded into. +Denote by H = 1/2 + Z the half-integers. Since Z acts on H by addition, we consider the +cosets H/rZ for a positive integer r. For p ∈ H we denote by p its coset in H/rZ and for +p ∈ Z, we denote by p its coset in Z/rZ. +Consider the following sets of sequences +SZ = {a : Z → {0, 1} | a(i) = 0 for i ≫ 0, a(i) = 1 for i ≪ 0} and +SH = {a : H → {0, 1} | a(i) = 0 for i ≫ 0, a(i) = 1 for i ≪ 0}. +We call SZ the sequences supported on integers and SH the sequences supported on half- +integers. Then we denote by F = F1 the Q(v)-vector space with basis SZ and by F +1/2 the +Q(v)-vector space with basis SH. We refer to F and F +1/2 simply as the Fock space. The +notation F1 is needed for type B to make the differentiation between Fock spaces clear in +that case. +Remark 3.1. While they are defined as maps we consider these as {0, 1}-sequences with +indices labelled by either Z or H. We refer to the value a(i) as the entry at position i and +say that the position is “empty” if a(i) = 0 and it is occupied if a(i) = 1. +Since a sequence in this set only has finitely many non-zero entry in its positive half and +only finitely many zero entry in its non-positive half, the following is well defined for a ∈ SZ +and equally for a ∈ SH +ch(a) = +� +i>0 +a(i) − +� +i≤0 +(1 − a(i)). +We call this the charge of a and the set of all sequences of charge N is denoted by SZ,N +respectively SH,N. The corresponding subspaces, the Fock space of charge N, are spanned +by sequences of charge N and denoted by FN = F1 +N and F +1/2 +N . +3.1. Moving operators and counting statistics. We define two basic operators on Fock +spaces that are moving a 1 entry to either the left or right. +Definition 3.2. For a ∈ SZ let i ∈ H and for a ∈ SH let i ∈ Z. +• Define a sequence b via b(i− 1/2)−b(i+ 1/2) = 1 and b(j) = a(j) for j ̸= i± 1/2. Then +eia = +� +b +if a(i + 1/2) − a(i − 1/2) = 1 +0 +otherwise. +• Define a sequence c via c(i + 1/2) − c(i − 1/2) = 1 and c(j) = a(j) for j ̸= i ± 1/2. +Then +fia = +� +c +if a(i − 1/2) − a(i + 1/2) = 1 +0 +otherwise. +These define linear operators on both F and on F +1/2, which we call the moving operators. +By definition ei and fi preserve the charge and thus restrict to linear operators on FN and +F +1/2 +N . We say that ei moves an entry 1 from position i + 1/2 to position i − 1/2 or is zero if +that is not possible, while fi moves an entry 1 in the opposite direction. +For the definition of the action of the quantum affine algebra in type A or the quantum +symmetric pair in other types, we need a number of counting statistics, which we introduce +now. They appear in different combination for all the actions. + +8 +M. EHRIG AND K. GAN +Definition 3.3. Fix r ∈ Z. For a ∈ SZ let j ∈ H and for a ∈ SZ let j ∈ Z. Then define +Re +r(j, a) = #{k ∈ j + rZ>0 | eka ̸= 0}, +Rf +r (j, a) = #{k ∈ j + rZ>0 | fka ̸= 0}, +Le +r(j, a) = #{k ∈ j − rZ>0 | eka ̸= 0}, +Lf +r(j, a) = #{k ∈ j − rZ>0 | fka ̸= 0}. +To shorten notation we also introduce +Re−f +r +(j, a) = Re +r(j, a) − Rf +r (j, a), +Rf−e +r +(j, a) = Rf +r (j, a) − Re +r(j, a), +Le−f +r +(j, a) = Le +r(j, a) − Lf +r(j, a), +Lf−e +r +(j, a) = Lf +r(j, a) − Le +r(j, a). +In addition, for a ∈ SZ let i ∈ H/rZ and for a ∈ SZ let i ∈ Z/rZ. Then define +T e +r (i, a) = #{j ∈ i | eja ̸= 0} and , +T f +r (i, a) = #{j ∈ i | fja ̸= 0}, +T e−f +r +(i, a) = T e +r (i, a) − T f +r (i, a), +T f−e +r +(i, a) = T f +r (i − a) − T e +r (i, a). +The cumbersome notation is necessary, since the action of the generators in the different +cases depend on both the sequence a as well as a position j such that an operator ej re- +spectively fj can be applied. In most cases the index r is r = ℓ, but in case of type B and +even ℓ it has to replaced by r = ℓ/2 for the action, hence the addition of the index r in the +definition. +Remark 3.4. Fix a sequence a in either SZ or SH. Then Re +r(j, a) counts how often, for +k ∈ j + rZ>0 (i.e. to the right of j), it happens that a(k − 1/2) = 0 and a(k + 1/2) = 1. +Similarly Rf +r (j, a) counts when a(k − 1/2) = 1 and a(k + 1/2) = 0 instead and Le +r(j, a) and +Lf +r(j, a) count these occurrences for k ∈ j − rZ>0, i.e. to the left of j. While the statistics +of the form T e +r (i, a) count where such situation occur on all of Z respectively H. This should +give an easy way to remember which of the operators counts what and in which direction of +a fixed position. +3.2. Type A. We briefly recall here the definition of the action on the Fock space in type +A. Instead of defining the Fock space via partitions as usual, we use sequences. +Fix X = ⊕N +i=1Zεi the a lattice of integral weights for glN(C). One could also use slN(C) +here, but the general linear case is more convenient from the view point of combinatorics. +In X we have +X+ = {(λ1, . . . , λN) ∈ X | λ1 ≥ λ2 ≥ . . . ≥ λN} and +P + = {λ1, . . . , λN) ∈ X+ | λN ≥ 0}, +the sets of dominant weights and polynomial weights of glN(C). One should consider P + +as the analogue of the dominant weights for slN(C). +Furthermore, fix the element ρ = +(0, −1, . . . , −(N − 1)). In contrast to slN(C) we have a choice here and we fix this particular +ρ to match up nicely with the usual definition of partitions and adding boxes of certain +residues. +To λ ∈ P + we associate the sequence aλ ∈ SZ given by +aλ(i) = + + + + + +1 +if i ≤ −N, +1 +if there exists 1 ≤ j ≤ N s.t. i = λj, +0 +otherwise. +i.e. we put a 1 at all positions that appear as entries in λ, since we added λ = λ + ρ, these +are all distinct, so this is well-defined. In addition we set all values to 1 for i ≤ −N. Note +that for the zero weight 0 = (0, −1, . . . , −(N −1)), hence the sequence associated to the zero +weight has value 1 for all non-positive integers and value 0 for all positive values. +Note that this embeds F(AN) into the Fock space F0 of charge zero. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +9 +Remark 3.5. To pass to the partition description of F0, see for example [Ari02] or [RT10], +fix a sequence a of charge 0. Let (λ1, λ2, . . .) be defined such that a(λ1) is the right most +entry equal to 1, a(λ2 − 1) is the second right most entry equal to 1 and so forth with +a(λi − (i − 1)) being the i-th right most entry equal to 1. Then by construction (λ1, λ2, . . .) +is a weakly decreasing sequence of non-negative integers that eventually stabilizes to 0 after +finitely many steps, hence a partition. +For p ∈ H/ℓZ and a ∈ SZ, define linear operators on F via +ˆEp a = +� +j∈p +vRe−f +ℓ +(j,a)ej a, +ˆFp a = +� +j∈p +vLf−e +ℓ +(j,a)fj a, and ˆKp a = vT f−e +ℓ +(p,a)a. +Restricting these operators to the subspace F0 gives then the well-known action on F0, see +for example [Ari02], by using Remark 3.5 to translate it to sequences instead of partitions. +Theorem 3.6. [Ari02, Theorem 10.6] The linear operators {ˆEp, ˆFp, ˆK±1 +p } define an action +of the quantum affine algebra Uv(ˆslℓ)′ on F0. +Note that there is a natural isomorphism between Ψm : F0 → Fm for any integer m, by +just shifting the sequence by m steps, i.e. Ψm(a)(i) = a(i − m). By construction +Ψm(ˆEpa) = ˆEp+mΨm(a), Ψm(ˆFpa) = ˆFp+mΨm(a), and Ψm(ˆKpa) = ˆKp+mΨm(a). +Since all relations of the quantum affine algebra are rotation invariant, this defines an action +of Uv(ˆslℓ)′ on all Fm, m ∈ Z, given by the same formulas. The only difference really being +for which index p, ˆFp does not act as zero on the “highest weight sequence” where all 1’s are +as far to the left as possible, i.e. the sequence on which all ˆEp′ act as zero. +3.3. Affine Weyl group combinatorics. To introduce alcove geometry for the affine Weyl +group we define XR = X ⊗Z R. Then for α ∈ Φ+ we consider the affine hyperplane +Hα,k = {x ∈ XR | (λ + ρ, α∨) = kℓα}. +The affine reflection at this affine hyperplane is precisely sα,k and the reflections at all such +hyperplanes give the action of Wℓ. We denote the set of all such affine hyperplanes by H and +conversely for a hyperplane H ∈ H we denote by sH the corresponding affine reflection. The +statements and definitions in this section can be found in most textbooks on affine reflection +groups, e.g. [Hum90]. +Definition 3.7. Consider the complement of all affine hyperplanes in H +Xreg +R += XR \ +� +H∈H +H. +A connected component of Xreg +R +is called an open alcove. The closure of a connected compo- +nent in XR is called a (closed) alcove. We denote the set of all (closed) alcoves by A. +Note that Wℓ acts simply transitively on the set A. Points in an open alcove have trivial +stabilizer, while points in the boundary of an open alcove have non-trivial stabilizers. +Each affine hyperplane H ∈ H defines two closed halfspaces. For a fixed alcove A ∈ A we +denote by H+ +A the half space that contains A. Furthermore for two alcoves A and A′ we call +a hyperplane H ∈ H between A and A′ if H+ +A ̸= H+ +A′. This give rise to the definition of a +distance function +d(A, A′) = #{H ∈ H | H between A and A′}, +for A, A′ ∈ A. +Lemma 3.8. Let H ∈ H be a hyperplane between alcoves A, A′ ∈ A. Then d(A, A′) > +d(sHA, A′) and d(A, A′) > d(A, sHA′). + +10 +M. EHRIG AND K. GAN +For an alcove A ∈ A consider the set of hyperplanes HA that intersect A in maximal +dimensions, these are called the walls of A. Then SA = {sH | H ∈ HA} is a generating set +of Wℓ as a Coxeter group. With respect to these generators we can see the distance as a +choice free substitute for the length function l with respect to SA, namely let A′ be another +alcove then +d(A, A′) = l(w) for wA = A′. +Remark 3.9. With the equality of distance and length, it follows that for a fixed alcove A +and H ∈ HA, the hyperplane H is the only hyperplane between A and sHA. +This leads to the combinatorics of (minimal) galleries. +Definition 3.10. Two alcoves A, A′ ∈ A are called adjacent if A′ = sHA for some H ∈ HA. +A sequence of alcove Γ = (A0, A1, A2, . . . , Ar) such that Ai is adjacent to Ai+1 is called an +(alcove) gallery from A0 to Ar. A gallery Γ = (A0, A1, A2, . . . , Ar) such that r = d(A0, Ar) +is called a minimal gallery. Note that the set of walls {Hi | Hi between Ai−1 and Ai} is +precisely the set of hyperplanes between A0 and Ar in case of a minimal gallery Γ. We call +these hyperplanes the hyperplanes that are crossed by the gallery Γ. +Since weights can be contained in a hyperplane, we need a more restrictive notion of +hyperplanes between two alcoves. +Definition 3.11. For λ ∈ X denote by Aλ an alcove that contains λ. For λ, µ ∈ X and +choices of alcoves Aλ and Aµ, we call a hyperplane H ∈ H strictly between Aλ and Aµ if H +is between Aλ and Aµ and H ∩ {λ, µ} = ∅. +Note that in Definition 3.11, the choice of an alcove Aλ is unique if and only if λ is in the +interior of an alcove. Otherwise multiple choices are possible. We make frequent use of the +following lemma to make a particularly good choice. +Lemma 3.12. Let λ, µ ∈ X such that λ ∈ Wℓµ. Then there are choices of alcoves Aλ and +Aµ such that a minimal gallery Γ from Aλ to Aµ only crosses hyperplanes that are strictly +between Aλ and Aµ +Proof. Let Aλ and Aµ be any choice for alcoves with λ ∈ Aλ and µ ∈ Aµ. Let Γ be a minimal +gallery and assume that there exists a hyperplane H that is between Aλ and Aµ but not +strictly between. Without loss of generality assume λ ∈ H (otherwise just rename), hence +λ ∈ sHAλ and so sHAλ is also a choice for an alcove containing λ. By Lemma 3.8 it holds +d(Aλ, Aµ) > d(sHAλ, Aµ). Thus we can replace Aλ by sHAλ and the distance between Aµ +and the new Aλ, i.e. the length of a minimal gallery, strictly decreases. Since the distance +is bounded below by 0 this process has to end after finitely many steps and at that point +all hyperplanes crossed by a minimal gallery are strictly between the final choices of Aλ and +Aµ. +□ +3.4. Affine quantum symmetric pair. For this section fix r > 3. To define the necessary +quantum symmetric pairs we consider sets of the form Z/rZ respectively H/rZ. The restric- +tion of r > 3 is so to not consider special cases for small r. In general one can define similar +algebras also for r = 2 and r = 3. The relations change slightly in those cases. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +11 +We consider the Dynkin diagram of type ˆAr−1 with indices either labeled by entries in +Z/rZ or H/rZ +0 +1 +r − 1 +(r−2)/2 +−(r+2)/2 +r/2 +Z/rZ for r even +0 +1 +r − 1 +(r−3)/2 +(r+3)/2 +(r−1)/2 +(r+1)/2 +Z/rZ for r odd +1/2 +r − 1/2 +3/2 +r − 3/2 +(r−3)/2 +(r+3)/2 +(r−1)/2 +(r+1)/2 +H/rZ for r even +1/2 +r − 1/2 +3/2 +r − 3/2 +(r−2)/2 +(r+2)/2 +r/2 +H/rZ for r odd +For both index sets we consider the automorphism Θ given by Θ(p) = −p. Thus in each +of the pictured Dynkin diagrams, this is the horizontal reflection along the dotted horizontal +line. Depending on whether r is even or odd, and whether considering the index set H/rZ +or Z/rZ, Θ have between zero and two fixed points on the left or right of the diagram. +In either case of Z/rZ and H/rZ we call cosets p and q linked if p = q ± 1, i.e. if the +corresponding nodes in the affine Dynkin diagram are connected by an edge. +To simplify the notations for the quantum symmetric pair we rewrite one type of quantum +Serre relation in form of a non-commutative polynomial +SRv(x, y) = x2y − [2]vxyx + yx2, +for non-commuting variables x, y over Q(v). In contrast to the usual quantum group Uv(ˆslr) +the quantum Serre relations for the generators of the quantum symmetric pair depend on +the type of the node in the Dynkin diagram of the generator, i.e. they are not invariant +under translation of indices. +Definition 3.13. A p ∈ Z/rZ respectively p ∈ H/rZ is called +(1) a fixed index if Θ(p) = p, +(2) a Θ-linked index if Θ(p) is linked to p, and +(3) a standard index if p is neither fixed not Θ-linked. +The affine quantum symmetric pair is then the following associative algebra. +Definition 3.14. [Kol14] Let I ∈ {Z/rZ, H/rZ}. +The affine quantum symmetric pair +Bv(I, Θ) is defined to be the associative algebra over Q(v) generated by {ˆBp | p ∈ I} and +{ˆLq | q ∈ I, q ̸= Θ(q)}, subject to the following relations: +For p, q not fixed it holds +ˆLpˆLq = ˆLqˆLp, +ˆLpˆLΘ(p) = 1. +For p standard, r Θ-linked, s fixed, and q not fixed it holds +ˆLq ˆBp = + + + + + + + + + + + + + +v2 ˆBpˆLq +if p = q +v−2 ˆBpˆLq +if p = Θ(q), +v−1 ˆBpˆLq +if p, q linked, +vˆBpˆLq +if p, Θ(q) linked, +ˆBqˆLq +otherwise, +ˆLq ˆBr = + + + + + + + + + + + + + +v3 ˆBrˆLq +if r = q, +v−3 ˆBrˆLq +if r = Θ(q), +v−1 ˆBrˆLq +if r, q linked, q ̸= Θ(r) +vˆBrˆLq +if r, Θ(q) linked, q ̸= r +ˆBrˆLq +otherwise, + +12 +M. EHRIG AND K. GAN +ˆLq ˆBs = ˆBsˆLq. +The generators ˆBp and ˆBq commute, unless +ˆBp ˆBΘ(p) − ˆBΘ(p)ˆBp = +ˆLp − ˆLΘ(p) +v − v−1 +for p standard +or +SRv(ˆBp, ˆBq) = + + + + + + + +0 +p, q linked, not fixed, and Θ(p) ̸= q, +0 +p, q linked, p standard, q fixed, +ˆBq +p, q linked, p fixed, q standard, +−[2]v ˆBp(vˆLp + v−2ˆLΘ(p)) +p, q Θ-linked , q = Θ(p). +Deriving these definitions from [Kol14] needs some clarifications. The relations between +the ˆBp are given in [Kol14, Theorem 7.4], note that in our language all the ci that appear +in [Kol14] are equal to 1 and the elements Zj = ˆLj with j an index in the Dynkin diagram. +The commutator relations between ˆBp and ˆLq are found in [Kol14, (7.7)]. +It is easier to think of Bv(I, Θ) as being an affine analogue of the quantum symmetric pair +of type AIII in [Let03, Section 7]. The relations for the quantum symmetric pair of type AIII +are nearly the same as for the ordinary quantum group, as in our case. There is either one +special generator, corresponding to a fixed index, or two special generators, corresponding +to a pair of Θ-fixed indices that do not behave like usual quantum group generators. In +contrast to the non-affine case, there are now two such situations. In the Dynkin diagrams +above these are the two areas where the dotted line intersects the circle. +Remark 3.15. Comparing this definition to the ones for non-affine quantum symmet- +ric pairs in [ES18, Proposition 7.17 and Proposition 7.18] (or the idempotent version in +[BSWW18] in case there are no Θ-linked indices), we see that locally the relations are the +same. The pairs ˇEi and ˇFi in the relations of [ES18] are the analogues of ˆBp and ˆBΘ(p) for +p standard. Since our generators are ordered in a cyclic way it is not reasonable to choose +a “positive” and “negative” one in such pairs. The generator ˇB in [ES18, Proposition 7.18] +is the analogue of ˆBp for a fixed index and the pair of generators ˇB+ and ˇB− in [ES18, +Proposition 7.17] are the analogue of ˆBp and ˆBΘ(p) for p being Θ-linked. +In contrast to [ES18] we use the analogue of a semi-simple Cartan, while [ES18] uses the +analogue of a reductive Cartan. This is just for simplicity of notation. One could easily +modify the definition and write every generator ˆBq as a product of two generators in the vein +of the definitions of [ES18]. +The only place where the relations differ is the last of the deformed quantum Serre relations +in Definition 3.14, with p being Θ-linked and q = Θ(p). This is due to the fact that the linear +operators B1/2 and B−1/2 in [ES18, Definition 7.8] that give the action of ˇB+ and ˇB− are not +symmetric in their definition. This was needed to match the grading of graded category O in +[ES18], but this is not necessary in our situation. +4. Type C +For type CN (N > 2) we choose X = �N +i=1 Zεi, where the εi are the projection onto +the i-th diagonal entries for the Cartan subalgebra of diagonal matrices inside sp2n(C). The +W-invariant bilinear form (−, −) is given such that the εi form an orthonormal basis. As +the positive roots in a root system Φ in X, we choose: +Φ+ = {β± +i,j = εi ± εj | 1 ≤ i < j ≤ N} ∪ {βi = 2εi | 1 ≤ i ≤ N}. +The simple roots for this choice are αi = β− +i,i+1 for 1 ≤ i < N and αN = βN. +The +corresponding coroots in X are then +(β± +i,j)∨ = β± +i,j and β∨ +i = εi. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +13 +As defined in Section 2.3 the elements di = 1 for 1 ≤ i < N and dN = 2, since αN is the +only simple long root. This is extended to all positive roots by having dβ = 2 if β = βi for +some i and 1 otherwise. +With these choices the fundamental weights are given as ωi = ε1 + . . . + εi. And thus the +dominant integral weights are +X+ = {(λ1, . . . , λN) | λi ∈ Z, λi ≥ λi+1, and λN ≥ 0}, +where we write row vectors with respect to the basis {εi}1≤i≤N. Then +ρ = +N +� +i=1 +ωi = Nε1 + (N − 1)ε2 + . . . + εN ∈ X+ +and thus +X+ +ρ = {(λ1, . . . , λN) | λi ∈ Z, λi > λi+1, and λN > 0}. +(Recall the convention that λ = λ + ρ for λ ∈ X+.) +Since there are roots with dβ ̸= 1 we have to distinguish the case of ℓ odd and ℓ even as +described in Section 2.3. In case ℓ odd, Wℓ is the affine Weyl group of type CN, just scaled +by the factor ℓ. While in case ℓ even, Wℓ is the affine Weyl group of type BN scaled by a +factor ℓ for the dual root system, i.e. where βi is replaced by εi with coroot 2εi instead. +For the action on [Uq-mod] we consider the functor −⊗C∆q(ω1), which is taking the tensor +product with the specialization of the quantum analogue of the natural representation. +Since the natural representation is minuscule one obtains the following tensor product +decomposition in the generic case. +Proposition 4.1. [HK02, Proposition 8.6.3] Let λ ∈ X+. Then in Uv(g)-mod it holds +∆v(λ) ⊗ ∆v(ω1) ∼= +� +i:λ+εi∈X+ +∆v(λ + εi) ⊕ +� +i:λ−εi∈X+ +∆v(λ − εi). +In this case the tensor product decomposition is straight forward: If λ + εi is dominant +the corresponding irreducible module appears and similarly for λ − εi. +4.1. Fock space of sequences and operators. To embed F(CN) into FN, we map λ to +the sequence aλ such that +aλ(i) = + + + + + +1 +if i ≤ 0, +1 +if there exists 1 ≤ j ≤ N s.t. i = λj, +0 +otherwise. +Note that by construction all non-positive entries in the sequence a are 1 and there are +exactly N entries equal to 1 in the strictly positive part. Hence the sequence aλ has charge +N. The map is obviously injective. By abuse of notation we simply write λ for the sequence +as well and identify F(CN) with its image. +Looking at the linear operators from Definition 3.2, we immediately see the following. +Lemma 4.2. Let λ ∈ F(CN) and r ∈ H. +Then erλ ∈ F(CN) and furthermore erλ is +non-zero if and only if λ − εi is dominant for r = λi − 1/2. +Similarly, frλ ∈ F(CN), for r ̸= 1/2, and λ + εi is dominant if and only if frλ is non-zero +for r = λi + 1/2. +Note that all operators er preserve the subspace F(CN) with er acting as zero for all +r ≤ 1/2. On the other hand the operators fr keep the subspace invariant, except for f1/2. +The obvious reason being that f1/2 can move a 1 from position 0 to position 1, leaving the +subspace F(CN). + +14 +M. EHRIG AND K. GAN +4.2. Linkage and operators. We are now considering when two weights obtained from +λ by applying two of the operators from above are linked under the assumption that the +sequences stay inside F(CN). +Remark 4.3. In the following we often just say: assume µ = erλ is defined. By this we +mean that erλ is again a sequence coming from the embedding of F(CN). +Even though the roots of type C are different from type A, the following Lemma has +pretty much the same proof as in type A, as the roots of the form β− +ij are the only ones +playing a role. +Lemma 4.4. Let λ ∈ X+ +ρ . Assume µ = erλ and ν = esλ, for r ̸= s are defined. Then µ +and ν are linked if and only if r ∈ s + ℓZ. +Assume µ = frλ and ν = fsλ, for r ̸= s, are defined. Then µ and ν are linked if and only +if r ∈ s + ℓZ. +Proof. It holds µ = λ − εi where i is determined by λi = r + 1/2. Similarly es defines j such +that λj = s + 1/2. Since s ̸= r we can assume that i < j. +Assume now that the two weights are linked. +Using Lemma 3.12, fix alcoves Aµ and +Aν such that a minimal gallery Aµ = A0, . . . , At = Aν only crosses walls that are strictly +between Aµ and Aν. +It holds +(ν − µ, β∨) = + + + + + + + + + +2 +if β = β− +i,j, +1 +if β ∈ {β− +i,k, β+ +i,k, β+ +k,i | k ̸= j} ∪ {βi}, +−1 +if β ∈ {β− +k,i | k < i}, +0 +otherwise. +Since we assume that µ ̸= ν, but µ and ν are linked, the minimal gallery needs to have at +least length 1. Thus there exists a hyperplane strictly between Aλ and Aµ, otherwise the +two alcoves agree which is a contradiction to the simply transitive action of Wℓ on the set +of alcoves. +Looking at the values (ν − µ, β∨) above, the only root that can have an affine hyperplane +strictly between Aµ and Aν is β = β− +ij. Hence the length of the minimal gallery is exactly 1 +and it holds ν = sβ− +i,j,mµ for some m ∈ Z. Let H = Hβ− +ij,m be this hyperplane. Hence +1 + mℓ = (µ, (β− +i,j)∨) = λi − λj + 1. +So we obtain that λi = λj + mℓ and so r ∈ s + ℓZ. +Now assume that r ∈ s+ℓZ. Then λi−λj = mℓ for some m ∈ Z. Hence (ν, (β− +ij)∨) = 1+mℓ +and thus sβ− +ij,m(ν) = µ and so the weights µ and ν are linked. +The statement for the f-operators is done completely analogous. +□ +Lemma 4.4 can be used word for word to see that in type A linked weights are obtained +by operators with indices congruent modulo ℓ. +Lemma 4.5. Let λ ∈ X+ +ρ . Assume µ = erλ and ν = fsλ, for r + 1 ̸= s, are defined. Then +µ and ν are linked if and only if r ∈ −s + ℓZ. +Proof. It holds µ = λ − εi where i is determined by λi = r + 1/2. Similarly fs defines j such +that λj = s − 1/2. Since s ̸= r + 1 it holds that i ̸= j. +Assume that the weights are linked. Fix Aµ and Aν via Lemma 3.12 such that the minimal +gallery only crosses hyperplanes that are strictly between the two alcoves. +Then we check that the only possibility for |(µ − ν, β∨)| > 1 is (µ − ν, β∨) = 2 for the +choice β = β+ +i,j (Technically β+ +i,j is only defined for i < j, so for i > j we just set β+ +i,j = β+ +j,i). + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +15 +Thus the unique affine hyperplane strictly between Aµ and Aν is of the form Hβ+ +i,j,m and so +it holds ν = sβ+ +i,j,mµ. In formulas for the weights we thus get +1 + mℓ = (µ, (β+ +i,j)∨) = λi + λj + 1. +So we obtain λi = −λj + mℓ and thus so r ∈ −s + ℓZ. +Assume that r ∈ −s + ℓZ, then λi + λj = mℓ for some m ∈ Z. Hence (ν, (β+ +ij)∨) = 1 + mℓ +and thus sβ+ +ij,m(ν) = µ and so the weights µ and ν are linked. +□ +This is the main difference to type A. The relationship of the form r ∈ −s + ℓZ forces us +to fix a type of origin. Hence the embedding of F(CN) depends on N in this case. +Lemma 4.6. Let ℓ be odd and λ ∈ X+ +ρ . Assume µ = er−1λ and ν = frλ are defined. Then +µ and ν are linked if and only if r ∈ 1/2 + ℓZ. +Proof. In this case there exists i such that µ = λ − εi and ν = λ + εi for i determined by +λi = r − 1/2. +Assume that the weights are linked and fix alcoves Aµ and Aν such that a minimal gallery +only crosses hyperplanes strictly between the alcoves, using Lemma 3.12. +Checking the values of (ν − µ, β∨), there are multiple choices for hyperplanes that can be +strictly between the chosen alcoves. Namely we have +(ν − µ, β∨) = + + + + + + + + + +2 +if β = βi, +2 +if β ∈ {β− +i,j, β+ +i,j, β+ +j,i | j ̸= i}, +−2 +if β ∈ {β− +j,i | j ̸= i}, +0 +otherwise. +Denote by H1 = Hγ1,m1, . . . , Ht = Hγt,mt be the hyperplanes crossed by the minimal gallery. +By construction each Hi is strictly between Aµ and Aν, hence for each γi it holds (µ−ν, γ∨ +i ) = +±2 and especially (λ, γ∨ +i ) = miℓ, i.e. λ ∈ Hi. +We now go through the different cases for γ1. +Case: γ1 = βi: In this case it holds that +1 + m1ℓ = (ν, β∨ +i ) = λi + 1. +Thus λi ∈ ℓZ and so r ∈ 1/2 + ℓZ. Since in this case it immediately holds sβi,m1ν = µ it +follows that sβi,m1Aν = Aµ and so the minimal gallery had length 1. +Case: γ1 = β− +i,j for some j > i: In this case the equation is +1 + m1ℓ = (ν, (β− +i,j)∨) = λi − λj + 1. +Thus λi = λj + m1ℓ and it holds sβ− +i,j,m1ν = λ + εj. For every hyperplane Hq for q > 1 and +γq ̸= β+ +i,j it holds mqℓ = (λ, γ∨ +p ) = (λ + εj, γ∨ +p ), hence λ + εj ∈ Hq. Thus applying sγq,mq +leaves λ + εj invariant. Since λ + εj ̸= µ, there must exists p > 1 such that γp = β+ +i,j. In +which case +1 + mpℓ = (λ + εi, (β+ +i,j)∨) = (λ + εj, (β+ +i,j)∨) = λi + λj + 1. +Hence λi = −λj + mpℓ. Thus 2λi = (mp + m1)ℓ. Since 2λi is an even integer and ℓ is odd, +(mp + m1) is even and so λi ∈ ℓZ and equivalently r ∈ 1/2 + ℓZ. +Case: γ1 = β+ +i,j for some j > i: This is nearly identical to the previous case. The only +difference is that one first obtains λi ∈ −λj + ℓZ, applying the first reflection gives λ − εj, +and the second used hyperplane is for β− +i,j and one obtains λi ∈ λj + ℓZ. The rest of the +argument is then the same. +Case: γ1 = β− +j,i for some j < i: In this case we start with +−1 + m1ℓ = (ν, (β− +j,i)∨) = λj − λi − 1. + +16 +M. EHRIG AND K. GAN +Hence λi ∈ λj + ℓZ and sγ1,m1(ν) = λ + εj and the rest is as in the cases before. +Case: γ1 = β+ +j,i for some j < i: This similarly follows as the previous cases. +For the converse, assume r ∈ 1/2 + ℓZ. Then λi = mℓ for some m ∈ Z. Hence (ν, (βi)∨) = +1 + mℓ and thus sβi,m(ν) = µ and so the weights µ and ν are linked. +□ +For the case of ℓ odd Lemma 4.6, gives a special case of Lemma 4.5 with a slightly more +delicate proof. This special case will lead to generators for the quantum symmetric pair of +non-standard indices. The special case for ℓ even is handled in the next statement. +Lemma 4.7. Let ℓ be even and λ ∈ X+ +ρ . Assume µ = er−1λ and ν = frλ are defined. Then +µ and ν are linked if and only if r ∈ 1/2 + (ℓ/2)Z. +Proof. The arguments in the proof follow the ones for Lemma 4.6 so we only point out the +differences. +Case: +γ1 = βi: In contrast to Lemma 4.6 we now obtain that λi ∈ (ℓ/2)Z and so +r ∈ 1/2 + (ℓ/2)Z. This is of course due to ℓβi = ℓ/2. +In all other cases we obtain 2λi ∈ ℓZ as before, but now we can simply divide ℓ by 2 and +obtain λi ∈ (ℓ/2)Z, which in turn implies r ∈ 1/2 + (ℓ/2)Z. +For the converse, assume r ∈ 1/2 + (ℓ/2)Z. +Then λi = mℓ/2 for some m ∈ Z. +Hence +(ν, (βi)∨) = 1 + mℓ/2 and thus sβi,m(ν) = µ since ℓβi = ℓ/2 and so the weights µ and ν are +linked. +□ +In contrast to Lemma 4.6, in Lemma 4.7 we see that for ℓ even the situation that er−1 +and fr produce linked weights happens for two types of positions, for a 1 at a position in ℓZ +or at a position in ℓ/2 + ℓZ. This is reflected in the existence of two Θ-linked pairs of indices +in the Dynkin diagram for H/ℓH and ℓ even in Section 3.4. +4.3. Quantum symmetric pair action. We now define operators on FN as follows. Recall +the automorphism Θ : H/ℓZ → H/ℓZ from Section 3.4 that changes the sign. +Definition 4.8. Let a ∈ SZ and p ∈ H/ℓZ. For Θ(p) ̸= p we define +ˆBp a = vT e−f +ℓ +(Θ(p),a) � +j∈p +vRe−f +ℓ +(j,a)ej a + +� +j∈−p +vLf−e +ℓ +(j,a)fj a and +ˆLpa = vT f−e +ℓ +(p,a)vT e−f +ℓ +(Θ(p),a)a. +For Θ(p) = p, i.e. p = ℓ/2, we define +ˆBℓ/2a = v−1vT e−f +ℓ +(ℓ/2,a) � +j∈ℓ/2 +vRe−f +ℓ +(j,a)eja + +� +j∈ℓ/2 +vLf−e +ℓ +(j,a)fja. +This defines linear operators on F that restrict to FN for any N. +As mentioned at the very end of Section 3.2, the action of Uv(ˆslℓ)′ is defined on any FN +with the same definitions as for F0. Thus by comparing we get the following. +Lemma 4.9. Let a ∈ SZ and p ∈ H/ℓZ. For Θ(p) ̸= p +ˆBp a = ˆEp ˆK−1 +−p a + ˆF−p a +and +ˆLpa = ˆKp ˆK−1 +−pa +and +ˆBℓ/2 a = v−1ˆEℓ/2 ˆK−1 +ℓ/2 a + ˆFℓ/2 a +The definition of the linear operators ˆBp is the reason why we do not embed into the Fock +space of charge 0 (as one does in type A). The linear operator ˆBp involves the ei’s for i ∈ p, +but also fj’s for −j ∈ p. Thus the definition is not invariant under arbitrary translation as +the linear operators for the type A action, only under translation by multiples of ℓ. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +17 +Remark 4.10. Note that for λ ∈ F(CN) it holds ˆBpλ ∈ F(CN) if −1/2 /∈ p. For ˆBℓ/2 +one just needs to consider the summand using f1/2 that can produce a sequence that is not +contained in F(CN). This is exactly parallel to the situation in type AN−1 where one can +apply an operator ˆFp, that in the language of Young diagrams, creates a box in row N + 1, +thus leaving the span of polynomial weights for glN. +For the relations of the operators we then get the following. +Proposition 4.11. The linear operators {ˆBp | p ∈ H/ℓZ} and {ˆLp | p ∈ H/ℓZ, Θ(p) ̸= p} +satisfy the relations given in Definition 3.14 by substituting the operators for the generators +with the same name. +Proof. Since we did not specify how to make all the choices in [Kol14], we give a short sketch +of how to quickly check that the relations are indeed satisfied. +The relations between the ˆLp are immediate by definition. +They all multiply a basis +element with a fixed scalar and the scalars for ˆLp and ˆL−p are inverse to each other. +The commuting relations between ˆLq and ˆBp are a straight-forward calculation using +Lemma 4.9 above. +That ˆBp and ˆBq commute unless for the specified index choices is also immediate from +the definition. Either the summands of ˆBp and ˆBq modify a basis vector at positions that +are not neighboured, i.e. the cosets are not linked and so the summands commute. Or, in +case that Θ(p) and q are linked, their summands can modify the same position by moving a +1 into different directions or moving two 1’s into the same position. In this case the product +of the summands is just always zero, hence they commute. +The commutator relation between ˆBp and ˆBΘ(p) is a direct and simple computation using +Lemma 4.9. +Finally the deformed and non-deformed Serre relations follow from [ES18] (or precisely +their use of [Let03]), since for a linked pair of indices p and q the Serre relation are indepen- +dent of the rest of the Dynkin diagram, the calculation only involves the relation between +the standard generators of the quantized enveloping algebra for the indices p, q, Θ(p), and +Θ(q), which are local and hence the same in the affine case and in [ES18], since ℓ > 3. +□ +Thus putting everything together we obtain the statement. +Theorem 4.12. There exists an action of Bv(H/ℓZ, Θ) on FN such that for λ ∈ X+ the +decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules +is obtained from +� +p∈H/ℓZ +ˆBpλ +by projecting onto the subspace F(CN) and evaluating v = 1. Furthermore if [∆q(µ)] and +[∆q(ν)] appear in the decomposition with µ and ν linked, then there exists a unique p ∈ H/ℓZ +such that µ and ν appear in ˆBpλ. +Proof. The action is the one from Proposition 4.11. That the decomposition is given by +the sum of all operators is Proposition 4.1 together with Proposition 2.5 to obtain the +translation of the decomposition into weight combinatorics and then Lemma 4.2 together +with the definition of the operators themselves. +That classes of two Weyl modules with linked weights are obtained from a unique operator +ˆBp is then Lemmas 4.4, 4.5, 4.7, and 4.6, . +□ +We want to address now the question of embedding into a single Fock space for different +ranks. +In type A, every F(AN) can be embedded into F0. This was natural on a combinatorial +level due to the choice of gl instead of sl and hence the ability to choose the element ρ. The + +18 +M. EHRIG AND K. GAN +choice was made such that the weight 0 always gets mapped to the sequence with all 1’s +in the non-positive half and 0’s in the positive half. From the point of view of the affine +operators and the action, any shift of the origin, i.e. an embedding into a different Fk makes +no difference, since the operators only involves entries that are congruent mod ℓ. Hence an +embedding into a different Fk is just related to a shift in the used operators. On the Lie +theory side this would just correspond to a different choice of ρ. +We can make a similar construction in type C if we restrict to certain N. Although this +is more artificial since on the Lie theory side, the element ρ is fixed. +Definition 4.13. Let N = mℓ + k for m ≥ 0 and 0 ≤ k < ℓ. Then we define a map from +FN to Fk via a(m)(i) = a(i + mℓ) for a ∈ SZ,N. We call a(m) the sequence shifted by m +ℓ-steps. +In this way we can view F(CN) as a subspace of Fk, which we call the shifted embedding +of F(CN). +That the map in Definition 4.13 is well-defined follows immediately with a simple cal- +culation for the charge. Since it intertwines the action of ei and ei−mℓ (and similar of fi +and fi−mℓ), the affine operators ˆBp and ˆLp defined on FN and Fk as restrictions from F +commute with the map of shifting a sequence by m steps. In terms of weights this is the +same as looking at the weight λ + ρ − mℓ(1, . . . , 1) which for most m is not dominant. +Thus using this embedding for a fixed 0 ≤ k < ℓ we can formulate this as follows. +Proposition 4.14. Let a ∈ SZ,k and r ≥ 0. Then for m ≫ 0, a = λ(m) for λ a dominant +weight for Uq of type Cmℓ+k. Furthermore there are Laurent polynomials dλ,µ(v) with non- +negative integer coefficients for µ dominant for Uq such that +� +(H/ℓZ)r +ˆBp1 · · · ˆBprλ(m) = +� +µ dominant for Uq +dλ,µ(v)µ(m), +and +[∆q(λ) ⊗ ∆q(ω1)⊗r] = +� +µ dominant for Uq +dλ,µ(1)[∆q(µ)]. +Proof. For a we consider the left most 0 in the non-positive part of the sequence. If this zero +is at position −n′ then using operators of the form ei this can be reduced to a sequence b +that has no 0’s in the non-positive part and exactly k 1’s at the positions 1, . . . , k. Then b +is the image of 0 under the shifted embedding of F(Cm′ℓ+k) for m′ℓ > n′. Then by applying +operators fi in the reverse order to the sequence of operators ei before one obtains a dominant +weight λ with a = λ(m′). If there was no 0 in the non-positive part then one can just use +m′ = 0. +A dominant weight λ for Uq of rank m′ℓ + k can be viewed as a dominant weight for Uq +of rank mℓ + k for m > m′, by just filling it up with 0 at the last (m − m′)ℓ entries. Choose +m > m′ such that mℓ > n′ + r. By our discussion about which fi can leave the embedded +subspace FN, after Lemma 4.2, we see that no product ˆBp1 · · · ˆBpr applied to a can create +a weight that is not in the shifted embedding of F(Cmℓ+k), since at most the 1’s at the +position −n′ − 1, . . . , −n′ − r can be moved to the right. Hence the statement then follows +from Theorem 4.12. +□ +Note that the proof of Proposition 5.13 gives a clear bound for what m needs to be, it is +just necessary that mℓ > n′ + r for n′ the position of the left most entry 0. +Remark 4.15. Note that the definition of the operators ˆBp is not uniquely determined. As +already mentioned in Remark 3.15, one can change some coefficients in the definition of the +generators and obtain a slightly modified algebra. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +19 +Since the definition in type A is not unique as well, compare for example the definition +in [RT10, Theorem 3.1] and in [Ari02, Section 10.1], one can make similar modifications in +type C. +5. Type B +In type BN (N > 1), we choose εi ∈ h∗ the projection onto the i-th diagonal entries for +the Cartan subalgebra of diagonal matrices inside so2n+1(C). The W-invariant bilinear form +(−, −) is given such that the εi form an orthonormal basis. For the positive roots we choose +Φ+ = {β± +i,j = εi ± εj | 1 ≤ i < j ≤ N} ∪ {βi = εi | 1 ≤ i ≤ N} +and similar to type C, the simple roots are αi = β− +i,i+1 for 1 ≤ i < N, but here αN = βN. +The corresponding coroots are +(β± +i,j)∨ = β± +i,j and β∨ +i = 2εi. +Following Section 2.3 we have di = 2 for 1 ≤ i < N and dN = 1, since αN is the only simple +short root in this case. This gets extended to all positive roots via dβ = 2 for β = βi for +some i and dβ = 1 otherwise. +The noticeable change is in the integral weight lattice. +The fundamental weights are +ωi = ε1 + . . . + εi for i < N and ωN = 1/2(ε1 + . . . + εN). Hence for the integral weights +X = �N +i+1 Zωi, the dominant weights X+ that can be naturally divided into two subsets +X +1/2,+ = {(λ1, . . . , λN) | λi ∈ Z, λi ≥ λi+1, and λN ≥ 0} and +X1,+ = {(λ1, . . . , λN) | λi ∈ H, λi ≥ λi+1, and λN ≥ 0}, +written as row vectors with respect to {εi}1≤i≤N inside h∗. We call X +1/2,+ the integer weights, +not to be confused with the integral weights, and X1,+ the half-integer weights. +Summing up the fundamental weights we get +ρ = +N +� +i=1 +ωi = (N − 1/2)ε1 + (N − 3/2)ε2 + . . . + 1/2εN ∈ X +1/2,+. +Thus adding ρ we obtain the sets to define sequences, namely +X +1/2,+ +ρ += {(λ1, . . . , λN) | λi ∈ H, λi > λi+1, and λN > 0}, +X1,+ +ρ += {(λ1, . . . , λN) | λi ∈ Z, λi > λi+1, and λN > 0}, +again recalling that λ = λ + ρ for λ ∈ X+. +Remark 5.1. This makes clear our convention of naming the set of integer weights X +1/2,+ +and the set of half-integer weights X1,+ (and not the other way around). After adding ρ, +elements in X +1/2,+ +ρ +have entries in H and elements of X1,+ +ρ +have entries in Z. Thus the +naming convention makes it easy to recognise what the domains for the sequences in the +different cases are. +Again the parity of ℓ is important. For ℓ odd, Wℓ is the affine Weyl group of type BN, +scaled by the factor ℓ. In case ℓ even, Wℓ is the affine Weyl group of type CN scaled by a +factor ℓ for the dual root system, i.e. where one would replace β± +ij in the root system by +1/2β± +ij. +For the action on Uq-mod we use the functor − ⊗C ∆q(ω1), again the tensor product with +the the specialization of the quantum analogue of the natural representation. + +20 +M. EHRIG AND K. GAN +Remark 5.2. Note that in contrast to the case of Uv(g) of type A or C, ∆v(ω1) is not a +tensor generator of Uv(g)-mod, hence there are other possible finite dimensional represen- +tations, like the specialization of the spin representation ∆q(ωN), that can give interesting +actions on Uq-mod. +In type B the natural representation is not minuscule. Hence the tensor product decom- +position has a slight complication. +Proposition 5.3. [HK02, Proposition 8.6.3] For λ ∈ X+ it holds in Uv(g)-mod +∆v(λ) ⊗ ∆v(ω1) ∼= +� +i:λ+εi∈X+ +∆v(λ + εi) ⊕ +� +i:λ−εi∈X+ +∆v(λ − εi) ⊕ ∆v(λ)⊕δpos, +where δpos = 0 if λN = 0 and δpos = 1 if λN > 0. +Thus in type B the tensor product decomposition for weights in X1,+ is straight-forward +with the only difference to type C being the appearance of a summand ∆v(λ) itself. Even +the sequences are looking very similar to the type C case. +For weights in X +1/2,+ on the other hand, the tensor product decomposition rule depends +on the explicit weight of the Weyl module, namely the summand ∆v(λ) only appears if +λN > 0. In addition we are dealing with sequences on half-integers in this case. +5.1. Fock space of sequences and operators. We decompose the Fock space as +F(BN) = F1(BN) ⊕ F +1/2(BN), +with F1(BN) having basis λ ∈ X1,+ +ρ +and F +1/2(BN) having basis λ ∈ X +1/2,+ +ρ +. For F1(BN) we +can use the same definition of sequences and Fock spaces as in type C, and embed F1(BN) +into FN as in Section 4.1. +For the case of F +1/2(BN) we use the sequences SH and corresponding Fock space F +1/2 from +Section 3. Hence we map λ ∈ X +1/2,+ +ρ +to the sequence aλ with +aλ(i) = + + + + + +1 +if i < 0, +1 +if there exists 1 ≤ j ≤ N s.t. i = λj, +0 +otherwise +to embed F +1/2(BN) into F +1/2. As before it is clear that the image is contained in the subspace +F +1/2 +N of sequences of charge N. We continue to write λ for the sequence as well. +The analogue of Lemma 4.2 holds in type B as well, i.e. +ei preserves the subpsaces +F1(BN) and F +1/2(BN), fi does so for i /∈ {0, 1/2}, and fi does not preserve the subpsaces for +i ∈ {0, 1/2}. +In contrast to type C, the cases of ℓ odd and even have to be treated separately. +5.2. Linkage and operators for odd ℓ. Throughout this section we assume that +ℓ > 3 is odd. +The proofs in this section have a very similar flavour to the ones in Section 4.2. Unfortunately +there are some subtle differences. We start with the statements that work for both the integer +and the half-integer weight cases. +Lemma 5.4. Let λ ∈ X+ +ρ and ℓ odd. Assume that µ = erλ and ν = esλ are defined for +r ̸= s. Then µ and ν are linked if and only if r ∈ s + ℓZ. +Assume that µ′ = frλ and ν′ = fsλ are defined for r ̸= s. Then µ′ and ν′ are linked if and +only if r ∈ s + ℓZ. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +21 +Proof. As in Lemma 4.4, µ = λ − εi with λi = r + 1/2 and ν = λ − εj with λj = s + 1/2 and +we assume that i < j. +Assume now that µ and ν are linked. Use Lemma 3.12 to obtain alcoves Aµ and Aν with +a minimal gallery only crossing walls strictly between the alcoves. The absolute value of +(ν − µ, β∨) can be 2 in case of β ∈ {β− +ij, βi, βj}. In case that a hyperplane of the form Hβ− +ij,m +is strictly between the alcoves, it follows as in Lemma 4.4 that r ∈ s + ℓZ. +Hence assume that there is no hyperplane of the form Hβ− +ij,m strictly between the alcoves. +Since ν − µ = εi − εj there has to be both a hyperplane of the form Hβi,m′ and of the form +Hβj,m′′ strictly between the alcoves since we assume that the weights are linked. This implies +2λi = m′ℓ and 2λj = m′′ℓ. Hence 2(λi − λj) = (m′ − m′′)ℓ. Since ℓ is odd (m′ − m′′) has to +be even and so Hβ− +ij,(m′−m′′)/2 is strictly between the alcoves, a contradiction. +Conversely assume that r ∈ s + ℓZ. Then λi − λj is a multiple of ℓ. Hence (λ, (β− +ij)∨) = +λi − λj is a multiple of ℓ and so (ν, (β− +ij)∨) = 1 + mℓ for some m ∈ Z. Thus sβ− +ij,m(ν) = µ +and so the weights are linked. +The analogous statement for fi and fj holds with the analogous proof. +□ +For the case of mixed operators we get accordingly. +Lemma 5.5. Let λ ∈ X+ +ρ and ℓ odd. Assume that µ = erλ and ν = fsλ are defined for +r + 1 ̸= s. Then µ and ν are linked if and only if r ∈ −s + ℓZ. +Proof. The proof is nearly identical to Lemma 5.4. Fix i ̸= j via µ = λ − εi and ν = λ + εj. +Then the roots that give absolute values strictly bigger than 1 for (ν − µ, β∨) = (εi + εj, β∨) +are β ∈ {β+ +ij, βi, βj}. +Assume that the weights are linked. Fixing the alcoves as usual and the corresponding +minimal gallery, we see that if Hβ+ +ij,m is strictly between the alcoves for some m then the +statement follows immediately. If one assumes that such a hyperplane does not exist, the +contradiction is obtained as in the proof of Lemma 5.4. +If it holds r ∈ −s+ℓZ, then (λ, (β+ +ij)∨) = λi+λj is a multiple of ℓ and so (ν, (β− +ij)∨) = 1+mℓ +for some m ∈ Z. Thus sβ+ +ij,m(ν) = µ and so the weights are linked. +□ +For the analogue of Lemmas 4.6 and 4.7 we get the following. +Lemma 5.6. Let ℓ odd. Assume λ ∈ X1,+ +ρ +and that µ = er−1λ and ν = frλ are defined. +Then µ and ν are linked if and only if r ∈ 1/2 + ℓZ. +Assume λ ∈ X +1/2,+ +ρ +and that µ = er−1λ and ν = frλ are defined. Then µ and ν are linked +if and only if r ∈ (ℓ+1)/2 + ℓZ. +Proof. Fix i via µ = λ − εi respectively ν = λ + εi, i.e. with λi = r − 1/2. +Assume that µ and ν are linked. Again use Lemma 3.12 to obtain alcoves Aµ and Aν +with a minimal gallery only crossing walls strictly between the alcoves. The possible positive +roots such that the absolute value |(ν − µ, β∨)| > 1 are β = βi for the value 4 and for the +value 2 it is β = β± +ij for i < j or β = β± +ji for j < i . +Let A0, . . . , At be the alcoves in the minimal gallery connecting Aµ and Aν and H1, . . . , Ht +the crossed hyperplanes in order, all strictly between the alcoves by Lemma 3.12. Fix positive +roots and integers such that Hp = Hγp,mp. Recall that for all these it holds |(ν − µ, γ∨ +p )| > 1. +Case: γ1 = βi and r = 1: Then (ν, β∨ +i ) = 2 + m1ℓ since we need to have sβi,m1(ν) = µ. +Thus 2λi + 2 = 2 + m1ℓ. In case that λ ∈ X1,+ +ρ +, it holds that m1 must be even (since ℓ is +odd) and so λi ∈ ℓZ or equivalently r ∈ 1/2 + ℓZ. If on the other hand λ ∈ X +1/2,+ +ρ +, then m1 +must be odd and we obtain λi ∈ ℓ/2 + ℓZ or equivalently r ∈ (ℓ+1)/2 + ℓZ. + +22 +M. EHRIG AND K. GAN +Case: γ1 = β− +ij for some j: Then it holds that (ν, (β− +ij)∨) = 1+m1ℓ, sβ− +ij,m1(ν) = λ+εj ̸= +µ, and λi−λj = m1ℓ. Now for q > 1 it holds (λ+εj, γ∨ +q ) ∈ ℓZ and thus sγq,mq(λ+εj) = λ+εj +unless γq ∈ {βi, β+ +ij}. Hence either βi or β+ +ij has to appear as a γq for q > 1. Let p > 1 be +the minimal such that γp ∈ {βi, β+ +ij}. +If γp = β+ +ij then +(λ + εj, (β+ +ij)∨) = (ν, (β+ +ij)∨) = 1 + mpℓ. +Hence λi + λj = mpℓ and so 2λi = (m1 + mp)ℓ. As above, for λ ∈ X1,+ +ρ +, it follows that +r ∈ 1/2 + ℓZ and for λ ∈ X +1/2,+ +ρ +it follows that r ∈ (ℓ+1)/2 + ℓZ. Furthermore, we see that +γq ̸= βi for all q since we already have sβ+ +ij,m′(λ + εj) = µ in this case. +If γp = βi then +(λ + εj, (βi)∨) = (ν, (βi)∨) − 2. +But Hβi,mp is assumed to be strictly between the alcoves, thus (ν, (βi)∨) = k + mpℓ for +k ∈ {1, 2, 3} (k = 4 is not possible since µ would lie on Hβi,mp in that case). Then (λ + +εj, (βi)∨) = k − 2 + mpℓ. +(1) If k = 1, then Hβi,mp was already crossed, which cannot be the case. +(2) If k = 2 then λ + εj is on Hβi,mp and invariant under sβi,mp, hence β+ +ij has to appear +in the sequence of γ’s that follow and one argues as above. +(3) If k = 3, it holds sβi,mp(λ + εj) = λ + εj − εi. But (λ + εj − εi, γ∨ +q ) = (µ, γ∨ +q ) for +q > p except for γq = β+ +ij and for β+ +ij it holds (λ+εj −εi, (β+ +ij)∨) = (λ, (β+ +ij)∨). Hence +λ + εj − εi is already in the same half-space as µ for all Hq with q > p and γq ̸= β+ +ij +and it is on the hyperplane Hq for q > p and γq = β+ +ij. Thus the only hyperplane +left to cross is Hp+1 and it must hold that γp+1 = β+ +ij, but sβ+ +ij,mp+1(λ + εj − εi) = +λ + εj − εi ̸= µ, which is a contradiction to the construction of the gallery. +Remaining cases: The remaining cases for γ1 are all done in the same way as the previous +case with slight modifications in the signs that appear, but the same general arguments. +Conversely for λ ∈ X1,+ +ρ +and r ∈ 1/2 + ℓZ, λi = mℓ for some m. Hence (λ + εi, β∨ +i ) = +2λi + 2 = 2 + 2mℓ. Thus sβi,2m(ν) = µ and so ν and µ are linked. +For λ ∈ X +1/2,+ +ρ +and r ∈ (ℓ+1)/2 + ℓZ, λi = ℓ/2 + mℓ. Hence (λ + εi, β∨ +i ) = 2λi + 2 = +2 + (2m + 1)ℓ. Thus sβi,2m+1(ν) = µ and so ν and µ are linked. +□ +In contrast to type C, λ can be linked to a weight µ obtained by applying an operator. +Lemma 5.7. Let λ ∈ X1,+ +ρ +and ℓ is odd. Assume that µ = erλ is defined. Then µ and λ +are linked if and only if r ∈ ℓ/2 + ℓZ. +Assume that ν = frλ is defined. Then ν and λ are linked if and only if r ∈ ℓ/2 + ℓZ. +Proof. Fix i via µ = λ − εi with λi = r + 1/2. +Assume now that µ and λ are linked. The absolute value of (λ − µ, β∨) can only be 2 for +β = βi. Hence there exists a hyperplane Hβi,m strictly between and so (λ, β∨ +i ) = 2λi = 1+mℓ. +Since 2λi is an even integer and ℓ is odd, m is odd as well. So we get r = ℓ/2+ m−1 +2 ℓ ∈ ℓ/2+ℓZ. +If r ∈ ℓ/2 + ℓZ then λi = (ℓ+1)/2 + mℓ for some m ∈ Z. Hence (λ, β∨ +i ) = 1 + (2m + 1)ℓ and +so λ and µ are linked. +The case of ν follows, considering that in this case erν = λ. +□ +The proof for the next Lemma is completely analogous to Lemma 5.7. +Lemma 5.8. Let λ ∈ X +1/2,+ +ρ +and ℓ is odd. Assume that µ = erλ is defined. Then µ and λ +are linked if and only if r ∈ ℓZ. +Assume that ν = frλ is defined. Then ν and λ are linked if and only if r ∈ ℓZ. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +23 +5.3. Quantum symmetric pair action for odd ℓ. As in the previous section we are +assuming throughout the section that +ℓ > 3 is odd. +Recall the various operators and counting statistics from Section 3.1. In contrast to type +C, we need to use operators for both types of index sets. +The definition of the linear operators is very close to the ones of type C in Section 4.3. +with one difference for the elements of H/ℓZ respectively Z/ℓZ that are fixed by Θ as defined +in Section 3.4. Note that for ℓ odd, Θ(i) = i implies that i = ℓ/2 for i ∈ H/ℓZ and i = 0 for +i ∈ Z/ℓZ. This corresponds to the two fixed nodes in the odd Dynkin diagrams in Section +3.4. +Definition 5.9. Assume ℓ odd. For a ∈ SZ let p ∈ H/ℓZ and for a ∈ SH let p ∈ Z/ℓZ. If +Θ(p) ̸= p define +ˆBp a = vT e−f +ℓ +(−p,a) � +j∈p +vRe−f +ℓ +(j,a)ej a + +� +j∈−p +vLf−e +ℓ +(j,a)fj a. +ˆLpa = vT f−e +ℓ +(p,a)vT e−f +ℓ +(−p,a)a. +If Θ(p) = p define +ˆBpa = v−1vT e−f +ℓ +(p,a) � +j∈p +vRe−f +ℓ +(j,a)eja + +� +j∈p +vLf−e +ℓ +(j,a)fja + vT e−f +ℓ +(p,a)a. +Finally, for a ∈ SH and z ∈ 1/2 + ℓZ define +ˆB[z] +0 a = v−1vT e−f +ℓ +(0,a) � +j∈0 +vRe−f +ℓ +(j,a)eja + +� +j∈0 +vLf−e +ℓ +(j,a)fja + δa(z),0vT e−f +ℓ +(0,a)a, +where δa(z),0 = 1− a(z). This defines linear operators on F and F +1/2 that restrict to FN and +F +1/2 +N +for any N. +The linear operator ˆB[z] +0 is not contained in the image of the affine quantum symmetric pair. +It is needed to describe the relationship with the tensor product decomposition. Comparing +this with the statements in Lemma 4.9, the same identifications hold except for the case +Θ(p) = p. +Lemma 5.10. Let a ∈ SZ (respectively a ∈ SH) and p ∈ H/ℓZ (respectively p ∈ Z/ℓZ) with +Θ(i) = i then +ˆBp a = v−1ˆEp ˆK−1 +p +a + ˆFp a + ˆK−1 +p a +and for a ∈ SH and z ∈ 1/2 + ℓZ +ˆB[z] +p a = v−1ˆEp ˆK−1 +p +a + ˆFp a + δa(z),0 ˆK−1 +p a +Remark 5.11. In [Kol14] a quantum symmetric pair using a generator of the form v−1ˆEp ˆK−1 +p + +ˆFp + ˆK−1 +p +is called a non-standard quantum symmetric pair. While the one that uses only +generators of the form that appeared in type C, i.e. without the extra summand ˆK−1 +p +are +called standard quantum symmetric pairs. +Acting with the generators on F1(BN) respectively F1(BN) has the same possibility of +having a single summand that is not in the subspace. +Remark 5.12. In case of FN the operators ˆBi leave the subspace F1(BN) invariant, except +for −1/2 ∈ i. For the operator ˆB1/2 the reason is again that it contains a summand f1/2 that +does not leave F1(BN) invariant. While for F +1/2 +N the operators ˆB0 and ˆB[1/2] +0 +do not leave the +subspace invariant because of the summand f0. + +24 +M. EHRIG AND K. GAN +For the relations of the operators we then get the following. +Proposition 5.13. Assume ℓ odd and I ∈ {Z/ℓZ, H/ℓZ}. The linear operators {ˆBp | p ∈ I} +and {ˆLp | p ∈ I, Θ(p) ̸= p} satisfy the relations of Bv(I, Θ) given in Definition 3.14 by +substituting the operators for the generators with the same name. +Proof. All relations in Definition 3.14 not involving the operator ˆBp for p fixed are precisely +the same as in type C and thus hold by Proposition 4.11. Note that for p fixed, the new +operator ˆBp only differs from the description in Lemma 4.9 by adding a term of the form +ˆK−1 +p . The commutator relation with all ˆLq is then obvious. The commutator relation with +“distant” ˆBq is also obvious. +Hence the only one left are the non-trivial quantum Serre +relations. There are only two that involve a fixed index, which are both a simple few line +calculation to verify. +□ +As in type C this all combines then to the following. For the case of weights in X1,+ the +statement is the complete analogue to Theorem 4.12 with precisely the same proof. +Theorem 5.14. Assume ℓ odd. There exists an action of Bv(H/ℓZ, Θ) on FN such that for +λ ∈ X1,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of +Weyl modules is obtained from +� +p∈H/ℓZ +ˆBpλ +by projecting onto the subspace F(CN) and evaluating v = 1. +Furthermore if [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then +there exists a unique p ∈ H/ℓZ such that µ and ν appear in ˆBpλ. +In case of λ ∈ X +1/2,+, the dependence of the tensor product decomposition rule on λN +makes this less clean. +Theorem 5.15. Assume ℓ odd. There exists an action of Bv(Z/ℓZ, Θ) on F +1/2 +N +such that +for λ ∈ X +1/2,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes +of Weyl modules is obtained from +� +p∈Z/ℓZ +ˆBpλ +if λN > 0 and from +� +p∈Z/ℓZ,p̸=0 +ˆBpλ + ˆB[1/2] +0 +λ +if λN = 0, by projecting onto the subspace F(CN) and evaluating v = 1. +If [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a +unique p ∈ Z/ℓZ such that µ and ν appear in ˆBpλ. +Proof. The only difference to Theorem 5.14 is that in case of λ ∈ X +1/2,+ the class of the +tensor product [∆q(λ) ⊗ ∆q(ω1)] contains a summand [∆q(λ)] if and only if λN > 0, But +ˆB0λ always contain a non-zero multiple of λ as a summand, hence one has to apply ˆB[1/2] +0 +instead in case λN = 0. +□ +Let 0 ≤ k < ℓ and N = mℓ + k for m ≥ 0. One can use the map from Definition 4.13 to +embed F1(Bmℓ+k) into F1 +k. In this case the analogue of Proposition 4.14 follows immediately. +In case of a ∈ SH,N the map from Definition 4.13 embeds F +1/2(Bmℓ+k) into F +1/2 +k . To +formulate the corresponding statement, define ˆB[z] +p = ˆBp if p ̸= 0 for z ∈ 1/2 + ℓZ. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +25 +Proposition 5.16. Let a ∈ SH,k and r ≥ 0. Then for m ≫ 0, a = λ(m) for λ ∈ X +1/2,+ +for Uq of type Bmℓ+k. Furthermore there are Laurent polynomials dλ,µ(v) with non-negative +integer coefficients for µ ∈ X +1/2,+ such that +� +(Z/ℓZ)r +ˆB[z] +p1 · · · ˆB[z] +pr λ(m) = +� +µ dominant for Uq +dλ,µ(v)µ(m), +with z = 1/2 − mℓ and +[∆q(λ) ⊗ ∆q(ω1)⊗r] = +� +µ dominant for Uq +dλ,µ(1)[∆q(µ)]. +Proof. Note that the only difference to Proposition 4.14 is that instead of ˆB0 one has to use +ˆB[z] +0 . We already know that � +(Z/ℓZ)r ˆBp1 · · · ˆBprλ(m) is contained in the shifted embedding +of F +1/2(Bmℓ+k) for m ≫ 0, hence also the summand � +(Z/ℓZ)r ˆB[z] +p1 · · · ˆB[z] +pr λ(m). +In case m ≫ 0 all weights ν such that [∆q(ν)] appears in [∆q(λ)⊗∆q(ω1)⊗s] (for 0 ≤ s < r) +have the property that νk+mℓ = 0, hence ν(1/2) = 1. Hence the [∆q(ν)⊗∆q(ω1)] never contain +the summand [∆q(ν)]. Thus ˆB[1/2] +0 +ν has to be used to get the tensor product decomposition +by Theorem 5.15. After the shift this becomes the operator ˆB[z] +0 ν. +□ +5.4. Linkage and operators for even ℓ. The first difference in case ℓ even is that ℓβ = 2 +for the roots of the form β = β± +ij. These were responsible in the odd case that all indices +had to be taken modulo ℓ to obtain the operators and that one had to group ep and fq for +p + q ∈ ℓZ. Now this is replaced by the analogue statements with ℓ/2 everywhere. Thus we +have to make the following assumption throughout the section +ℓ is even and ℓ/2 > 3. +As before this is needed to avoid quantum symmetric pairs for small ℓ/2, the general argu- +ments for linkage work the same. +Namely one obtains the two statements. +Lemma 5.17. Let λ ∈ X+ +ρ and ℓ even. Assume that µ = erλ and ν = esλ are defined for +r ̸= s. Then µ and ν are linked if and only if r ∈ s + (ℓ/2)Z. +Assume that µ′ = frλ and ν′ = fsλ are defined for r ̸= s. Then µ′ and ν′ are linked if and +only if r ∈ s + (ℓ/2)Z. +Lemma 5.18. Let λ ∈ X+ +ρ and ℓ even. Assume that µ = erλ and ν = fsλ are defined for +r + 1 ̸= s. Then µ and ν are linked if and only if r ∈ −s + (ℓ/2)Z. +The analogue of Lemma 5.6 now depend on whether ℓ/2 is itself odd or even. This makes +sense if one looks at the corresponding Dynkin diagrams in Section 2.3 for r = ℓ/2. In case +of X1,+ +ρ +the Dynkin diagram with ℓ/2 nodes always has the property that 1/2 is Θ-linked, but +it depends on the parity of ℓ/2 whether ℓ/4 is fixed in case ℓ/2 is even or ℓ−2/4 is also Θ-linked +in case ℓ/2 is odd. Similarly for X +1/2,+ +ρ +the Dynkin diagram with ℓ/2 nodes always has 0 is +fixed, but it depends on the parity of ℓ/2 whether the “opposite” side of the Dynkin diagram +has another fixed label or a pair of Θ-linked labels. Thus the analogue of Lemma 5.5 has +four possible combinations. +Lemma 5.19. Let ℓ even. +(1) Assume ℓ/2 odd, λ ∈ X1,+ +ρ +, and that µ = er−1λ and ν = frλ are defined. Then µ and +ν are linked if and only if r ∈ 1/2 + (ℓ/2)Z. +(2) Assume ℓ/2 even, λ ∈ X1,+ +ρ +, and that µ = er−1λ and ν = frλ are defined. If µ and ν +are linked then r ∈ 1/2 + (ℓ/2)Z or r ∈ ℓ+2/4 + (ℓ/2)Z. + +26 +M. EHRIG AND K. GAN +(3) Assume ℓ/2 odd, λ ∈ X +1/2,+ +ρ +, and that µ = er−1λ and ν = frλ are defined. If µ and ν +are linked then r ∈ ℓ+2/4 + (ℓ/2)Z. +(4) Assume ℓ/2 even, λ ∈ X +1/2,+ +ρ +, and that µ = er−1λ and ν = frλ are defined. Then µ +and ν are not linked. +Proof. We only sketch the proof here, since for each situation one has to go through the +same procedure as in the proof of Lemma 5.5. +For ℓ/2 odd and λ ∈ X1,+ +ρ +the proof is nearly word for word the same, hence the result is +also the same. +For ℓ/2 even and λ ∈ X1,+ +ρ +one obtains that r ∈ 1/2 + (ℓ/2)Z in case of the minimal +gallery having length one and γ1 = βi. But in the remaining cases that γ1 is different one +obtains either r ∈ 1/2 + (ℓ/2)Z if m1 + mp appearing in the proof of Lemma 5.5 is even or +r ∈ (ℓ+2)/4 + (ℓ/2)Z if m1 + mp is odd. In case that r ∈ (ℓ+2)/4 + (ℓ/2)Z one does not obtain +the converse. The argument using a reflection at a hyperplane for βi does not work here, +since ℓβi = ℓ. +In the case ℓ/2 odd and λ ∈ X +1/2,+ +ρ +the case of a minimal gallery of length 1 gives a +contradiction, while the other cases give r ∈ ℓ+2/4 + (ℓ/2)Z. Note that if ℓ/2 is odd, ℓ+2/4 is +an integer, so the r is well-defined. +Finally, the case ℓ/2 even and λ ∈ X +1/2,+ +ρ +with the assumption of linked weights always +gives a contradiction. +□ +Thus we are left with comparing λ and the image under one of the operators ei or fi. In +the case of λ ∈ X1,+ +ρ +the situation becomes quite easy. +Lemma 5.20. Let λ ∈ X1,+ +ρ +and ℓ is even. Assume that µ = erλ is defined. Then µ and λ +are not linked. Assume that ν = frλ is defined. Then ν and λ are not linked. +Proof. Fix i via µ = λ − εi and r + 1/2 = λi. Then as in the proof of Lemma 5.7 one only +has to consider the root βi. And one assumes that the weights are linked with hyperplane +Hβi,m strictly between the alcoves having to exist. Hence one obtains +(λ, β∨ +i ) = 2λi = 1 + mℓ +for some m ∈ Z. But 2λi is an even integer, while 1 + mℓ is always odd, hence this is a +contradiction. +Again the second case follows since if ν = frλ is defined, then erν = λ holds. +□ +In case of λ ∈ X +1/2,+ +ρ +, the situation resembles the case of ℓ odd. +Lemma 5.21. Let λ ∈ X +1/2,+ +ρ +and ℓ is even. Assume that µ = erλ is defined. Then µ and +λ are linked if and only if r ∈ (ℓ/2)Z. +Assume that ν = frλ is defined. Then ν and λ are linked if and only if r ∈ (ℓ/2)Z. +Proof. We proceed like in the proof of Lemma 5.20 and obtain the same equation +(λ, β∨ +i ) = 2λi = 1 + mℓ +for some m ∈ Z. But now 2λi is an odd integer, hence λ = 1/2 + mℓ/2 which then implies +r ∈ (ℓ/2)Z. +Conversely if r ∈ (ℓ/2)Z, then (λ, β∨ +i ) = 2λi = 1 + mℓ for some m ∈ Z, hence sβi,m(λ = µ +and λ and µ are linked. +As before the case for fr is done by using that erν = λ under the assumptions. +□ + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +27 +5.5. Quantum symmetric pair action for even ℓ. Throughout the section we make the +assumption that +ℓ is even and ℓ/2 > 3. +In this situation the action of the quantum symmetric pair depends on two different choices. +First whether ℓ/2 is even or odd and second whether one is looking at weights in X1,+ +ρ +or +X +1/2,+ +ρ +. We just list the definitions of the linear operators in the different situations. Note +that we obtain all possible quantum symmetric pairs for the affine type A Dynkin diagram +that we described before, i.e. we can have either two fixed labels, two Θ-linked label pairs, +or one of each and in that case we will see that they differ in the sense that one of them is +a standard quantum symmetric pair, the other one is a non-standard quantum symmetric +pair. +We are considering now Z/(ℓ/2)Z and H/(ℓ/2)Z with the usual automorphism Θ. +Definition 5.22. Assume ℓ even. For a ∈ SZ let p ∈ H/(ℓ/2)Z and for a ∈ SH let p ∈ +Z/(ℓ/2)Z. Then define for p ̸= Θ(p) +ˆBp a = v +T e−f +ℓ/2 (−p,a) � +j∈p +v +Re−f +ℓ/2 (j,a)ej a + +� +j∈−p +v +Lf−e +ℓ/2 (j,a)fj a. +ˆLp a = v +T f−e +ℓ/2 (p,a)v +T e−f +ℓ/2 (−p,a)a. +For p = Θ(p), but 0 /∈ p define +ˆBpa = v−1v +T e−f +ℓ/2 (p,a) � +j∈p +v +Re−f +ℓ/2 (j,a)eja + +� +j∈p +v +Lf−e +ℓ/2 (j,a)fja, +while for 0 ∈ i z ∈ 1/2 + (ℓ/2)Z define +ˆBpa = v−1v +T e−f +ℓ/2 (p,a) � +j∈p +v +Re−f +ℓ/2 (j,a)eja + +� +j∈p +v +Lf−e +ℓ/2 (j,a)fja + v +T e−f +ℓ/2 (p,a)a, and +ˆB[z] +0 a = v−1v +T e−f +ℓ/2 (0,a) � +j∈0 +v +Re−f +ℓ/2 (j,a)eja + +� +j∈0 +v +Lf−e +ℓ/2 (j,a)fja + δa(z),0v +T e−f +ℓ/2 (0,a)a. +This defines linear operators on F and F +1/2 that restrict to FN and F +1/2 +N +for any N. +As one can see, in case that a ∈ SZ and ℓ/2 is even there is no fixed index p ∈ H/(ℓ/2)Z, +instead there are two pairs of indices that are Θ-linked, namely ±1/2 and ℓ±2/4. +In case that a ∈ SZ and ℓ/2 is odd, the pair ±1/2 is still Θ-linked, but ℓ/4 is a fixed index. +But the quantum symmetric pair is a standard quantum symmetric pair, like in type C. +In case that a ∈ SH and ℓ/2 is even, both 0 and ℓ/4 are fixed, but the generator for 0 makes +this a non-standard quantum symmetric pair like in type B for odd ℓ. +In case that a ∈ SH and ℓ/2 is odd, we again have a non-standard operator for the fixed +index 0 and a pair of Θ-linked indices with ℓ/4. +We skip the analogue of Lemma 5.10 in this case, it is just a combination of Lemmas 4.9 +and 5.10 depending on whether a generator for a fixed index is defined as in type C or as in +type B. Thus we get the corresponding actions +Proposition 5.23. Assume ℓ even and I ∈ {Z/(ℓ/2)Z, H/(ℓ/2)Z}. +The linear operators +{ˆBp | p ∈ I} and {ˆLp | p ∈ I, Θ(p) ̸= p} satisfy the relations of Bv(I, Θ) given in Definition +3.14 by substituting the operators for the generators with the same name. +Depending on the definition of the operators this follows either from the situation in type +C or from the case ℓ odd in type B. Then the corresponding theorems for the action and +the decomposition of the tensor product can then be immediately formulated as follows. + +28 +M. EHRIG AND K. GAN +Theorem 5.24. Assume ℓ even. There exists an action of Bv(H/(ℓ/2)Z, Θ) on FN such that +for λ ∈ X1,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes +of Weyl modules is obtained from +� +p∈H/(ℓ/2)Z +ˆBpλ + λ +by projecting onto the subspace F(CN) and evaluating v = 1. +Furthermore if [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then +there exists a unique p ∈ H/ℓZ such that µ and ν appear in ˆBpλ. +Proof. This is the exact analogue of Theorem 5.14. But in case of ℓ/2 even there is no operator +that produces a multiple of λ and in case of ℓ/2 odd the linear operator for the fixed index is +standard, i.e. also does not produce a multiple of λ. But since λ ∈ X1,+, the tensor product +always includes the class of [∆q(λ)] in the Grothendieck group, hence one has to naively add +it. +□ +And the analogue of Theorem 5.15 is the following. +Theorem 5.25. Assume ℓ even. There exists an action of Bv(Z/(ℓ/2)Z, Θ) on F +1/2 +N +such +that for λ ∈ X +1/2,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the +classes of Weyl modules is obtained from +� +p∈Z/(ℓ/2)Z +ˆBpλ +if λN > 0 and from +� +p∈Z/(ℓ/2)Z,p̸=0 +ˆBpλ + ˆB[1/2] +0 +λ +if λN = 0, by projecting onto the subspace F(CN) and evaluating v = 1. +If [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a +unique p ∈ Z/ℓZ such that µ and ν appear in ˆBpλ. +Note in case of ℓ/2 even and for λ ∈ X +1/2,+, there exist the operator ˆBℓ/4 that has a fixed +index, but it is of the same form as in type C, hence does not produce a multiple λ. Hence +only ˆB0λ needs to be replaced by ˆB[1/2] +0 +λ. +We skip the discussion of an analogues of Proposition 4.14 and Proposition 5.16. The +construction is analogous to type C for the space Fk and analogous, including the same +manipulations, as in type B (with ℓ odd) for F +1/2 +k . +6. Type D and beyond +The type D case is part of the second authors Master thesis and the details will be +published separately. We give a rough summary of what happens in this situation. +In type DN (N > 3), the choices of εi and the invariant bilinear form are precisely the +same as in type B. The positive roots can be chosen as +Φ+ = {β± +i,j = εi ± εj | 1 ≤ i < j ≤ N}. +Note that all roots are short and hence equal to their own coroots and ℓβ = ℓ for all positive +roots. Integral weights have a similar structure as in type B with +X1,+ = {(λ1, . . . , λN) | λi ∈ Z, λi ≥ λi+1, and λN−1 ≥ |λN|} and +X +1/2,+ = {(λ1, . . . , λN) | λi ∈ H, λi ≥ λi+1, and λN−1 ≥ |λN|}. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +29 +With ρ = (N − 1)ε1 + (N − 2)ε2 + . . . + εN−1 ∈ X +1/2,+ we thus get +X1,+ +ρ += {(λ1, . . . , λN) | λi ∈ H, λi > λi+1, and λN−1 ≥ |λN|}, +X +1/2,+ +ρ += {(λ1, . . . , λN) | λi ∈ Z, λi > λi+1, and λN−1 ≥ |λN|}. +In analogy to type B, the combinatorial Fock space of type D decomposes into two summands +F +1/2(DN) and F1(DN). Furthermore the tensor product decomposition in type D is the same +as in type C in Proposition 4.1, especially there is no complication with a ∆v(λ) summand +as in type B. +Remark 6.1. The condition for λ to be dominant does not depend on λN itself, but rather on +its absolute value. Thus (λ1, . . . , λN−1, λN)+εN is dominant if and only if (λ1, . . . , λN−1, −λN)− +εN is dominant. In terms of sequences this can be interpreted as saying that a 1 should be +placed at position |λN| and the sign of λN needs to be recorded as well. +In case of X +1/2,+ +ρ +, following Remark 6.1 one sees that one can embed F +1/2(DN) into F +1/2,+ +N +⊕ +F +1/2,− +N +, where both spaces are isomorphic to F +1/2 +N . +For this let λ′ be equal to λ except +λ′ +N = |λN|. Then λ is mapped to (aλ, 0) if λN > 0 and to (0, aλ′) otherwise. The moving +operators from Definition 3.2 are defined on F +1/2,+ +N +⊕ F +1/2,− +N +, since they are defined on F +1/2 +N . +The interplay between moving operators and linkage, is analogous to types C. Hence the +operators ˆBp and ˆLp are given as in Definition 4.8, up to changing the index set from H/ℓZ +to Z/ℓZ, and give an action of Bv(Z/ℓZ, Θ). +In addition, one introduces an operator ˆB that goes between F +1/2,+ +N +and F +1/2,− +N +, since +(λ1, . . . , λN−1, 1/2) − εN = (λ1, . . . , λN−1, −1/2), but both are represented by the same se- +quence, but in different components. +One defines ˆB(a, 0) = (0, a) if a(1/2) = 1 and the +operator is zero otherwise, and similarly ˆB(0, a) = (a, 0) if a(1/2) = 1 and zero otherwise. It +follows that only ˆB and ˆB0 can produce linked weights, hence one needs to consider ˆB0 + ˆB. +The analogue of Theorem 5.15 holds, with the difference that ˆB0 is replaced by ˆB0 + ˆB. +The limit construction from Proposition 4.14 does not work in type D as one cannot regard +a dominant weight λ with λN < 0 as a dominant weight for a bigger rank. If one restricts +to this case then one essentially recovers the combinatorics of type C. +The case of X1,+ +ρ +brings additional complications with it, since λN = 0 does not allow for +a “good” embedding into a sum of Fock spaces F1,+ ⊕ F1,−, where these spaces are defined +as in the half-integer case. One instead embeds λ with λN = 0 diagonally. This allows for +an analogue of Theorem 4.12, but depending on the starting weight, (λ, 0), (0, λ) or (λ, λ) +represents the class of the Weyl module [∆q(λ)] and similar for the occurring Weyl modules +in the decompositions. The limit construction does not work for the same reason as above. +Remark 6.2. As a concluding remark in type D one can note that the combinatorial cal- +culations are easier than in type B and C, since it is of simply-laced type. But the action +of the quantum symmetric pair is further away from the combinatorics of the tensor product +multiplicities and the limit construction is not possible. +This kind of complication in type D has some analogy in [ES18]. Here the type D situation +is investigated in terms of category O and certain generators of the quantum symmetric pair +do not act as indecomposable translation functors, while in the analogous situation in type +B they would. +We focused in this paper on non-exceptional Lie algebras. The general ideas can of course +be transferred to exceptional cases as well. There is in general no clear analogue of the +representation ∆q(ω1), i.e. a “natural” representation. For example, in type G2 one can +use the specialization and quantization of the natural representation of so8(C), since so8(C) +contains the Lie algebra of type G2 as a fixed point Lie algebra. +But there is nothing +analogous for the other exceptional types. + +30 +M. EHRIG AND K. GAN +As for the analogue of the quantum symmetric pair, it is reasonable to expect that in +type G2 the acting algebra has some relationship to the quantum symmetric pair and an +automorphism of order 3. This is in analogy to the fact that the quantum symmetric pair +is constructed from the quantum affine algebra using an automorphism of order 2. It is a +priori not clear what the corresponding object in other exceptional cases would look like or +come from. +References +[And03] +Andersen, H. H. (2003). The strong linkage principle for quantum groups at roots of 1. J. Algebra +260:2–15. DOI: 10.1016/S0021-8693(02)00618-X +[APW91] +Andersen, H. H., Polo, P., Wen, K. X. (1991). Representations of quantum algebras. Invent. Math. +104(1):1–59. DOI: 10.1007/BF01245066 +[Ari02] +Ariki, S. (2002). Representations of quantum algebras and combinatorics of Young tableaux. Prov- +idence, RI: American Mathematical Society. +[AST18] +Andersen, H. H., Stroppel, C., Tubbenhauer, D. (2018). 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In Representations of finite +dimensional algebras and related topics in Lie theory and geometry, Volume 40 of Fields Institute +Communications Providence, RI: American Mathematical Society, pp 261–276. + +COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS +31 +M.E.: Beijing Institute of Technology, School of Mathematics and Statistics, Liangxiang +Campus of Beijing Institute of Technology, Fangshan District, 100288 Beijing, China +Email address: michael.ehrig@bit.edu.cn +K.G.: Beijing Institute of Technology, School of Mathematics and Statistics, Liangxiang +Campus of Beijing Institute of Technology, Fangshan District, 100288 Beijing, China +Email address: 3120191408@bit.edu.cn + diff --git a/YdE1T4oBgHgl3EQfcQQe/content/tmp_files/load_file.txt b/YdE1T4oBgHgl3EQfcQQe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc5296dd3b3b404b042c0496ee66f8fc7692ee83 --- /dev/null +++ b/YdE1T4oBgHgl3EQfcQQe/content/tmp_files/load_file.txt @@ -0,0 +1,1540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf,len=1539 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='03181v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='QA] 9 Jan 2023 COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS MICHAEL EHRIG AND KAIXUAN GAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A way to construct the natural representation of the quantized affine algebra Uv(ˆslℓ) is via the deformed Fock space by Misra and Miwa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This relates the classes of Weyl modules for Uq(slN) were q is a root of unity to the action of Uv(ˆslℓ) as N tends towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this paper we investigate the situation outside of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In classical types, we construct embeddings of the Grothendieck group of finite dimensional Uq(g)-modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Preliminaries 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fock spaces, affine Weyl groups, and quantum symmetric pairs 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type C 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type B 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type D and beyond 28 References 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Introduction The Fock space (of charge zero) F0 arises in mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the context of representation theory it gives a particularly nice realisation of a representation for an affine Kac-Moody algebras (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [Kac90, Chapter 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With a basis labelled by all partitions, the basis elements can naturally be interpreted as the classes of irreducible finite dimensional highest weight modules L(λ) for glN(C) with dominant polynomial highest weights when N tends towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One major step is to consider F0 = F0 ⊗Q Q(v) the Q(v)-deformation of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Here and for the rest of the paper v is an indeterminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case Misra and Miwa [MM90], following the work of Hayahsi [Hay90] defined an action of the quantized universal enveloping algebra Uv(ˆslℓ) on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The action is of a particularly nice form in the sense that the image of a partition under a Chevalley generators has, as coefficients, monomials in v that are combinatorially easy to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In turn F0 can be realized via the affine Hecke algebra, in this way it obtains the struc- ture of a KL-module depending on a parameter ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This leads to a second natural basis, the Kazhdan-Lusztig basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One now adopts the point of view that the standard basis of partitions corresponds to classes of Weyl modules ∆q(λ) for the quantum group Uq(glN) spe- cialized at a 2ℓ’s root of unity and N tending towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The Kazhdan-Lusztig basis corresponds to the classes of simple modules in the root of unity case and the transition matrix between the two basis evaluated at 1 gives the multiplicities for the modules in terms of evaluations of parabolic affine Kazhdan-Lusztig modules, see [LT00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' From a Lie theo- retic point of view the coefficients for the action of Chevalley generators can be connected 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN to Shapovalov determinants as shown in [RT10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that here and in the rest of the paper q is a root of unity such that q2 has order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The combinatorial Fock space was defined in [LRS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It is isomorphic, after extension of scalars, to the Grothendieck group of finite dimensional representations for Uq(g), where g is an arbitrary finite dimensional semi-simple complex Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This comes naturally equipped with a basis corresponding to dominant weights or via the isomorphism to the classes of Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Depending on the order ℓ, [LRS19] establishes the structure of a KL-module on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This in turn yields a Kazhdan-Lusztig type basis that corresponds to the classes of simple modules with the transition matrix being given by parabolic affine Kazhdan-Lusztig polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In type AN these Fock space models can be embedded in each other for growing N and in the direct limit give the space F0 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This paper is a step to generalize the setup in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Outside of type A, we embed the combinatorial Fock space into a traditional Fock space, this plays the role of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is done via the combinatorics of certain sequences that are a natural generalisation of Young tableaux in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We then define an action of an affine quantum symmetric pair related to Uv(ˆslℓ) on this Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Such algebras were studied and classified in the non-affine situation (see [Let02] and [Let03]) and in the Kac-Moody setting (see [Kol14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This action is compatible with the linkage principal for the quantum group at a root of unity and describes certain tensor product multiplicities for Weyl modules in a similar way as the action of the quantum affine algebra does in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Roughly said the action of generators describes, in the Grothendieck group, the image of a Weyl module under the translation functor given by taking the tensor product with the “natural” representation for a fixed rank of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In Section 2 we recall the necessary definitions for weight combinatorics for quantum groups, the definition of the combinatorial Fock space from [LRS19], and some facts about quantum groups at a root of unity and their representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In Section 3 we introduce the combinatorial definition of the analogue of F0 in the type A setting as a space of certain sequences as well as most of the combinatorics for sequences that are needed later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We briefly recall the type A situation translated into these combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We introduce some notions for affine Weyl groups and alcove combinatorics needed later on, as well as the affine quantum symmetric pair that acts outside of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In Section 4 we investigate the type C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is the easiest case with the least amount of complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We first specialize the combinatorics to this case, then relate linear operators on the Fock space to the linkage principal in type C and then define the action of the affine quantum symmetric pair (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12 we describe the relationship between the action and tensor product multiplicities and in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 the analogue of letting N tend to infinity here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Section 5 has then the same structure but for type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that here one has to distinguish between the case of ℓ even and ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The definition of the action can be found in Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relationship with tensor product multiplicities is divided into Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='24, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The situation of letting N tend towards infinity is described in and before Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='16 for ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The same results for ℓ even can be derived in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Finally in Section 6 we give a sketch of how to apply this construction to type D and what the corresponding results look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We also elaborate on why the construction is not fully detailed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We thank Daniel Tubbenhauer for comments and remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 12050410261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Preliminaries In this part we review the fundamental definitions for the three main objects involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The weight combinatorics of semi-simple Lie algebras, the combinatorial Fock space, and the quantum group at a root of unity corresponding to a non-exceptional semi-simple complex Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Throughout the paper we use an integer ℓ that is the order of a root of unity in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In general we assume ℓ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is done to avoid special cases of quantum symmetric pairs in case of small ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lie algebras and weight combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We fix g a finite dimensional semi-simple complex Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the following we are only interested in the non-exceptional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence from now on we make the assumption that g is of type AN, BN, CN or DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote this type by XN in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To not having to deal with the various isomorphisms in small ranks we assume N > 1 in type BN, N > 2 in type CN, and N > 3 in type DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' All constructions outside of Sections 4 and 5 can be made for exceptional Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We go into more details why we are restricting to the classical cases in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix a Cartan subalgebra h and a Borel subalgebra b containing h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With the choice of h we denote by Φ the root system of g and by Φ+ the positive roots with respect to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By W we denote the Weyl group corresponding to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix a non-degenerate W-invariant bilinear form (−, −) : h∗ × h∗ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For α ∈ Φ+, the corresponding coroot is defined by α∨ = 2α/(α, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We label the simple roots in Φ+ by α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , αN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The lattice of integral weights is denoted by X = {λ ∈ h∗ | (λ, α∨) ∈ Z, for α ∈ Φ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The elements ωi ∈ X such that (ωi, α∨ j ) = δij are called the fundamental weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Inside the weight lattice we fix the set of dominant weights X+ = {λ ∈ X | (λ, α∨) ≥ 0, for α ∈ Φ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' On X we have the action of the Weyl group W with the reflection sα given by sα(λ) = λ − (λ, α∨)α for α ∈ Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The combinatorial Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix the element ρ = 1 2 � α∈Φ+ α and the set of ρ- shifted dominant weights X+ ρ = X+ + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote by Q(v) the rational functions over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Following [LRS19, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1], the combinatorial Fock space F(XN) of type XN is the Q(v)-vector space with basis {λ = λ + ρ | λ ∈ X+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In [LRS19] the Fock space is originally defined to be generated as a Z[v, v−1]- module by elements indexed by X modulo relations, but [LRS19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1] shows that the elements indexed by X+ form a basis as a free Z[v, v−1]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We simple shift the indexing set of the basis elements by ρ and extend scalars to the field Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We do not make use of the description, but it should be noted that one of the main and most intricate results of [LRS19] is to endow the combinatorial Fock space F(XN) with the structure of a KL-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For this it is alternatively constructed by starting from the affine Hecke algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For this purpose an action of an affine Weyl group has to be introduced and this action depends on an integer ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus in contrast to [LRS19] we drop the label ℓ in the notation for the Fock space, as we do not make use of the description via affine Hecke algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Quantum group at a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Starting with a fixed semi-simple complex Lie algebra of type XN we define the corresponding quantized enveloping algebra, following [Lus10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For 1 ≤ i, j ≤ N we set aij = (αj, α∨ i ) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For 1 ≤ i ≤ N we fix di = 1 if αi is short root and di = 2 if αi is a long root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let vi = vdi ∈ Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is extended to all roots, by setting dα = di if α and αi are in the same Weyl group orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that in type AN and type DN all simple roots are short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the non simply-laced cases we have the Dynkin diagrams BN : and CN : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence in type BN there is precisely one short simple root αN corresponding to the right most vertex in the diagram above, while in type CN there is precisely one long simple root, again corresponding to the right-most vertex in the diagram above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , dN), with di corresponding to the i-th node from the left in the diagrams above, we have in type BN the vector (2, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , 2, 1) and in type CN the vector (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To fix the notation for the quantum integers we define for n ∈ Z≥0 and k ∈ Z the following elements in Z[v, v−1] [n]v = vn − v−n v − v−1 , [n]v!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' = n � m=1 vm − v−m v − v−1 , and � k n � v = n � m=1 vk+1−m − v−k−1+m vm − v−m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We use the same notation if we substitute v for vi or an element of a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The quantized enveloping algebra Uv(g) is the associative algebra over Q(v) generated by elements {Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' K±1 i | 1 ≤ i ≤ N} subject to the following relations for all 1 ≤ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' j ≤ N (1) KiK−1 i = 1 = K−1 i Ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' KiKj = KjKi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (2) KiEj = vaij i EjKi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' KiFj = v−aij i FjKi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (3) EiFj − FjEi = δij Ki−K−1 i vi−v−1 i (commutator relation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (4) �1−aij k=0 (−1)kE(1−aij−k) i EjE(k) i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' for i ̸= j (quantum Serre relations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (5) �1−aij k=0 (−1)kF (1−aij−k) i FjF (k) i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' for i ̸= j (quantum Serre relations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' where E(k) i = Ek i /[k]vi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' and F (k) i = F k i /[k]vi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' are the divided powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a Uv(g)-module M and λ ∈ X we fix the λ-weight space as Mλ = � m ∈ M | Kim = v(λ,α∨ i ) i m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote by Uv(g)-mod the full subcategory of finite dimensional Uv(g)-modules consisting of modules M of type 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' those finite dimensional modules satisfying M = � λ∈X Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote by [Uv(g)-mod] the Grothendieck group of Uv(g)-mod and to avoid unnecessary clutter of notations, we assume that the scalars of the Grothendieck group are extended from Z to Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Following [Lus90], we fix the subring A = Z[v, v−1] ⊂ Q(v) and denote by UA(g) the A-form of Uv(g), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' the A subalgebra of Uv(g) generated by divided powers E(k) i , F (k) i , K±1 i and the elements � Ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' c k � = k � s=1 Kivc+1−s i − K−1 i vs−1−c i vs i − v−s i for c ∈ Z, k ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that for k = 1 and c = 0 this is exactly equal to [Ei, Fi], while other elements of this form appear in generalised versions of commutator relation for divided powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 5 Fix q ∈ C with q2 is primitive ℓ-th root of unity and consider Uq = UA(g) ⊗A C, where v ∈ A acts by multiplication with q in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a Uq-module M and λ ∈ X we set Mλ = � m ∈ M | Kim = qdi(λ,α∨ i )m, � Ki;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 0 k � m = � (λ, α∨ i ) k � qdi m � We refer to Uq simply as the quantum group at a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In analogy to the generic case, we denote by Uq-mod the full subcategory of finite dimen- sional Uq-modules consisting of modules M such that M = � λ∈X Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote again by [Uq-mod] the Grothendieck group and assume that the scalars are extended from Z to Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In both cases of Uv(g)-mod and Uq-mod we have the character of a module given as ch(M) = � λ∈X dim(Mλ)eλ for formal symbols eλ and the dimension taken over the re- spective fields Q(v) and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Replacing the dimension by the rank over A, one obtains the character of a UA(g)-module that is free as an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the generic case, for λ ∈ X+, we denote by ∆v(λ) = Lv(λ) the irreducible highest weight module of highest weight λ and we fix xλ ∈ Lv(λ) a highest weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Following [Tan04, Section 7], this can be lifted to the A-form by defining ∆A(λ) to be the UA(g)- submodule of ∆v(λ) generated by xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Alternatively, [APW91] construct the module as a quotient of the Verma module defined for UA(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ∆A(λ) is a free A-module with ∆v(λ) = ∆A(λ) ⊗A Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Especially one obtains, see [APW91, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='22], ch(∆v(λ)) = ch(∆A(λ)), both of them given by Weyl’s character formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As noted in [APW91, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='25] this does not follow the usual tradition from algebraic groups to construct Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In those setting one would induce from the opposite Borel to obtain the dual Weyl module and then use the duality to obtain the Weyl module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This more involved approach has many advantages, but since we are only interested in the classes of Weyl modules in the Grothendieck group we use this simpler construction here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since ∆A(λ) is free over A, we can set ∆q(λ) = ∆A(λ) ⊗A C and naturally ch(∆q(λ)) = ch(∆A(λ)) = ch(∆v(λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (1) The module ∆q(λ) is called the Weyl module with highest weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By construction it is a quotient of the corresponding Verma module Mq(λ) of highest weight λ and ∆q(λ) has a unique irreducible quotient Lq(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that every simple object in Uq-mod can be obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since ∆q(λ) has Lq(λ) as its head and by highest weight theory all other composition factors are of the form Lq(µ) for λ − µ a positive, non-zero, sum of positive roots, the classes [∆q(λ)] for λ ∈ X+ form a basis of the Grothendieck group [Uq-mod].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The Weyl modules play the key role for our situation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The following observation is well-known to experts, but since it is central to our arguments we formulate it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ, µ ∈ X+ and consider the tensor product decomposition ∆v(λ) ⊗Q(v) ∆v(µ) ∼= r � i=1 ∆v(νi) in Uv(g)-mod for some dominant weights ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then in the Grothendieck group [Uq-mod] it holds [∆q(λ) ⊗C ∆q(µ)] = r � i=1 [∆q(νi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Consider the tensor product ∆A(λ) ⊗A ∆A(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then, by (1), it holds ch(∆q(λ) ⊗ ∆q(µ)) = ch(∆A(λ) ⊗A ∆A(µ)) = ch(∆v(λ) ⊗Q(v) ∆A(µ)) Since the classes of Weyl modules form a basis of [Uq-mod] and (1), we thus obtain that the decomposition of the character in the generic case, which is just the tensor product decomposition, gives the decomposition in the Grothendieck group in the root of unity case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ We only need the statement about the Grothendieck group from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5, but a stronger statement in the category also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With the same notations as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5, ∆q(λ)⊗C∆q(µ) has a filtration with subquotient isomorphic to Weyl modules of the form ∆q(νi) for ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , νr some dominant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The existence of a filtration with Weyl modules as subquotient can be derived from the literature in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In [AST18] (see arXiv-Appendix, Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1) this is worked out in the simply-laced case and for dual Weyl modules (hence one needs to apply a duality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In [Par94, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3] this can be found more general under the name of good filtrations, which then dualizes to a filtration by Weyl modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To determine which Weyl modules appear in the filtration one uses the same argument about characters as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to Uv(g)-mod which is a semi-simple category, Uq-mod is not semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a criterion to decide when two simple modules respectively two Weyl modules can be in the same block, one introduces an action of an affine reflection group on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Namely for α ∈ Φ+ define ℓα = ℓ/gcd(ℓ, dα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then denote by Wℓ the group generated by the affine reflections of the form sα,k · λ = sα · λ + kℓαα, for α ∈ Φ+, k ∈ Z and w · λ = w(λ + ρ) − ρ for w ∈ W the dot action of W on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that in our situation dα is either 1 or 2, hence in case that ℓ is odd ℓα = ℓ for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In which case Wℓ is the affine Weyl group attached to W, except that the action is scaled by a factor ℓ after the shift by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that ℓ is even ℓα = ℓ/2 for a long root α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case the group Wℓ is acting as the affine Weyl group for the dual root system, shifted by ρ and scaled by a factor ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call two weights λ, µ ∈ X+ linked, if there exists an element w ∈ Wℓ such that λ = w·µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This correlates to extensions between simple modules as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [And03, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3] Let λ, µ ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If Ext1(Lq(λ), Lq(µ)) ̸= 0 then λ and µ are linked and not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since ∆q(λ) for λ ∈ X+ is indecomposable, we thus get that if ∆q(λ) and ∆q(µ) are in the same block, then λ and µ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As mentioned, the Lq(λ) for λ ∈ X+ form a complete set of irreducible modules in Uq-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We fix the following identification from now on to view classes of Weyl modules as basis elements of the corresponding combinatorial Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There is a Q(v)-vector space isomorphism between F(XN) and [Uq-mod] ⊗Z Q(v), mapping the basis vector λ = λ + ρ to [∆q(λ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that the interpretation of F(XN) as a KL-module in [LRS19] allows to identify the KL-basis of F(XN) with the basis given by the classes of irreducible modules in [Uq-mod]⊗Z Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fock spaces, affine Weyl groups, and quantum symmetric pairs In this section we introduce the necessary combinatorics of sequences to define the spaces that the F(XN) are embedded into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Denote by H = 1/2 + Z the half-integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since Z acts on H by addition, we consider the cosets H/rZ for a positive integer r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For p ∈ H we denote by p its coset in H/rZ and for p ∈ Z, we denote by p its coset in Z/rZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Consider the following sets of sequences SZ = {a : Z → {0, 1} | a(i) = 0 for i ≫ 0, a(i) = 1 for i ≪ 0} and SH = {a : H → {0, 1} | a(i) = 0 for i ≫ 0, a(i) = 1 for i ≪ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call SZ the sequences supported on integers and SH the sequences supported on half- integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then we denote by F = F1 the Q(v)-vector space with basis SZ and by F 1/2 the Q(v)-vector space with basis SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We refer to F and F 1/2 simply as the Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The notation F1 is needed for type B to make the differentiation between Fock spaces clear in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' While they are defined as maps we consider these as {0, 1}-sequences with indices labelled by either Z or H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We refer to the value a(i) as the entry at position i and say that the position is “empty” if a(i) = 0 and it is occupied if a(i) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since a sequence in this set only has finitely many non-zero entry in its positive half and only finitely many zero entry in its non-positive half, the following is well defined for a ∈ SZ and equally for a ∈ SH ch(a) = � i>0 a(i) − � i≤0 (1 − a(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call this the charge of a and the set of all sequences of charge N is denoted by SZ,N respectively SH,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The corresponding subspaces, the Fock space of charge N, are spanned by sequences of charge N and denoted by FN = F1 N and F 1/2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Moving operators and counting statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We define two basic operators on Fock spaces that are moving a 1 entry to either the left or right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a ∈ SZ let i ∈ H and for a ∈ SH let i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Define a sequence b via b(i− 1/2)−b(i+ 1/2) = 1 and b(j) = a(j) for j ̸= i± 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then eia = � b if a(i + 1/2) − a(i − 1/2) = 1 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Define a sequence c via c(i + 1/2) − c(i − 1/2) = 1 and c(j) = a(j) for j ̸= i ± 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then fia = � c if a(i − 1/2) − a(i + 1/2) = 1 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' These define linear operators on both F and on F 1/2, which we call the moving operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By definition ei and fi preserve the charge and thus restrict to linear operators on FN and F 1/2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We say that ei moves an entry 1 from position i + 1/2 to position i − 1/2 or is zero if that is not possible, while fi moves an entry 1 in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the definition of the action of the quantum affine algebra in type A or the quantum symmetric pair in other types, we need a number of counting statistics, which we introduce now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' They appear in different combination for all the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a ∈ SZ let j ∈ H and for a ∈ SZ let j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then define Re r(j, a) = #{k ∈ j + rZ>0 | eka ̸= 0}, Rf r (j, a) = #{k ∈ j + rZ>0 | fka ̸= 0}, Le r(j, a) = #{k ∈ j − rZ>0 | eka ̸= 0}, Lf r(j, a) = #{k ∈ j − rZ>0 | fka ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To shorten notation we also introduce Re−f r (j, a) = Re r(j, a) − Rf r (j, a), Rf−e r (j, a) = Rf r (j, a) − Re r(j, a), Le−f r (j, a) = Le r(j, a) − Lf r(j, a), Lf−e r (j, a) = Lf r(j, a) − Le r(j, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In addition, for a ∈ SZ let i ∈ H/rZ and for a ∈ SZ let i ∈ Z/rZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then define T e r (i, a) = #{j ∈ i | eja ̸= 0} and , T f r (i, a) = #{j ∈ i | fja ̸= 0}, T e−f r (i, a) = T e r (i, a) − T f r (i, a), T f−e r (i, a) = T f r (i − a) − T e r (i, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The cumbersome notation is necessary, since the action of the generators in the different cases depend on both the sequence a as well as a position j such that an operator ej re- spectively fj can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In most cases the index r is r = ℓ, but in case of type B and even ℓ it has to replaced by r = ℓ/2 for the action, hence the addition of the index r in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix a sequence a in either SZ or SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then Re r(j, a) counts how often, for k ∈ j + rZ>0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' to the right of j), it happens that a(k − 1/2) = 0 and a(k + 1/2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Similarly Rf r (j, a) counts when a(k − 1/2) = 1 and a(k + 1/2) = 0 instead and Le r(j, a) and Lf r(j, a) count these occurrences for k ∈ j − rZ>0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' to the left of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' While the statistics of the form T e r (i, a) count where such situation occur on all of Z respectively H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This should give an easy way to remember which of the operators counts what and in which direction of a fixed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We briefly recall here the definition of the action on the Fock space in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Instead of defining the Fock space via partitions as usual, we use sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix X = ⊕N i=1Zεi the a lattice of integral weights for glN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One could also use slN(C) here, but the general linear case is more convenient from the view point of combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In X we have X+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) ∈ X | λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' ≥ λN} and P + = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) ∈ X+ | λN ≥ 0}, the sets of dominant weights and polynomial weights of glN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One should consider P + as the analogue of the dominant weights for slN(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore, fix the element ρ = (0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , −(N − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to slN(C) we have a choice here and we fix this particular ρ to match up nicely with the usual definition of partitions and adding boxes of certain residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To λ ∈ P + we associate the sequence aλ ∈ SZ given by aλ(i) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 if i ≤ −N, 1 if there exists 1 ≤ j ≤ N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' i = λj, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' we put a 1 at all positions that appear as entries in λ, since we added λ = λ + ρ, these are all distinct, so this is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In addition we set all values to 1 for i ≤ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that for the zero weight 0 = (0, −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , −(N −1)), hence the sequence associated to the zero weight has value 1 for all non-positive integers and value 0 for all positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that this embeds F(AN) into the Fock space F0 of charge zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 9 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To pass to the partition description of F0, see for example [Ari02] or [RT10], fix a sequence a of charge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=') be defined such that a(λ1) is the right most entry equal to 1, a(λ2 − 1) is the second right most entry equal to 1 and so forth with a(λi − (i − 1)) being the i-th right most entry equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then by construction (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=') is a weakly decreasing sequence of non-negative integers that eventually stabilizes to 0 after finitely many steps, hence a partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For p ∈ H/ℓZ and a ∈ SZ, define linear operators on F via ˆEp a = � j∈p vRe−f ℓ (j,a)ej a, ˆFp a = � j∈p vLf−e ℓ (j,a)fj a, and ˆKp a = vT f−e ℓ (p,a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Restricting these operators to the subspace F0 gives then the well-known action on F0, see for example [Ari02], by using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 to translate it to sequences instead of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [Ari02, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6] The linear operators {ˆEp, ˆFp, ˆK±1 p } define an action of the quantum affine algebra Uv(ˆslℓ)′ on F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that there is a natural isomorphism between Ψm : F0 → Fm for any integer m, by just shifting the sequence by m steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Ψm(a)(i) = a(i − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By construction Ψm(ˆEpa) = ˆEp+mΨm(a), Ψm(ˆFpa) = ˆFp+mΨm(a), and Ψm(ˆKpa) = ˆKp+mΨm(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since all relations of the quantum affine algebra are rotation invariant, this defines an action of Uv(ˆslℓ)′ on all Fm, m ∈ Z, given by the same formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The only difference really being for which index p, ˆFp does not act as zero on the “highest weight sequence” where all 1’s are as far to the left as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' the sequence on which all ˆEp′ act as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Affine Weyl group combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To introduce alcove geometry for the affine Weyl group we define XR = X ⊗Z R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then for α ∈ Φ+ we consider the affine hyperplane Hα,k = {x ∈ XR | (λ + ρ, α∨) = kℓα}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The affine reflection at this affine hyperplane is precisely sα,k and the reflections at all such hyperplanes give the action of Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote the set of all such affine hyperplanes by H and conversely for a hyperplane H ∈ H we denote by sH the corresponding affine reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The statements and definitions in this section can be found in most textbooks on affine reflection groups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [Hum90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Consider the complement of all affine hyperplanes in H Xreg R = XR \\ � H∈H H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A connected component of Xreg R is called an open alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The closure of a connected compo- nent in XR is called a (closed) alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We denote the set of all (closed) alcoves by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that Wℓ acts simply transitively on the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Points in an open alcove have trivial stabilizer, while points in the boundary of an open alcove have non-trivial stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Each affine hyperplane H ∈ H defines two closed halfspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a fixed alcove A ∈ A we denote by H+ A the half space that contains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore for two alcoves A and A′ we call a hyperplane H ∈ H between A and A′ if H+ A ̸= H+ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This give rise to the definition of a distance function d(A, A′) = #{H ∈ H | H between A and A′}, for A, A′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let H ∈ H be a hyperplane between alcoves A, A′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then d(A, A′) > d(sHA, A′) and d(A, A′) > d(A, sHA′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN For an alcove A ∈ A consider the set of hyperplanes HA that intersect A in maximal dimensions, these are called the walls of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then SA = {sH | H ∈ HA} is a generating set of Wℓ as a Coxeter group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With respect to these generators we can see the distance as a choice free substitute for the length function l with respect to SA, namely let A′ be another alcove then d(A, A′) = l(w) for wA = A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With the equality of distance and length, it follows that for a fixed alcove A and H ∈ HA, the hyperplane H is the only hyperplane between A and sHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This leads to the combinatorics of (minimal) galleries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Two alcoves A, A′ ∈ A are called adjacent if A′ = sHA for some H ∈ HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A sequence of alcove Γ = (A0, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , Ar) such that Ai is adjacent to Ai+1 is called an (alcove) gallery from A0 to Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A gallery Γ = (A0, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , Ar) such that r = d(A0, Ar) is called a minimal gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that the set of walls {Hi | Hi between Ai−1 and Ai} is precisely the set of hyperplanes between A0 and Ar in case of a minimal gallery Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call these hyperplanes the hyperplanes that are crossed by the gallery Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since weights can be contained in a hyperplane, we need a more restrictive notion of hyperplanes between two alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For λ ∈ X denote by Aλ an alcove that contains λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For λ, µ ∈ X and choices of alcoves Aλ and Aµ, we call a hyperplane H ∈ H strictly between Aλ and Aµ if H is between Aλ and Aµ and H ∩ {λ, µ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11, the choice of an alcove Aλ is unique if and only if λ is in the interior of an alcove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Otherwise multiple choices are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We make frequent use of the following lemma to make a particularly good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ, µ ∈ X such that λ ∈ Wℓµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then there are choices of alcoves Aλ and Aµ such that a minimal gallery Γ from Aλ to Aµ only crosses hyperplanes that are strictly between Aλ and Aµ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let Aλ and Aµ be any choice for alcoves with λ ∈ Aλ and µ ∈ Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let Γ be a minimal gallery and assume that there exists a hyperplane H that is between Aλ and Aµ but not strictly between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Without loss of generality assume λ ∈ H (otherwise just rename), hence λ ∈ sHAλ and so sHAλ is also a choice for an alcove containing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8 it holds d(Aλ, Aµ) > d(sHAλ, Aµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus we can replace Aλ by sHAλ and the distance between Aµ and the new Aλ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' the length of a minimal gallery, strictly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since the distance is bounded below by 0 this process has to end after finitely many steps and at that point all hyperplanes crossed by a minimal gallery are strictly between the final choices of Aλ and Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Affine quantum symmetric pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For this section fix r > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To define the necessary quantum symmetric pairs we consider sets of the form Z/rZ respectively H/rZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The restric- tion of r > 3 is so to not consider special cases for small r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In general one can define similar algebras also for r = 2 and r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relations change slightly in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 11 We consider the Dynkin diagram of type ˆAr−1 with indices either labeled by entries in Z/rZ or H/rZ 0 1 r − 1 (r−2)/2 −(r+2)/2 r/2 Z/rZ for r even 0 1 r − 1 (r−3)/2 (r+3)/2 (r−1)/2 (r+1)/2 Z/rZ for r odd 1/2 r − 1/2 3/2 r − 3/2 (r−3)/2 (r+3)/2 (r−1)/2 (r+1)/2 H/rZ for r even 1/2 r − 1/2 3/2 r − 3/2 (r−2)/2 (r+2)/2 r/2 H/rZ for r odd For both index sets we consider the automorphism Θ given by Θ(p) = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus in each of the pictured Dynkin diagrams, this is the horizontal reflection along the dotted horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Depending on whether r is even or odd, and whether considering the index set H/rZ or Z/rZ, Θ have between zero and two fixed points on the left or right of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In either case of Z/rZ and H/rZ we call cosets p and q linked if p = q ± 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' if the corresponding nodes in the affine Dynkin diagram are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To simplify the notations for the quantum symmetric pair we rewrite one type of quantum Serre relation in form of a non-commutative polynomial SRv(x, y) = x2y − [2]vxyx + yx2, for non-commuting variables x, y over Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to the usual quantum group Uv(ˆslr) the quantum Serre relations for the generators of the quantum symmetric pair depend on the type of the node in the Dynkin diagram of the generator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' they are not invariant under translation of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A p ∈ Z/rZ respectively p ∈ H/rZ is called (1) a fixed index if Θ(p) = p, (2) a Θ-linked index if Θ(p) is linked to p, and (3) a standard index if p is neither fixed not Θ-linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The affine quantum symmetric pair is then the following associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [Kol14] Let I ∈ {Z/rZ, H/rZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The affine quantum symmetric pair Bv(I, Θ) is defined to be the associative algebra over Q(v) generated by {ˆBp | p ∈ I} and {ˆLq | q ∈ I, q ̸= Θ(q)}, subject to the following relations: For p, q not fixed it holds ˆLpˆLq = ˆLqˆLp, ˆLpˆLΘ(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For p standard, r Θ-linked, s fixed, and q not fixed it holds ˆLq ˆBp = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 v2 ˆBpˆLq if p = q v−2 ˆBpˆLq if p = Θ(q), v−1 ˆBpˆLq if p, q linked, vˆBpˆLq if p, Θ(q) linked, ˆBqˆLq otherwise, ˆLq ˆBr = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 v3 ˆBrˆLq if r = q, v−3 ˆBrˆLq if r = Θ(q), v−1 ˆBrˆLq if r, q linked, q ̸= Θ(r) vˆBrˆLq if r, Θ(q) linked, q ̸= r ˆBrˆLq otherwise, 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN ˆLq ˆBs = ˆBsˆLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The generators ˆBp and ˆBq commute, unless ˆBp ˆBΘ(p) − ˆBΘ(p)ˆBp = ˆLp − ˆLΘ(p) v − v−1 for p standard or SRv(ˆBp, ˆBq) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0 p, q linked, not fixed, and Θ(p) ̸= q, 0 p, q linked, p standard, q fixed, ˆBq p, q linked, p fixed, q standard, −[2]v ˆBp(vˆLp + v−2ˆLΘ(p)) p, q Θ-linked , q = Θ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Deriving these definitions from [Kol14] needs some clarifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relations between the ˆBp are given in [Kol14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4], note that in our language all the ci that appear in [Kol14] are equal to 1 and the elements Zj = ˆLj with j an index in the Dynkin diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The commutator relations between ˆBp and ˆLq are found in [Kol14, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It is easier to think of Bv(I, Θ) as being an affine analogue of the quantum symmetric pair of type AIII in [Let03, Section 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relations for the quantum symmetric pair of type AIII are nearly the same as for the ordinary quantum group, as in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There is either one special generator, corresponding to a fixed index, or two special generators, corresponding to a pair of Θ-fixed indices that do not behave like usual quantum group generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to the non-affine case, there are now two such situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the Dynkin diagrams above these are the two areas where the dotted line intersects the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Comparing this definition to the ones for non-affine quantum symmet- ric pairs in [ES18, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='17 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='18] (or the idempotent version in [BSWW18] in case there are no Θ-linked indices), we see that locally the relations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The pairs ˇEi and ˇFi in the relations of [ES18] are the analogues of ˆBp and ˆBΘ(p) for p standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since our generators are ordered in a cyclic way it is not reasonable to choose a “positive” and “negative” one in such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The generator ˇB in [ES18, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='18] is the analogue of ˆBp for a fixed index and the pair of generators ˇB+ and ˇB− in [ES18, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='17] are the analogue of ˆBp and ˆBΘ(p) for p being Θ-linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to [ES18] we use the analogue of a semi-simple Cartan, while [ES18] uses the analogue of a reductive Cartan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is just for simplicity of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One could easily modify the definition and write every generator ˆBq as a product of two generators in the vein of the definitions of [ES18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The only place where the relations differ is the last of the deformed quantum Serre relations in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14, with p being Θ-linked and q = Θ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is due to the fact that the linear operators B1/2 and B−1/2 in [ES18, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8] that give the action of ˇB+ and ˇB− are not symmetric in their definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This was needed to match the grading of graded category O in [ES18], but this is not necessary in our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type C For type CN (N > 2) we choose X = �N i=1 Zεi, where the εi are the projection onto the i-th diagonal entries for the Cartan subalgebra of diagonal matrices inside sp2n(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The W-invariant bilinear form (−, −) is given such that the εi form an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As the positive roots in a root system Φ in X, we choose: Φ+ = {β± i,j = εi ± εj | 1 ≤ i < j ≤ N} ∪ {βi = 2εi | 1 ≤ i ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The simple roots for this choice are αi = β− i,i+1 for 1 ≤ i < N and αN = βN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The corresponding coroots in X are then (β± i,j)∨ = β± i,j and β∨ i = εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 13 As defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3 the elements di = 1 for 1 ≤ i < N and dN = 2, since αN is the only simple long root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is extended to all positive roots by having dβ = 2 if β = βi for some i and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' With these choices the fundamental weights are given as ωi = ε1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' And thus the dominant integral weights are X+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi ≥ λi+1, and λN ≥ 0}, where we write row vectors with respect to the basis {εi}1≤i≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ρ = N � i=1 ωi = Nε1 + (N − 1)ε2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + εN ∈ X+ and thus X+ ρ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi > λi+1, and λN > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (Recall the convention that λ = λ + ρ for λ ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=') Since there are roots with dβ ̸= 1 we have to distinguish the case of ℓ odd and ℓ even as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case ℓ odd, Wℓ is the affine Weyl group of type CN, just scaled by the factor ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' While in case ℓ even, Wℓ is the affine Weyl group of type BN scaled by a factor ℓ for the dual root system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' where βi is replaced by εi with coroot 2εi instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the action on [Uq-mod] we consider the functor −⊗C∆q(ω1), which is taking the tensor product with the specialization of the quantum analogue of the natural representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since the natural representation is minuscule one obtains the following tensor product decomposition in the generic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [HK02, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3] Let λ ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then in Uv(g)-mod it holds ∆v(λ) ⊗ ∆v(ω1) ∼= � i:λ+εi∈X+ ∆v(λ + εi) ⊕ � i:λ−εi∈X+ ∆v(λ − εi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case the tensor product decomposition is straight forward: If λ + εi is dominant the corresponding irreducible module appears and similarly for λ − εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fock space of sequences and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To embed F(CN) into FN, we map λ to the sequence aλ such that aλ(i) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 if i ≤ 0, 1 if there exists 1 ≤ j ≤ N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' i = λj, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that by construction all non-positive entries in the sequence a are 1 and there are exactly N entries equal to 1 in the strictly positive part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the sequence aλ has charge N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The map is obviously injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By abuse of notation we simply write λ for the sequence as well and identify F(CN) with its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Looking at the linear operators from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2, we immediately see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ F(CN) and r ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then erλ ∈ F(CN) and furthermore erλ is non-zero if and only if λ − εi is dominant for r = λi − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Similarly, frλ ∈ F(CN), for r ̸= 1/2, and λ + εi is dominant if and only if frλ is non-zero for r = λi + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that all operators er preserve the subspace F(CN) with er acting as zero for all r ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' On the other hand the operators fr keep the subspace invariant, except for f1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The obvious reason being that f1/2 can move a 1 from position 0 to position 1, leaving the subspace F(CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Linkage and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We are now considering when two weights obtained from λ by applying two of the operators from above are linked under the assumption that the sequences stay inside F(CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the following we often just say: assume µ = erλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By this we mean that erλ is again a sequence coming from the embedding of F(CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Even though the roots of type C are different from type A, the following Lemma has pretty much the same proof as in type A, as the roots of the form β− ij are the only ones playing a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume µ = erλ and ν = esλ, for r ̸= s are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume µ = frλ and ν = fsλ, for r ̸= s, are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It holds µ = λ − εi where i is determined by λi = r + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Similarly es defines j such that λj = s + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since s ̸= r we can assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume now that the two weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12, fix alcoves Aµ and Aν such that a minimal gallery Aµ = A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , At = Aν only crosses walls that are strictly between Aµ and Aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It holds (ν − µ, β∨) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2 if β = β− i,j, 1 if β ∈ {β− i,k, β+ i,k, β+ k,i | k ̸= j} ∪ {βi}, −1 if β ∈ {β− k,i | k < i}, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since we assume that µ ̸= ν, but µ and ν are linked, the minimal gallery needs to have at least length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus there exists a hyperplane strictly between Aλ and Aµ, otherwise the two alcoves agree which is a contradiction to the simply transitive action of Wℓ on the set of alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Looking at the values (ν − µ, β∨) above, the only root that can have an affine hyperplane strictly between Aµ and Aν is β = β− ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the length of the minimal gallery is exactly 1 and it holds ν = sβ− i,j,mµ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let H = Hβ− ij,m be this hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence 1 + mℓ = (µ, (β− i,j)∨) = λi − λj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' So we obtain that λi = λj + mℓ and so r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Now assume that r ∈ s+ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then λi−λj = mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (ν, (β− ij)∨) = 1+mℓ and thus sβ− ij,m(ν) = µ and so the weights µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The statement for the f-operators is done completely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4 can be used word for word to see that in type A linked weights are obtained by operators with indices congruent modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume µ = erλ and ν = fsλ, for r + 1 ̸= s, are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ −s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It holds µ = λ − εi where i is determined by λi = r + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Similarly fs defines j such that λj = s − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since s ̸= r + 1 it holds that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that the weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix Aµ and Aν via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12 such that the minimal gallery only crosses hyperplanes that are strictly between the two alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then we check that the only possibility for |(µ − ν, β∨)| > 1 is (µ − ν, β∨) = 2 for the choice β = β+ i,j (Technically β+ i,j is only defined for i < j, so for i > j we just set β+ i,j = β+ j,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 15 Thus the unique affine hyperplane strictly between Aµ and Aν is of the form Hβ+ i,j,m and so it holds ν = sβ+ i,j,mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In formulas for the weights we thus get 1 + mℓ = (µ, (β+ i,j)∨) = λi + λj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' So we obtain λi = −λj + mℓ and thus so r ∈ −s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that r ∈ −s + ℓZ, then λi + λj = mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (ν, (β+ ij)∨) = 1 + mℓ and thus sβ+ ij,m(ν) = µ and so the weights µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ This is the main difference to type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relationship of the form r ∈ −s + ℓZ forces us to fix a type of origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the embedding of F(CN) depends on N in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let ℓ be odd and λ ∈ X+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case there exists i such that µ = λ − εi and ν = λ + εi for i determined by λi = r − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that the weights are linked and fix alcoves Aµ and Aν such that a minimal gallery only crosses hyperplanes strictly between the alcoves, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Checking the values of (ν − µ, β∨), there are multiple choices for hyperplanes that can be strictly between the chosen alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Namely we have (ν − µ, β∨) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2 if β = βi, 2 if β ∈ {β− i,j, β+ i,j, β+ j,i | j ̸= i}, −2 if β ∈ {β− j,i | j ̸= i}, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Denote by H1 = Hγ1,m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , Ht = Hγt,mt be the hyperplanes crossed by the minimal gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By construction each Hi is strictly between Aµ and Aν, hence for each γi it holds (µ−ν, γ∨ i ) = ±2 and especially (λ, γ∨ i ) = miℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' λ ∈ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We now go through the different cases for γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = βi: In this case it holds that 1 + m1ℓ = (ν, β∨ i ) = λi + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus λi ∈ ℓZ and so r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since in this case it immediately holds sβi,m1ν = µ it follows that sβi,m1Aν = Aµ and so the minimal gallery had length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = β− i,j for some j > i: In this case the equation is 1 + m1ℓ = (ν, (β− i,j)∨) = λi − λj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus λi = λj + m1ℓ and it holds sβ− i,j,m1ν = λ + εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For every hyperplane Hq for q > 1 and γq ̸= β+ i,j it holds mqℓ = (λ, γ∨ p ) = (λ + εj, γ∨ p ), hence λ + εj ∈ Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus applying sγq,mq leaves λ + εj invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since λ + εj ̸= µ, there must exists p > 1 such that γp = β+ i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In which case 1 + mpℓ = (λ + εi, (β+ i,j)∨) = (λ + εj, (β+ i,j)∨) = λi + λj + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence λi = −λj + mpℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus 2λi = (mp + m1)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since 2λi is an even integer and ℓ is odd, (mp + m1) is even and so λi ∈ ℓZ and equivalently r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = β+ i,j for some j > i: This is nearly identical to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The only difference is that one first obtains λi ∈ −λj + ℓZ, applying the first reflection gives λ − εj, and the second used hyperplane is for β− i,j and one obtains λi ∈ λj + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The rest of the argument is then the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = β− j,i for some j < i: In this case we start with −1 + m1ℓ = (ν, (β− j,i)∨) = λj − λi − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Hence λi ∈ λj + ℓZ and sγ1,m1(ν) = λ + εj and the rest is as in the cases before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = β+ j,i for some j < i: This similarly follows as the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the converse, assume r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then λi = mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (ν, (βi)∨) = 1 + mℓ and thus sβi,m(ν) = µ and so the weights µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ For the case of ℓ odd Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6, gives a special case of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 with a slightly more delicate proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This special case will lead to generators for the quantum symmetric pair of non-standard indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The special case for ℓ even is handled in the next statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let ℓ be even and λ ∈ X+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ 1/2 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The arguments in the proof follow the ones for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6 so we only point out the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = βi: In contrast to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6 we now obtain that λi ∈ (ℓ/2)Z and so r ∈ 1/2 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is of course due to ℓβi = ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In all other cases we obtain 2λi ∈ ℓZ as before, but now we can simply divide ℓ by 2 and obtain λi ∈ (ℓ/2)Z, which in turn implies r ∈ 1/2 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the converse, assume r ∈ 1/2 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then λi = mℓ/2 for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (ν, (βi)∨) = 1 + mℓ/2 and thus sβi,m(ν) = µ since ℓβi = ℓ/2 and so the weights µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ In contrast to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6, in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7 we see that for ℓ even the situation that er−1 and fr produce linked weights happens for two types of positions, for a 1 at a position in ℓZ or at a position in ℓ/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is reflected in the existence of two Θ-linked pairs of indices in the Dynkin diagram for H/ℓH and ℓ even in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Quantum symmetric pair action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We now define operators on FN as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Recall the automorphism Θ : H/ℓZ → H/ℓZ from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4 that changes the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let a ∈ SZ and p ∈ H/ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For Θ(p) ̸= p we define ˆBp a = vT e−f ℓ (Θ(p),a) � j∈p vRe−f ℓ (j,a)ej a + � j∈−p vLf−e ℓ (j,a)fj a and ˆLpa = vT f−e ℓ (p,a)vT e−f ℓ (Θ(p),a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For Θ(p) = p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' p = ℓ/2, we define ˆBℓ/2a = v−1vT e−f ℓ (ℓ/2,a) � j∈ℓ/2 vRe−f ℓ (j,a)eja + � j∈ℓ/2 vLf−e ℓ (j,a)fja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This defines linear operators on F that restrict to FN for any N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As mentioned at the very end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2, the action of Uv(ˆslℓ)′ is defined on any FN with the same definitions as for F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus by comparing we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let a ∈ SZ and p ∈ H/ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For Θ(p) ̸= p ˆBp a = ˆEp ˆK−1 −p a + ˆF−p a and ˆLpa = ˆKp ˆK−1 −pa and ˆBℓ/2 a = v−1ˆEℓ/2 ˆK−1 ℓ/2 a + ˆFℓ/2 a The definition of the linear operators ˆBp is the reason why we do not embed into the Fock space of charge 0 (as one does in type A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The linear operator ˆBp involves the ei’s for i ∈ p, but also fj’s for −j ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus the definition is not invariant under arbitrary translation as the linear operators for the type A action, only under translation by multiples of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 17 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that for λ ∈ F(CN) it holds ˆBpλ ∈ F(CN) if −1/2 /∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For ˆBℓ/2 one just needs to consider the summand using f1/2 that can produce a sequence that is not contained in F(CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is exactly parallel to the situation in type AN−1 where one can apply an operator ˆFp, that in the language of Young diagrams, creates a box in row N + 1, thus leaving the span of polynomial weights for glN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the relations of the operators we then get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The linear operators {ˆBp | p ∈ H/ℓZ} and {ˆLp | p ∈ H/ℓZ, Θ(p) ̸= p} satisfy the relations given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 by substituting the operators for the generators with the same name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since we did not specify how to make all the choices in [Kol14], we give a short sketch of how to quickly check that the relations are indeed satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The relations between the ˆLp are immediate by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' They all multiply a basis element with a fixed scalar and the scalars for ˆLp and ˆL−p are inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The commuting relations between ˆLq and ˆBp are a straight-forward calculation using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' That ˆBp and ˆBq commute unless for the specified index choices is also immediate from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Either the summands of ˆBp and ˆBq modify a basis vector at positions that are not neighboured, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' the cosets are not linked and so the summands commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Or, in case that Θ(p) and q are linked, their summands can modify the same position by moving a 1 into different directions or moving two 1’s into the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case the product of the summands is just always zero, hence they commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The commutator relation between ˆBp and ˆBΘ(p) is a direct and simple computation using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Finally the deformed and non-deformed Serre relations follow from [ES18] (or precisely their use of [Let03]), since for a linked pair of indices p and q the Serre relation are indepen- dent of the rest of the Dynkin diagram, the calculation only involves the relation between the standard generators of the quantized enveloping algebra for the indices p, q, Θ(p), and Θ(q), which are local and hence the same in the affine case and in [ES18], since ℓ > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ Thus putting everything together we obtain the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There exists an action of Bv(H/ℓZ, Θ) on FN such that for λ ∈ X+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules is obtained from � p∈H/ℓZ ˆBpλ by projecting onto the subspace F(CN) and evaluating v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore if [∆q(µ)] and [∆q(ν)] appear in the decomposition with µ and ν linked, then there exists a unique p ∈ H/ℓZ such that µ and ν appear in ˆBpλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The action is the one from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' That the decomposition is given by the sum of all operators is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1 together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 to obtain the translation of the decomposition into weight combinatorics and then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2 together with the definition of the operators themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' That classes of two Weyl modules with linked weights are obtained from a unique operator ˆBp is then Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ We want to address now the question of embedding into a single Fock space for different ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In type A, every F(AN) can be embedded into F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This was natural on a combinatorial level due to the choice of gl instead of sl and hence the ability to choose the element ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN choice was made such that the weight 0 always gets mapped to the sequence with all 1’s in the non-positive half and 0’s in the positive half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' From the point of view of the affine operators and the action, any shift of the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' an embedding into a different Fk makes no difference, since the operators only involves entries that are congruent mod ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence an embedding into a different Fk is just related to a shift in the used operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' On the Lie theory side this would just correspond to a different choice of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We can make a similar construction in type C if we restrict to certain N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Although this is more artificial since on the Lie theory side, the element ρ is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let N = mℓ + k for m ≥ 0 and 0 ≤ k < ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then we define a map from FN to Fk via a(m)(i) = a(i + mℓ) for a ∈ SZ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call a(m) the sequence shifted by m ℓ-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this way we can view F(CN) as a subspace of Fk, which we call the shifted embedding of F(CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' That the map in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13 is well-defined follows immediately with a simple cal- culation for the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since it intertwines the action of ei and ei−mℓ (and similar of fi and fi−mℓ), the affine operators ˆBp and ˆLp defined on FN and Fk as restrictions from F commute with the map of shifting a sequence by m steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In terms of weights this is the same as looking at the weight λ + ρ − mℓ(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , 1) which for most m is not dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus using this embedding for a fixed 0 ≤ k < ℓ we can formulate this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let a ∈ SZ,k and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then for m ≫ 0, a = λ(m) for λ a dominant weight for Uq of type Cmℓ+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore there are Laurent polynomials dλ,µ(v) with non- negative integer coefficients for µ dominant for Uq such that � (H/ℓZ)r ˆBp1 · · · ˆBprλ(m) = � µ dominant for Uq dλ,µ(v)µ(m), and [∆q(λ) ⊗ ∆q(ω1)⊗r] = � µ dominant for Uq dλ,µ(1)[∆q(µ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a we consider the left most 0 in the non-positive part of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If this zero is at position −n′ then using operators of the form ei this can be reduced to a sequence b that has no 0’s in the non-positive part and exactly k 1’s at the positions 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then b is the image of 0 under the shifted embedding of F(Cm′ℓ+k) for m′ℓ > n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then by applying operators fi in the reverse order to the sequence of operators ei before one obtains a dominant weight λ with a = λ(m′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If there was no 0 in the non-positive part then one can just use m′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' A dominant weight λ for Uq of rank m′ℓ + k can be viewed as a dominant weight for Uq of rank mℓ + k for m > m′, by just filling it up with 0 at the last (m − m′)ℓ entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Choose m > m′ such that mℓ > n′ + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' By our discussion about which fi can leave the embedded subspace FN, after Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2, we see that no product ˆBp1 · · · ˆBpr applied to a can create a weight that is not in the shifted embedding of F(Cmℓ+k), since at most the 1’s at the position −n′ − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , −n′ − r can be moved to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the statement then follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ Note that the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13 gives a clear bound for what m needs to be, it is just necessary that mℓ > n′ + r for n′ the position of the left most entry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that the definition of the operators ˆBp is not uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As already mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15, one can change some coefficients in the definition of the generators and obtain a slightly modified algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 19 Since the definition in type A is not unique as well, compare for example the definition in [RT10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1] and in [Ari02, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1], one can make similar modifications in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type B In type BN (N > 1), we choose εi ∈ h∗ the projection onto the i-th diagonal entries for the Cartan subalgebra of diagonal matrices inside so2n+1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The W-invariant bilinear form (−, −) is given such that the εi form an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the positive roots we choose Φ+ = {β± i,j = εi ± εj | 1 ≤ i < j ≤ N} ∪ {βi = εi | 1 ≤ i ≤ N} and similar to type C, the simple roots are αi = β− i,i+1 for 1 ≤ i < N, but here αN = βN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The corresponding coroots are (β± i,j)∨ = β± i,j and β∨ i = 2εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Following Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3 we have di = 2 for 1 ≤ i < N and dN = 1, since αN is the only simple short root in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This gets extended to all positive roots via dβ = 2 for β = βi for some i and dβ = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The noticeable change is in the integral weight lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The fundamental weights are ωi = ε1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + εi for i < N and ωN = 1/2(ε1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + εN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence for the integral weights X = �N i+1 Zωi, the dominant weights X+ that can be naturally divided into two subsets X 1/2,+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi ≥ λi+1, and λN ≥ 0} and X1,+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ H, λi ≥ λi+1, and λN ≥ 0}, written as row vectors with respect to {εi}1≤i≤N inside h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We call X 1/2,+ the integer weights, not to be confused with the integral weights, and X1,+ the half-integer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Summing up the fundamental weights we get ρ = N � i=1 ωi = (N − 1/2)ε1 + (N − 3/2)ε2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + 1/2εN ∈ X 1/2,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus adding ρ we obtain the sets to define sequences, namely X 1/2,+ ρ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ H, λi > λi+1, and λN > 0}, X1,+ ρ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi > λi+1, and λN > 0}, again recalling that λ = λ + ρ for λ ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This makes clear our convention of naming the set of integer weights X 1/2,+ and the set of half-integer weights X1,+ (and not the other way around).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' After adding ρ, elements in X 1/2,+ ρ have entries in H and elements of X1,+ ρ have entries in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus the naming convention makes it easy to recognise what the domains for the sequences in the different cases are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Again the parity of ℓ is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For ℓ odd, Wℓ is the affine Weyl group of type BN, scaled by the factor ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case ℓ even, Wℓ is the affine Weyl group of type CN scaled by a factor ℓ for the dual root system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' where one would replace β± ij in the root system by 1/2β± ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the action on Uq-mod we use the functor − ⊗C ∆q(ω1), again the tensor product with the the specialization of the quantum analogue of the natural representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that in contrast to the case of Uv(g) of type A or C, ∆v(ω1) is not a tensor generator of Uv(g)-mod, hence there are other possible finite dimensional represen- tations, like the specialization of the spin representation ∆q(ωN), that can give interesting actions on Uq-mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In type B the natural representation is not minuscule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the tensor product decom- position has a slight complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' [HK02, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3] For λ ∈ X+ it holds in Uv(g)-mod ∆v(λ) ⊗ ∆v(ω1) ∼= � i:λ+εi∈X+ ∆v(λ + εi) ⊕ � i:λ−εi∈X+ ∆v(λ − εi) ⊕ ∆v(λ)⊕δpos, where δpos = 0 if λN = 0 and δpos = 1 if λN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus in type B the tensor product decomposition for weights in X1,+ is straight-forward with the only difference to type C being the appearance of a summand ∆v(λ) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Even the sequences are looking very similar to the type C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For weights in X 1/2,+ on the other hand, the tensor product decomposition rule depends on the explicit weight of the Weyl module, namely the summand ∆v(λ) only appears if λN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In addition we are dealing with sequences on half-integers in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fock space of sequences and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We decompose the Fock space as F(BN) = F1(BN) ⊕ F 1/2(BN), with F1(BN) having basis λ ∈ X1,+ ρ and F 1/2(BN) having basis λ ∈ X 1/2,+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For F1(BN) we can use the same definition of sequences and Fock spaces as in type C, and embed F1(BN) into FN as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the case of F 1/2(BN) we use the sequences SH and corresponding Fock space F 1/2 from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence we map λ ∈ X 1/2,+ ρ to the sequence aλ with aλ(i) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 if i < 0, 1 if there exists 1 ≤ j ≤ N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' i = λj, 0 otherwise to embed F 1/2(BN) into F 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As before it is clear that the image is contained in the subspace F 1/2 N of sequences of charge N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We continue to write λ for the sequence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The analogue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2 holds in type B as well, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' ei preserves the subpsaces F1(BN) and F 1/2(BN), fi does so for i /∈ {0, 1/2}, and fi does not preserve the subpsaces for i ∈ {0, 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to type C, the cases of ℓ odd and even have to be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Linkage and operators for odd ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Throughout this section we assume that ℓ > 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The proofs in this section have a very similar flavour to the ones in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Unfortunately there are some subtle differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We start with the statements that work for both the integer and the half-integer weight cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ and ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ and ν = esλ are defined for r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ′ = frλ and ν′ = fsλ are defined for r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ′ and ν′ are linked if and only if r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4, µ = λ − εi with λi = r + 1/2 and ν = λ − εj with λj = s + 1/2 and we assume that i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume now that µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12 to obtain alcoves Aµ and Aν with a minimal gallery only crossing walls strictly between the alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The absolute value of (ν − µ, β∨) can be 2 in case of β ∈ {β− ij, βi, βj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that a hyperplane of the form Hβ− ij,m is strictly between the alcoves, it follows as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4 that r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence assume that there is no hyperplane of the form Hβ− ij,m strictly between the alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since ν − µ = εi − εj there has to be both a hyperplane of the form Hβi,m′ and of the form Hβj,m′′ strictly between the alcoves since we assume that the weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This implies 2λi = m′ℓ and 2λj = m′′ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence 2(λi − λj) = (m′ − m′′)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since ℓ is odd (m′ − m′′) has to be even and so Hβ− ij,(m′−m′′)/2 is strictly between the alcoves, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Conversely assume that r ∈ s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then λi − λj is a multiple of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (λ, (β− ij)∨) = λi − λj is a multiple of ℓ and so (ν, (β− ij)∨) = 1 + mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus sβ− ij,m(ν) = µ and so the weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The analogous statement for fi and fj holds with the analogous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ For the case of mixed operators we get accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ and ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ and ν = fsλ are defined for r + 1 ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ −s + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The proof is nearly identical to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix i ̸= j via µ = λ − εi and ν = λ + εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then the roots that give absolute values strictly bigger than 1 for (ν − µ, β∨) = (εi + εj, β∨) are β ∈ {β+ ij, βi, βj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that the weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fixing the alcoves as usual and the corresponding minimal gallery, we see that if Hβ+ ij,m is strictly between the alcoves for some m then the statement follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If one assumes that such a hyperplane does not exist, the contradiction is obtained as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If it holds r ∈ −s+ℓZ, then (λ, (β+ ij)∨) = λi+λj is a multiple of ℓ and so (ν, (β− ij)∨) = 1+mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus sβ+ ij,m(ν) = µ and so the weights are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ For the analogue of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7 we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume λ ∈ X1,+ ρ and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume λ ∈ X 1/2,+ ρ and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ (ℓ+1)/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix i via µ = λ − εi respectively ν = λ + εi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' with λi = r − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ and ν are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Again use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12 to obtain alcoves Aµ and Aν with a minimal gallery only crossing walls strictly between the alcoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The possible positive roots such that the absolute value |(ν − µ, β∨)| > 1 are β = βi for the value 4 and for the value 2 it is β = β± ij for i < j or β = β± ji for j < i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , At be the alcoves in the minimal gallery connecting Aµ and Aν and H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , Ht the crossed hyperplanes in order, all strictly between the alcoves by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix positive roots and integers such that Hp = Hγp,mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Recall that for all these it holds |(ν − µ, γ∨ p )| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Case: γ1 = βi and r = 1: Then (ν, β∨ i ) = 2 + m1ℓ since we need to have sβi,m1(ν) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus 2λi + 2 = 2 + m1ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that λ ∈ X1,+ ρ , it holds that m1 must be even (since ℓ is odd) and so λi ∈ ℓZ or equivalently r ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If on the other hand λ ∈ X 1/2,+ ρ , then m1 must be odd and we obtain λi ∈ ℓ/2 + ℓZ or equivalently r ∈ (ℓ+1)/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Case: γ1 = β− ij for some j: Then it holds that (ν, (β− ij)∨) = 1+m1ℓ, sβ− ij,m1(ν) = λ+εj ̸= µ, and λi−λj = m1ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Now for q > 1 it holds (λ+εj, γ∨ q ) ∈ ℓZ and thus sγq,mq(λ+εj) = λ+εj unless γq ∈ {βi, β+ ij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence either βi or β+ ij has to appear as a γq for q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let p > 1 be the minimal such that γp ∈ {βi, β+ ij}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If γp = β+ ij then (λ + εj, (β+ ij)∨) = (ν, (β+ ij)∨) = 1 + mpℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence λi + λj = mpℓ and so 2λi = (m1 + mp)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As above, for λ ∈ X1,+ ρ , it follows that r ∈ 1/2 + ℓZ and for λ ∈ X 1/2,+ ρ it follows that r ∈ (ℓ+1)/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore, we see that γq ̸= βi for all q since we already have sβ+ ij,m′(λ + εj) = µ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If γp = βi then (λ + εj, (βi)∨) = (ν, (βi)∨) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But Hβi,mp is assumed to be strictly between the alcoves, thus (ν, (βi)∨) = k + mpℓ for k ∈ {1, 2, 3} (k = 4 is not possible since µ would lie on Hβi,mp in that case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then (λ + εj, (βi)∨) = k − 2 + mpℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (1) If k = 1, then Hβi,mp was already crossed, which cannot be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (2) If k = 2 then λ + εj is on Hβi,mp and invariant under sβi,mp, hence β+ ij has to appear in the sequence of γ’s that follow and one argues as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (3) If k = 3, it holds sβi,mp(λ + εj) = λ + εj − εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But (λ + εj − εi, γ∨ q ) = (µ, γ∨ q ) for q > p except for γq = β+ ij and for β+ ij it holds (λ+εj −εi, (β+ ij)∨) = (λ, (β+ ij)∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence λ + εj − εi is already in the same half-space as µ for all Hq with q > p and γq ̸= β+ ij and it is on the hyperplane Hq for q > p and γq = β+ ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus the only hyperplane left to cross is Hp+1 and it must hold that γp+1 = β+ ij, but sβ+ ij,mp+1(λ + εj − εi) = λ + εj − εi ̸= µ, which is a contradiction to the construction of the gallery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remaining cases: The remaining cases for γ1 are all done in the same way as the previous case with slight modifications in the signs that appear, but the same general arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Conversely for λ ∈ X1,+ ρ and r ∈ 1/2 + ℓZ, λi = mℓ for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (λ + εi, β∨ i ) = 2λi + 2 = 2 + 2mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus sβi,2m(ν) = µ and so ν and µ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For λ ∈ X 1/2,+ ρ and r ∈ (ℓ+1)/2 + ℓZ, λi = ℓ/2 + mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (λ + εi, β∨ i ) = 2λi + 2 = 2 + (2m + 1)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus sβi,2m+1(ν) = µ and so ν and µ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ In contrast to type C, λ can be linked to a weight µ obtained by applying an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X1,+ ρ and ℓ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and λ are linked if and only if r ∈ ℓ/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that ν = frλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ν and λ are linked if and only if r ∈ ℓ/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix i via µ = λ − εi with λi = r + 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume now that µ and λ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The absolute value of (λ − µ, β∨) can only be 2 for β = βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence there exists a hyperplane Hβi,m strictly between and so (λ, β∨ i ) = 2λi = 1+mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Since 2λi is an even integer and ℓ is odd, m is odd as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' So we get r = ℓ/2+ m−1 2 ℓ ∈ ℓ/2+ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If r ∈ ℓ/2 + ℓZ then λi = (ℓ+1)/2 + mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence (λ, β∨ i ) = 1 + (2m + 1)ℓ and so λ and µ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The case of ν follows, considering that in this case erν = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ The proof for the next Lemma is completely analogous to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X 1/2,+ ρ and ℓ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and λ are linked if and only if r ∈ ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that ν = frλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ν and λ are linked if and only if r ∈ ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Quantum symmetric pair action for odd ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As in the previous section we are assuming throughout the section that ℓ > 3 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Recall the various operators and counting statistics from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In contrast to type C, we need to use operators for both types of index sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The definition of the linear operators is very close to the ones of type C in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' with one difference for the elements of H/ℓZ respectively Z/ℓZ that are fixed by Θ as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that for ℓ odd, Θ(i) = i implies that i = ℓ/2 for i ∈ H/ℓZ and i = 0 for i ∈ Z/ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This corresponds to the two fixed nodes in the odd Dynkin diagrams in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a ∈ SZ let p ∈ H/ℓZ and for a ∈ SH let p ∈ Z/ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If Θ(p) ̸= p define ˆBp a = vT e−f ℓ (−p,a) � j∈p vRe−f ℓ (j,a)ej a + � j∈−p vLf−e ℓ (j,a)fj a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' ˆLpa = vT f−e ℓ (p,a)vT e−f ℓ (−p,a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If Θ(p) = p define ˆBpa = v−1vT e−f ℓ (p,a) � j∈p vRe−f ℓ (j,a)eja + � j∈p vLf−e ℓ (j,a)fja + vT e−f ℓ (p,a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Finally, for a ∈ SH and z ∈ 1/2 + ℓZ define ˆB[z] 0 a = v−1vT e−f ℓ (0,a) � j∈0 vRe−f ℓ (j,a)eja + � j∈0 vLf−e ℓ (j,a)fja + δa(z),0vT e−f ℓ (0,a)a, where δa(z),0 = 1− a(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This defines linear operators on F and F 1/2 that restrict to FN and F 1/2 N for any N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The linear operator ˆB[z] 0 is not contained in the image of the affine quantum symmetric pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It is needed to describe the relationship with the tensor product decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Comparing this with the statements in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9, the same identifications hold except for the case Θ(p) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let a ∈ SZ (respectively a ∈ SH) and p ∈ H/ℓZ (respectively p ∈ Z/ℓZ) with Θ(i) = i then ˆBp a = v−1ˆEp ˆK−1 p a + ˆFp a + ˆK−1 p a and for a ∈ SH and z ∈ 1/2 + ℓZ ˆB[z] p a = v−1ˆEp ˆK−1 p a + ˆFp a + δa(z),0 ˆK−1 p a Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In [Kol14] a quantum symmetric pair using a generator of the form v−1ˆEp ˆK−1 p + ˆFp + ˆK−1 p is called a non-standard quantum symmetric pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' While the one that uses only generators of the form that appeared in type C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' without the extra summand ˆK−1 p are called standard quantum symmetric pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Acting with the generators on F1(BN) respectively F1(BN) has the same possibility of having a single summand that is not in the subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case of FN the operators ˆBi leave the subspace F1(BN) invariant, except for −1/2 ∈ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the operator ˆB1/2 the reason is again that it contains a summand f1/2 that does not leave F1(BN) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' While for F 1/2 N the operators ˆB0 and ˆB[1/2] 0 do not leave the subspace invariant because of the summand f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 24 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN For the relations of the operators we then get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ odd and I ∈ {Z/ℓZ, H/ℓZ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The linear operators {ˆBp | p ∈ I} and {ˆLp | p ∈ I, Θ(p) ̸= p} satisfy the relations of Bv(I, Θ) given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 by substituting the operators for the generators with the same name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' All relations in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 not involving the operator ˆBp for p fixed are precisely the same as in type C and thus hold by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that for p fixed, the new operator ˆBp only differs from the description in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9 by adding a term of the form ˆK−1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The commutator relation with all ˆLq is then obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The commutator relation with “distant” ˆBq is also obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the only one left are the non-trivial quantum Serre relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There are only two that involve a fixed index, which are both a simple few line calculation to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ As in type C this all combines then to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For the case of weights in X1,+ the statement is the complete analogue to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12 with precisely the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There exists an action of Bv(H/ℓZ, Θ) on FN such that for λ ∈ X1,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules is obtained from � p∈H/ℓZ ˆBpλ by projecting onto the subspace F(CN) and evaluating v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore if [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a unique p ∈ H/ℓZ such that µ and ν appear in ˆBpλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case of λ ∈ X 1/2,+, the dependence of the tensor product decomposition rule on λN makes this less clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There exists an action of Bv(Z/ℓZ, Θ) on F 1/2 N such that for λ ∈ X 1/2,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules is obtained from � p∈Z/ℓZ ˆBpλ if λN > 0 and from � p∈Z/ℓZ,p̸=0 ˆBpλ + ˆB[1/2] 0 λ if λN = 0, by projecting onto the subspace F(CN) and evaluating v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a unique p ∈ Z/ℓZ such that µ and ν appear in ˆBpλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The only difference to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 is that in case of λ ∈ X 1/2,+ the class of the tensor product [∆q(λ) ⊗ ∆q(ω1)] contains a summand [∆q(λ)] if and only if λN > 0, But ˆB0λ always contain a non-zero multiple of λ as a summand, hence one has to apply ˆB[1/2] 0 instead in case λN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ Let 0 ≤ k < ℓ and N = mℓ + k for m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One can use the map from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13 to embed F1(Bmℓ+k) into F1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this case the analogue of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case of a ∈ SH,N the map from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='13 embeds F 1/2(Bmℓ+k) into F 1/2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' To formulate the corresponding statement, define ˆB[z] p = ˆBp if p ̸= 0 for z ∈ 1/2 + ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 25 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let a ∈ SH,k and r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then for m ≫ 0, a = λ(m) for λ ∈ X 1/2,+ for Uq of type Bmℓ+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore there are Laurent polynomials dλ,µ(v) with non-negative integer coefficients for µ ∈ X 1/2,+ such that � (Z/ℓZ)r ˆB[z] p1 · · · ˆB[z] pr λ(m) = � µ dominant for Uq dλ,µ(v)µ(m), with z = 1/2 − mℓ and [∆q(λ) ⊗ ∆q(ω1)⊗r] = � µ dominant for Uq dλ,µ(1)[∆q(µ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that the only difference to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 is that instead of ˆB0 one has to use ˆB[z] 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We already know that � (Z/ℓZ)r ˆBp1 · · · ˆBprλ(m) is contained in the shifted embedding of F 1/2(Bmℓ+k) for m ≫ 0, hence also the summand � (Z/ℓZ)r ˆB[z] p1 · · · ˆB[z] pr λ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case m ≫ 0 all weights ν such that [∆q(ν)] appears in [∆q(λ)⊗∆q(ω1)⊗s] (for 0 ≤ s < r) have the property that νk+mℓ = 0, hence ν(1/2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the [∆q(ν)⊗∆q(ω1)] never contain the summand [∆q(ν)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus ˆB[1/2] 0 ν has to be used to get the tensor product decomposition by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' After the shift this becomes the operator ˆB[z] 0 ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Linkage and operators for even ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The first difference in case ℓ even is that ℓβ = 2 for the roots of the form β = β± ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' These were responsible in the odd case that all indices had to be taken modulo ℓ to obtain the operators and that one had to group ep and fq for p + q ∈ ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Now this is replaced by the analogue statements with ℓ/2 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus we have to make the following assumption throughout the section ℓ is even and ℓ/2 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As before this is needed to avoid quantum symmetric pairs for small ℓ/2, the general argu- ments for linkage work the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Namely one obtains the two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ and ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ and ν = esλ are defined for r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ s + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ′ = frλ and ν′ = fsλ are defined for r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ′ and ν′ are linked if and only if r ∈ s + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X+ ρ and ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ and ν = fsλ are defined for r + 1 ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ −s + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The analogue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='6 now depend on whether ℓ/2 is itself odd or even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This makes sense if one looks at the corresponding Dynkin diagrams in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='3 for r = ℓ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case of X1,+ ρ the Dynkin diagram with ℓ/2 nodes always has the property that 1/2 is Θ-linked, but it depends on the parity of ℓ/2 whether ℓ/4 is fixed in case ℓ/2 is even or ℓ−2/4 is also Θ-linked in case ℓ/2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Similarly for X 1/2,+ ρ the Dynkin diagram with ℓ/2 nodes always has 0 is fixed, but it depends on the parity of ℓ/2 whether the “opposite” side of the Dynkin diagram has another fixed label or a pair of Θ-linked labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus the analogue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 has four possible combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (1) Assume ℓ/2 odd, λ ∈ X1,+ ρ , and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are linked if and only if r ∈ 1/2 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (2) Assume ℓ/2 even, λ ∈ X1,+ ρ , and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If µ and ν are linked then r ∈ 1/2 + (ℓ/2)Z or r ∈ ℓ+2/4 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 26 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN (3) Assume ℓ/2 odd, λ ∈ X 1/2,+ ρ , and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If µ and ν are linked then r ∈ ℓ+2/4 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (4) Assume ℓ/2 even, λ ∈ X 1/2,+ ρ , and that µ = er−1λ and ν = frλ are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and ν are not linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We only sketch the proof here, since for each situation one has to go through the same procedure as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For ℓ/2 odd and λ ∈ X1,+ ρ the proof is nearly word for word the same, hence the result is also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For ℓ/2 even and λ ∈ X1,+ ρ one obtains that r ∈ 1/2 + (ℓ/2)Z in case of the minimal gallery having length one and γ1 = βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But in the remaining cases that γ1 is different one obtains either r ∈ 1/2 + (ℓ/2)Z if m1 + mp appearing in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5 is even or r ∈ (ℓ+2)/4 + (ℓ/2)Z if m1 + mp is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that r ∈ (ℓ+2)/4 + (ℓ/2)Z one does not obtain the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The argument using a reflection at a hyperplane for βi does not work here, since ℓβi = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the case ℓ/2 odd and λ ∈ X 1/2,+ ρ the case of a minimal gallery of length 1 gives a contradiction, while the other cases give r ∈ ℓ+2/4 + (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that if ℓ/2 is odd, ℓ+2/4 is an integer, so the r is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Finally, the case ℓ/2 even and λ ∈ X 1/2,+ ρ with the assumption of linked weights always gives a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ Thus we are left with comparing λ and the image under one of the operators ei or fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In the case of λ ∈ X1,+ ρ the situation becomes quite easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X1,+ ρ and ℓ is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and λ are not linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that ν = frλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ν and λ are not linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Fix i via µ = λ − εi and r + 1/2 = λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='7 one only has to consider the root βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' And one assumes that the weights are linked with hyperplane Hβi,m strictly between the alcoves having to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence one obtains (λ, β∨ i ) = 2λi = 1 + mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But 2λi is an even integer, while 1 + mℓ is always odd, hence this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Again the second case follows since if ν = frλ is defined, then erν = λ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ In case of λ ∈ X 1/2,+ ρ , the situation resembles the case of ℓ odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Let λ ∈ X 1/2,+ ρ and ℓ is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that µ = erλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then µ and λ are linked if and only if r ∈ (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume that ν = frλ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then ν and λ are linked if and only if r ∈ (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We proceed like in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='20 and obtain the same equation (λ, β∨ i ) = 2λi = 1 + mℓ for some m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But now 2λi is an odd integer, hence λ = 1/2 + mℓ/2 which then implies r ∈ (ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Conversely if r ∈ (ℓ/2)Z, then (λ, β∨ i ) = 2λi = 1 + mℓ for some m ∈ Z, hence sβi,m(λ = µ and λ and µ are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As before the case for fr is done by using that erν = λ under the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Quantum symmetric pair action for even ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Throughout the section we make the assumption that ℓ is even and ℓ/2 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In this situation the action of the quantum symmetric pair depends on two different choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' First whether ℓ/2 is even or odd and second whether one is looking at weights in X1,+ ρ or X 1/2,+ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We just list the definitions of the linear operators in the different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that we obtain all possible quantum symmetric pairs for the affine type A Dynkin diagram that we described before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' we can have either two fixed labels, two Θ-linked label pairs, or one of each and in that case we will see that they differ in the sense that one of them is a standard quantum symmetric pair, the other one is a non-standard quantum symmetric pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We are considering now Z/(ℓ/2)Z and H/(ℓ/2)Z with the usual automorphism Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For a ∈ SZ let p ∈ H/(ℓ/2)Z and for a ∈ SH let p ∈ Z/(ℓ/2)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then define for p ̸= Θ(p) ˆBp a = v T e−f ℓ/2 (−p,a) � j∈p v Re−f ℓ/2 (j,a)ej a + � j∈−p v Lf−e ℓ/2 (j,a)fj a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' ˆLp a = v T f−e ℓ/2 (p,a)v T e−f ℓ/2 (−p,a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For p = Θ(p), but 0 /∈ p define ˆBpa = v−1v T e−f ℓ/2 (p,a) � j∈p v Re−f ℓ/2 (j,a)eja + � j∈p v Lf−e ℓ/2 (j,a)fja, while for 0 ∈ i z ∈ 1/2 + (ℓ/2)Z define ˆBpa = v−1v T e−f ℓ/2 (p,a) � j∈p v Re−f ℓ/2 (j,a)eja + � j∈p v Lf−e ℓ/2 (j,a)fja + v T e−f ℓ/2 (p,a)a, and ˆB[z] 0 a = v−1v T e−f ℓ/2 (0,a) � j∈0 v Re−f ℓ/2 (j,a)eja + � j∈0 v Lf−e ℓ/2 (j,a)fja + δa(z),0v T e−f ℓ/2 (0,a)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This defines linear operators on F and F 1/2 that restrict to FN and F 1/2 N for any N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As one can see, in case that a ∈ SZ and ℓ/2 is even there is no fixed index p ∈ H/(ℓ/2)Z, instead there are two pairs of indices that are Θ-linked, namely ±1/2 and ℓ±2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that a ∈ SZ and ℓ/2 is odd, the pair ±1/2 is still Θ-linked, but ℓ/4 is a fixed index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But the quantum symmetric pair is a standard quantum symmetric pair, like in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that a ∈ SH and ℓ/2 is even, both 0 and ℓ/4 are fixed, but the generator for 0 makes this a non-standard quantum symmetric pair like in type B for odd ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case that a ∈ SH and ℓ/2 is odd, we again have a non-standard operator for the fixed index 0 and a pair of Θ-linked indices with ℓ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We skip the analogue of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10 in this case, it is just a combination of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='10 depending on whether a generator for a fixed index is defined as in type C or as in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus we get the corresponding actions Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ even and I ∈ {Z/(ℓ/2)Z, H/(ℓ/2)Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The linear operators {ˆBp | p ∈ I} and {ˆLp | p ∈ I, Θ(p) ̸= p} satisfy the relations of Bv(I, Θ) given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 by substituting the operators for the generators with the same name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Depending on the definition of the operators this follows either from the situation in type C or from the case ℓ odd in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then the corresponding theorems for the action and the decomposition of the tensor product can then be immediately formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 28 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There exists an action of Bv(H/(ℓ/2)Z, Θ) on FN such that for λ ∈ X1,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules is obtained from � p∈H/(ℓ/2)Z ˆBpλ + λ by projecting onto the subspace F(CN) and evaluating v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore if [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a unique p ∈ H/ℓZ such that µ and ν appear in ˆBpλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is the exact analogue of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But in case of ℓ/2 even there is no operator that produces a multiple of λ and in case of ℓ/2 odd the linear operator for the fixed index is standard, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' also does not produce a multiple of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But since λ ∈ X1,+, the tensor product always includes the class of [∆q(λ)] in the Grothendieck group, hence one has to naively add it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' □ And the analogue of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Assume ℓ even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There exists an action of Bv(Z/(ℓ/2)Z, Θ) on F 1/2 N such that for λ ∈ X 1/2,+ the decomposition of [∆q(λ) ⊗ ∆q(ω1)] in [Uq-mod] with respect to the classes of Weyl modules is obtained from � p∈Z/(ℓ/2)Z ˆBpλ if λN > 0 and from � p∈Z/(ℓ/2)Z,p̸=0 ˆBpλ + ˆB[1/2] 0 λ if λN = 0, by projecting onto the subspace F(CN) and evaluating v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If [∆q(µ)] and [∆q(ν)] appear in the decomposition for µ and ν linked, then there exists a unique p ∈ Z/ℓZ such that µ and ν appear in ˆBpλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note in case of ℓ/2 even and for λ ∈ X 1/2,+, there exist the operator ˆBℓ/4 that has a fixed index, but it is of the same form as in type C, hence does not produce a multiple λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence only ˆB0λ needs to be replaced by ˆB[1/2] 0 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We skip the discussion of an analogues of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The construction is analogous to type C for the space Fk and analogous, including the same manipulations, as in type B (with ℓ odd) for F 1/2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Type D and beyond The type D case is part of the second authors Master thesis and the details will be published separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We give a rough summary of what happens in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In type DN (N > 3), the choices of εi and the invariant bilinear form are precisely the same as in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The positive roots can be chosen as Φ+ = {β± i,j = εi ± εj | 1 ≤ i < j ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Note that all roots are short and hence equal to their own coroots and ℓβ = ℓ for all positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Integral weights have a similar structure as in type B with X1,+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi ≥ λi+1, and λN−1 ≥ |λN|} and X 1/2,+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ H, λi ≥ λi+1, and λN−1 ≥ |λN|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' COMBINATORIAL FOCK SPACES AND QUANTUM SYMMETRIC PAIRS 29 With ρ = (N − 1)ε1 + (N − 2)ε2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' + εN−1 ∈ X 1/2,+ we thus get X1,+ ρ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ H, λi > λi+1, and λN−1 ≥ |λN|}, X 1/2,+ ρ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN) | λi ∈ Z, λi > λi+1, and λN−1 ≥ |λN|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In analogy to type B, the combinatorial Fock space of type D decomposes into two summands F 1/2(DN) and F1(DN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Furthermore the tensor product decomposition in type D is the same as in type C in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1, especially there is no complication with a ∆v(λ) summand as in type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The condition for λ to be dominant does not depend on λN itself, but rather on its absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Thus (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN−1, λN)+εN is dominant if and only if (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN−1, −λN)− εN is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In terms of sequences this can be interpreted as saying that a 1 should be placed at position |λN| and the sign of λN needs to be recorded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In case of X 1/2,+ ρ , following Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='1 one sees that one can embed F 1/2(DN) into F 1/2,+ N ⊕ F 1/2,− N , where both spaces are isomorphic to F 1/2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For this let λ′ be equal to λ except λ′ N = |λN|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Then λ is mapped to (aλ, 0) if λN > 0 and to (0, aλ′) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The moving operators from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2 are defined on F 1/2,+ N ⊕ F 1/2,− N , since they are defined on F 1/2 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The interplay between moving operators and linkage, is analogous to types C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Hence the operators ˆBp and ˆLp are given as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='8, up to changing the index set from H/ℓZ to Z/ℓZ, and give an action of Bv(Z/ℓZ, Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' In addition, one introduces an operator ˆB that goes between F 1/2,+ N and F 1/2,− N , since (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN−1, 1/2) − εN = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' , λN−1, −1/2), but both are represented by the same se- quence, but in different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One defines ˆB(a, 0) = (0, a) if a(1/2) = 1 and the operator is zero otherwise, and similarly ˆB(0, a) = (a, 0) if a(1/2) = 1 and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It follows that only ˆB and ˆB0 can produce linked weights, hence one needs to consider ˆB0 + ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The analogue of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='15 holds, with the difference that ˆB0 is replaced by ˆB0 + ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The limit construction from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='14 does not work in type D as one cannot regard a dominant weight λ with λN < 0 as a dominant weight for a bigger rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' If one restricts to this case then one essentially recovers the combinatorics of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The case of X1,+ ρ brings additional complications with it, since λN = 0 does not allow for a “good” embedding into a sum of Fock spaces F1,+ ⊕ F1,−, where these spaces are defined as in the half-integer case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' One instead embeds λ with λN = 0 diagonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This allows for an analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='12, but depending on the starting weight, (λ, 0), (0, λ) or (λ, λ) represents the class of the Weyl module [∆q(λ)] and similar for the occurring Weyl modules in the decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The limit construction does not work for the same reason as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' As a concluding remark in type D one can note that the combinatorial cal- culations are easier than in type B and C, since it is of simply-laced type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But the action of the quantum symmetric pair is further away from the combinatorics of the tensor product multiplicities and the limit construction is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This kind of complication in type D has some analogy in [ES18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' Here the type D situation is investigated in terms of category O and certain generators of the quantum symmetric pair do not act as indecomposable translation functors, while in the analogous situation in type B they would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' We focused in this paper on non-exceptional Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The general ideas can of course be transferred to exceptional cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' There is in general no clear analogue of the representation ∆q(ω1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' a “natural” representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' For example, in type G2 one can use the specialization and quantization of the natural representation of so8(C), since so8(C) contains the Lie algebra of type G2 as a fixed point Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' But there is nothing analogous for the other exceptional types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 30 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' EHRIG AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' GAN As for the analogue of the quantum symmetric pair, it is reasonable to expect that in type G2 the acting algebra has some relationship to the quantum symmetric pair and an automorphism of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' This is in analogy to the fact that the quantum symmetric pair is constructed from the quantum affine algebra using an automorphism of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' It is a priori not clear what the corresponding object in other exceptional cases would look like or come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' References [And03] Andersen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' The strong linkage principle for quantum groups at roots of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' : Beijing Institute of Technology, School of Mathematics and Statistics, Liangxiang Campus of Beijing Institute of Technology, Fangshan District, 100288 Beijing, China Email address: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='ehrig@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='cn K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content=' : Beijing Institute of Technology, School of Mathematics and Statistics, Liangxiang Campus of Beijing Institute of Technology, Fangshan District, 100288 Beijing, China Email address: 3120191408@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE1T4oBgHgl3EQfcQQe/content/2301.03181v1.pdf'} diff --git a/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/2301.01569v1.pdf.txt b/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/2301.01569v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a15158a2ee5c765fee7a56d6f3d5dafdf89d7296 --- /dev/null +++ b/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/2301.01569v1.pdf.txt @@ -0,0 +1,2135 @@ +Learning Decorrelated Representations Efficiently +Using Fast Fourier Transform +Yutaro Shigeto* Masashi Shimbo∗ +Yuya Yoshikawa +Akikazu Takeuchi +{shigeto,shimbo,yoshikawa,takeuchi}@stair.center +STAIR Lab, Chiba Institute of Technology, Narashino, Chiba, Japan +Abstract +Barlow Twins and VICReg are self-supervised represen- +tation learning models that use regularizers to decorrelate +features. Although they work as well as conventional repre- +sentation learning models, their training can be computa- +tionally demanding if the dimension of projected represen- +tations is high; as these regularizers are defined in terms +of individual elements of a cross-correlation or covariance +matrix, computing the loss for 𝑑-dimensional projected rep- +resentations of 𝑛 samples takes 𝑂(𝑛𝑑2) time. In this paper, +we propose a relaxed version of decorrelating regulariz- +ers that can be computed in 𝑂(𝑛𝑑 log 𝑑) time by the fast +Fourier transform. We also propose an inexpensive trick to +mitigate the undesirable local minima that develop with the +relaxation. Models learning representations using the pro- +posed regularizers show comparable accuracy to existing +models in downstream tasks, whereas the training requires +less memory and is faster when 𝑑 is large. +1. Introduction +Unsupervised or self-supervised learning (SSL) of repre- +sentations [1,4–6,10,12,13,24,25,28,29] has become an in- +tegral part of deep learning applications in computer vision. +In SSL, a network is first pretrained on a “self-supervised” +pretext task for which labeled data can be readily obtained +on a large scale without human supervision. Lower layers +of the pretrained network are then reused for downstream +tasks, with the expectation that these layers produce generic +representations of the input that are also useful in down- +stream. +Most of the SSL models for visual representations em- +ploy a multi-view, Siamese network architecture. First, an +input image is converted by different random transforma- +tions that do not alter its original semantics. These two +augmented examples are then fed to a neural network (or +in some cases, two different networks) that consists of a +*Equal contributions. +backbone network cascased with a small projection net- +work (usually a multi-layer perceptron), to produce “twin” +projected embeddings of the original. Finally, the network +weights are trained so that the two embeddings (also called +“positive pair”) are similar, reflecting the fact that they rep- +resent the same original image. After training, the projec- +tion network is discarded, and only the backbone network is +reused for downstream tasks as the encoder of input images +that produces their representations. +One major issue in SSL is the representation collapse, +or the presence of meaningless solutions such that all ex- +amples are projected to a single vector embedding. Con- +trastive approaches [4,5,13,24,25] eliminate such solutions +by a loss term to repel embeddings produced from different +original images, or “negative pairs.” Because considering +all possible negative pairs is infeasible, negative sampling +is usually performed. +Some recent work has explored non-contrastive SSL +models. Among these, Barlow Twins [28] and VICReg [1] +use loss functions to penarlize the features of embeddings +with a small variance, which in turn discourages the col- +lapse. They further introduce regularization terms to decor- +relate features, by promoting off-diagonals of the correla- +tion/covariance matrices to be zero. Although these mod- +els perform as well as contrastive models, their regularizers +are computationally demanding for high-dimensional em- +beddings; since the regulaizers are defined in terms of indi- +vidual elements in covariance or cross-correlation matrices, +they require 𝑂(𝑛𝑑2) time to compute, where 𝑛 is the num- +ber of samples in a batch, and 𝑑 is the dimensionality of +the projected embeddings. This is unfortunate, as improved +performance is reported for both Barlow Twins and VICReg +as 𝑑 is increased [1,28]. +Contributions. +In this paper, we address the inefficiency +of Barlow Twins and VICReg mentioned above with a re- +laxed version of decorrelating regularizer. This regulariza- +tion does not require the calculation of the correlation/co- +variance matrices explicitly and can be computed in time +𝑂(𝑛𝑑 log 𝑑) by means of circular convolution and the fast +1 +arXiv:2301.01569v1 [cs.LG] 4 Jan 2023 + +Fourier transform (FFT). Undesirable local minima that can +develop with the use of the relaxed regularizer can be over- +come by feature permutation during training. +We show +that the resulting method achieves competitive performance +with Barlow Twins and VICReg on downstream tasks, with +substantially less computation time when 𝑑 is large. The +proposed method also reduces memory consumption, which +allows an increased batch size. +Notation. +We use the 0-based indexing for vector and ma- +trix components unless stated otherwise; thus, for a vector +x ∈ R𝑑, the component index ranges from 0 to 𝑑 − 1. We +denote by [x]𝑖 the 𝑖th component of vector x, and [M]𝑖 𝑗 +denotes the (𝑖, 𝑗)-element of matrix M. For a complex vec- +tor c, c denotes its componentwise complex conjugate. For +vectors x and y, x ◦ y denotes their componentwise product. +2. Related Work +Contrastive SSL. +Contrastive representation learning +uses positive and negative pairs of augmented samples [4,5, +7,9,13,22,24,25]. The commonly used InfoNCE loss [24] +consists of an alignment term, which maximizes similarity +between positive pairs, and a uniformity term, which mini- +mizes similarity between negative pairs [25]. SimCLR [4] +is one of the state-of-the-art methods in this category. How- +ever, to obtain good representations, SimCLR needs a large +number of negative pairs [4], or, in other words, a large +batch size 𝑛. This can be a computational bottleneck, as +the loss computation of SimCLR takes 𝑂(𝑛2𝑑) time where +𝑑 is the dimensionality of the projected embeddings. +Non-contrastive SSL by Asymmetric Architecture. +Recently, researchers have started exploring the possibility +of non-contrastive approaches to self-supervised represen- +tation learning, i.e., those that do not use negative pairs for +training. To overcome the representation collapse, models +such as BYOL [12] and SimSiam [6] employ asymmetric +network architectures, e.g., by suppressing gradient updates +and/or using the moving average of network parameters for +one view. These methods are heuristically motivated, as +they do not explicitly penalize the collapse but work well in +practice. +Non-contrastive SSL by Decorrelating Regularization. +Barlow Twins [28] was the first method to introduce a loss +function that explicitly penalizes collapsed embeddings. It +uses a regularizer based on cross-correlation matrices over +two views. VICReg [1] also introduced two regularizers, +but they are defined in terms of covariance matrices of indi- +vidual views. We review these methods in Sec. 3 in detail. +Non-contrastive SSL by Whitening. +Some authors [10, +15] used whitening to explicitly decorrelate features during +training, without resorting to regularization. [29] whitened +both the feature and sample covariances. +Because the +whitening procedures used in these approaches require the +computation of all the eigenvalues of the covariance ma- +trices, a training epoch takes time 𝑂(𝑑3) (or 𝑂(𝑑3 + 𝑛3) +with [29]), which can be problematic with large 𝑑 or 𝑛. +Use of Convolution in Machine Learning. +Convolution +is the basic building block of convolution neural networks +(CNNs). +CNNs take (linear) convolution of input vec- +tors with small learnable filter kernels to extract local fea- +tures. In contrast, we use convolution to compute summary +statistics of the covariance/cross-correlation matrices. Al- +though FFT reduces the asymptotic complexity of convolu- +tion computation, it is seldom used with CNNs, because the +size of kernels is typically too small to warrant speed-up by +FFT. +In other areas of machine learning, circular convolu- +tion and its non-commutative analogue, circular correla- +tion, have been applied to implement associative memory +[3, 18, 20]. The idea has recently been revived for knowl- +edge graph embeddings [16]. +3. SSL Models Using Decorrelating Regulariz- +ers +In this section, Barlow Twins [28] and VICReg [1] are +reviewed. Given a batch of 𝑛 original training examples, +these models apply two different transformations chosen +randomly to each example and input the transformed ex- +amples into a neural network to obtain twin embeddings +(views) of the original. +Let a(𝑘), b(𝑘) ∈ R𝑑 denote the +twin embeddings thus obtained for the 𝑘th example, and let +A = {a(𝑘)}𝑛 +𝑘=1, B = {b(𝑘)}𝑛 +𝑘=1 be the sets of embeddings for +individual views. The network is then trained to minimize +the loss function specific to each model. +Barlow Twins. +Barlow Twins optimizes the loss function +defined in terms of the cross-correlation matrix C(A, B) ∈ +R𝑑×𝑑 between views A and B: +𝐿BT = +𝑑−1 +∑︁ +𝑖=0 +(1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅off(C(A, B)), +(1) +where hyperparameter 𝜆 ≥ 0 controls the strength of reg- +ularization, and 𝑅off : R𝑑×𝑑 → R is a regularizer function +defined as +𝑅off(M) = +𝑑−1 +∑︁ +𝑖=0 +𝑑−1 +∑︁ +𝑗=0 +𝑗≠𝑖 +[M]2 +𝑖 𝑗. +(2) +2 + +The first term in Eq. (1) is minimized when the corre- +sponding features of two views are fully correlated, i.e., +[C(A, B)]𝑖𝑖 += +1 for 𝑖 += +0, . . . , 𝑑 − 1. +This term +can be efficiently computed in time 𝑂(𝑛𝑑). +Regularizer +𝑅off(C(A, B)) in the second term is for feature decorrela- +tion, as off-diagonal elements in C(A, B) are pushed toward +zero. This regularization term can be a computational bur- +den when 𝑑 is large, as it takes 𝑂(𝑛𝑑2) time to compute, +thanks to the 𝑑 × 𝑑 matrix C(A, B). +VICReg. +Let K(A), K(B) ∈ R𝑑×𝑑 be the covariance ma- +trices of A and B, respectively. In VICReg, the loss function +is defined as +𝐿VIC = 𝛼 +𝑛 +𝑛 +∑︁ +𝑘=1 +∥a(𝑘) − b(𝑘) ∥2 +2 ++ 𝜇 +𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) ++ 𝜈 +𝑑 (𝑅off(K(A)) + 𝑅off(K(B))) , +(3) +where 𝛼, 𝜇, 𝜈 ≥ 0 are hyperparameters to control the impor- +tance of individual terms, and 𝑅var is the regularizer defined +as +𝑅var(M) = +𝑑−1 +∑︁ +𝑖=0 +max(0, 𝛾 − +√︁ +[M]𝑖𝑖), +(4) +with the target standard deviation 𝛾 > 0. Function 𝑅off is +the same regularizer as used in Barlow Twins (Eq. (2)), but +applied to K(A) and K(B) instead of C(A, B). +The first term in Eq. (3) brings two views of the same ex- +ample closer. Regularizer 𝑅var penalizes collapsed embed- +dings with zero variances, whereas 𝑅off promotes the diver- +sity of features by encouraging the covariance of features to +be 0. The time complexity of caclculating 𝑅off(K(A)) and +𝑅off(K(B)) is 𝑂(𝑛𝑑2), just like 𝑅off(C(A, B)) in Barlow +Twins. +4. Proposed Method +We propose a weaker but efficiently computable alterna- +tive to the regularizer function 𝑅off used by Barlow Twins +and VICReg. In the following, we present our regularizer in +terms of cross-correlation matrix C(A, B), similarly to Bar- +low Twins. However, if applied to K(A) and K(B) instead, +this regulizer can be used as a drop-in replacement for 𝑅off +in the VICReg’s loss function. +4.1. Regularizer Based on Sums of Cross-correla- +tion +Recall that the loss of Barlow Twins is based on the +cross-correlation matrix C = C(A, B) of two views A = +{a(𝑘)}𝑛 +𝑘=1 and B = {b(𝑘)}𝑛 +𝑘=1. For brevity, assume that both +𝑐00 +𝑐01 +𝑐02 +𝑐10 +𝑐11 +𝑐12 +𝑐20 +𝑐21 +𝑐22 +𝑣0 +𝑣1 +𝑣2 +���� +���� +𝑣0 = 𝑐00 + 𝑐11 + 𝑐22 +𝑣1 = 𝑐01 + 𝑐12 + 𝑐20 +𝑣2 = 𝑐02 + 𝑐10 + 𝑐21 +Figure 1. A 3×3 cross-correlation matrix C = +� +𝑐𝑖 𝑗 +� +(𝑖, 𝑗 = 0, 1, 2) +and sumvec(C) = [𝑣0 𝑣1 𝑣2]T. +𝐴 and 𝐵 are standardized. Then, the cross-correlation ma- +trix is simply given by C = (1/(𝑛 − 1)) �𝑛 +𝑘=1 a(𝑘)b(𝑘)T. +Our regularizer is defined in terms of a 𝑑-dimensional +“summary” vector of the 𝑑 × 𝑑 cross-correlation matrix C. +This vector, denoted here by sumvec(C), is given compo- +nentwise by +[sumvec(C)]𝑖 = +𝑑−1 +∑︁ +𝑗=0 +[C] 𝑗,(𝑖+ 𝑗) mod 𝑑. +(5) +Note that the component indices are 0-based. The 0th com- +ponent [sumvec(C)]0 is the trace of C. Each of the remain- +ing 𝑑 − 1 components corresponds to a sum of 𝑑 different +off-diagonal elements of C, with no single element appear- +ing in two distinct sums. Thus, every element in C appears +exactly once in the summations in Eq. (5). The calculation +of a summary vector for a 3 × 3 covariance matrix is illus- +trated in Fig. 1. +Now, we define a regularizer in terms of all but the 0th +component of sumvec(C): +𝑅sum(C) = +𝑑−1 +∑︁ +𝑖=1 +∥[sumvec(C)]𝑖∥𝑞 +𝑞, +(6) +where hyperparameter 𝑞 ∈ {1, 2}. This function 𝑅sum can +be used as a drop-in replacement for 𝑅off in Barlow Twins’ +loss function (Eq. (1)). The 0th component [sumvec(C)]0 is +excluded from the summation in Eq. (6), because it is equal +to the sum of the diagonal elements of C, which are irrele- +vant to feature decorrelation; they do not appear in Barlow +Twin’s regularizer 𝑅off(C), either. +The regularizer 𝑅sum is weaker than 𝑅off in that it im- +poses constraints on the components of the summary vec- +tor, or the sums of 𝑑 elements of C, whereas 𝑅off constrains +individual elements. Indeed, 𝑅sum(C) is a lower bound of +𝑅off(C). However, as we discuss in Sec. 4.2, 𝑅sum allows +faster computation. Furthermore, in Sec. 4.3, we provide a +simple technique to mitigate the weakness of our regular- +izer. +4.2. Efficient Computation +Computing sumvec(C) by Eq. (5) requires cross- +correlation matrix C(A, B), whose calculation incurs the +3 + +same computational inefficiency as the regularizer in Bar- +low Twins. Fortunately, sumvec(C) can be calculated di- +rectly from the vectors in A and B without their cross- +correlation matrix calculated explicitly, by means of FFT. +To this end, we first need the definitions of involution and +circular convolution. +The involution [20] (also called flipping [21]) inv(x) of +a vector x ∈ R𝑑 is the vector obtained by reversing the or- +der of its 1st (not the 0th) to (𝑑 − 1)st components; i.e., +[inv(x)]𝑖 = [x] (𝑑−𝑖) mod 𝑑 for 𝑖 = 0, . . . , 𝑑 − 1. +For vectors x, y ∈ R𝑑, their circular convolution x ∗ y is +a 𝑑-dimensional vector with components +[x ∗ y]𝑖 = +𝑑−1 +∑︁ +𝑗=0 +� +x yT� +𝑗,(𝑖−𝑗) mod 𝑑 , +(7) +Due to this definition, circular convolution is known as the +“compressed outer product.” +Now, for each twin representations a(𝑘) ∈ A and b(𝑘) ∈ +B (𝑘 = 1, . . . , 𝑛), let us consider vector inv(a(𝑘)) ∗ b(𝑘) ∈ +R𝑑.1 Noting the indices altered by involution, we see that +this vector is given componentwise by +� +inv(a(𝑘)) ∗ b(𝑘)� +𝑖 = +𝑑−1 +∑︁ +𝑗=0 +� +a(𝑘)b(𝑘)T� +𝑗,(𝑖+ 𝑗) mod 𝑑 . +(8) +Substituting C = (1/(𝑛 − 1)) �𝑛 +𝑘=1 a(𝑘)b(𝑘)T into Eq. (5) +and using Eq. (8), we have +[sumvec(C)]𝑖 = +𝑑−1 +∑︁ +𝑗=0 +C +������������������������������������������������ +� +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +a(𝑘)b(𝑘)T +� +𝑗,(𝑖+ 𝑗) mod 𝑑 += +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +𝑑−1 +∑︁ +𝑗=0 +� +a(𝑘)b(𝑘)T� +𝑗,(𝑖+ 𝑗) mod 𝑑 += +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +� +inv(a(𝑘)) ∗ b(𝑘)� +𝑖 , +(9) +or, as a vector, +sumvec(C) = +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +inv(a(𝑘)) ∗ b(𝑘). +(10) +Now, let F and F−1 denote the (discrete) Fourier and the +inverse Fourier transforms, respectively. +Noting that F(inv(x)) = F(x) for any x ∈ R𝑑 (see e.g., +[21], Section 7.4.2) and using the celebrated convolution +1The vector inv(x) ∗ y is known as the circular (cross-)correlation of +x and y [18, 20, 21]. We opt not to use this term in this paper to avoid +confusion with the cross-correlation of random vectors, which is used in +Barlow Twins. +theorem F(x ∗ y) = F(x) ◦ F(y), we have +inv(a(𝑘)) ∗ b(𝑘) = F−1� +F(a(𝑘)) ◦ F(b(𝑘)) +� +(11) +where ◦ denotes componentwise product. Plugging Eq. (11) +into Eq. (10), we obtain +sumvec(C) += +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +inv(a(𝑘))∗b(𝑘) +������������������������������������������������������ +F−1� +F(a(𝑘)) ◦ F(b(𝑘)) +� += +1 +𝑛 − 1 F−1 +� 𝑛 +∑︁ +𝑘=1 +F(a(𝑘)) ◦ F(b(𝑘)) +� +. +(12) +Using this equation, we can calculate sumvec(C) directly +from the representation vectors in A and B, bypassing the +cumbersome calculation of C: First compute the Fourier +transform of all reprenstations a(𝑘) and b(𝑘), and simply +apply Eq. (12). Since the (inverse) Fourier transform of a +𝑑-dimensional vector can be done in time 𝑂(𝑑 log 𝑑) by the +FFT algorithm, and the computation of complex conjugates +and component products, and the sum of the vectors of 𝑛 +takes 𝑂(𝑛𝑑) time, the overall time to obtain sumvec(C) +is 𝑂(𝑛𝑑 log 𝑑). +The time to calculate 𝑅sum(C) is also +𝑂(𝑛𝑑 log 𝑑). This is a substantial improvement over Barlow +Twin’s 𝑂(𝑛𝑑2). The space requirement is 𝑂(𝑛𝑑), which +is optimal if we consider the same 𝑂(𝑛𝑑) space needed to +store input vectors A and B as part of the space complexity. +In contrast, Barlow Twins needs extra 𝑂(𝑑2) space to store +C. +4.3. Feature Permutation to Mitigate Undesirable +Local Minima +As seen from Eq. (5), the components of sumvec(C) +are the sums of 𝑑 elements in C, and the proposed regu- +larizer 𝑅sum (Eq. (6)) encourages these sums to be close to +zero. This is weaker than Barlow Twins’ regularizer 𝑅off +(Eq. (2)), which pushes individual elements of C towards +zero. Indeed, 𝑅sum(C) can be close to zero even if individ- +ual elements in C are not; i.e., the summands in Eq. (5) can +cancel each other, since they can be either positive or neg- +ative. As a result, undesirable local minima develop in the +parameter space, making our regularizer ineffective. +Here, we propose a simple trick to eliminate these local +minima: Randomly permute feature indices during train- +ing, so that the combination of features appearing in a sum +in sumvec(C) changes frequently. To see why this works, +consider minimizing 𝑅sum(C), regarding the elements of C +as independent variables. It is easy to see that the minimum +is attained by the solutions to a homogeneous system of lin- +4 + +ear equations: +[sumvec(C)]𝑖 +�������������������������������������� +𝑑−1 +∑︁ +𝑗=0 +[C] 𝑗,(𝑖+ 𝑗) mod 𝑑 = 0, +for 𝑖 = 1, . . . , 𝑑 − 1. +This is an underdetermined system, with only 𝑑 − 1 equa- +tions but with 𝑑(𝑑 − 1) unknowns, namely, [C] 𝑗ℓ; 𝑗, ℓ = +0, . . . , 𝑑 − 1, 𝑗 ≠ ℓ. This is why nontrivial solutions arise +such that [C] 𝑗ℓ ≠ 0, i.e., those in which summands with +opposite signs cancel each other in an equation and which +are undesirable for our purpose. +Now, by repeatedly permuting the feature indices and +minimizing the loss, we effectively introduce more and +more equations to the system, since permutation can pro- +duce different sets of linear equations over the unknowns, +and these new constraints eventually make non-trivial solu- +tions inadmissible. +For ease of implementation, we permute feature indices +randomly during training, instead of generating all permu- +tations systematically at once. Note that the permuted fea- +ture indices need not be identical across mini-batches, even +within a single epoch; indeed, in the experiments in Sec. 5, +we use a different random permutation of features in every +mini-batch in every epoch. +4.4. Feature Grouping to Control the Degree of Re- +laxation +Instead of computing a summary vector for an entire +cross-correlation matrix C, we can compute summaries at +a more fine-grained level. Specifically, we partition 𝑑 fea- +tures into groups of size 𝑏 each2. This partitioning induces +in C a total of ⌈𝑑/𝑏⌉2 block submatrices of size 𝑏 × 𝑏, +i.e., C = [C𝑖 𝑗] (𝑖, 𝑗 = 1, . . . , ⌈𝑑/𝑏⌉) with submatrices +C𝑖 𝑗 ∈ R𝑏×𝑏. We then define the regularizer by +𝑅(𝑏) +sum(C) = +⌈𝑑/𝑏⌉ +∑︁ +𝑖=1 +𝑏−1 +∑︁ +ℓ=1 +∥[sumvec(C𝑖𝑖)]ℓ∥𝑞 +𝑞 ++ +⌈𝑑/𝑏⌉ +∑︁ +𝑖, 𝑗=1 +𝑖≠ 𝑗 +𝑏−1 +∑︁ +ℓ=0 +��[sumvec(C𝑖 𝑗)]ℓ +��𝑞 +𝑞 , +(13) +As before, sumvec(C𝑖 𝑗) can be computed without ex- +plicitly computing C𝑖 𝑗 by means of involution, circu- +lar convolution (of subvectors of embeddings), and the +Fourier transform. +Calculating a single sumvec(C𝑖 𝑗) +takes 𝑂(𝑛𝑏 log 𝑏) time using FFT, and since there are +⌈𝑑/𝑏⌉2 blocks, the total time needed to compute 𝑅(𝑏) +sum is +𝑂((𝑛𝑑2/𝑏) log 𝑏). +2If 𝑑 is not divisible by 𝑏, pad dummy features that are constantly 0 in +the last group. +The block size hyperparameter 𝑏 controls the granular- +ity of the summary computation. In particular, when 𝑏 = 1, +the regularizer 𝑅(1) +sum(C) reduces 𝑅off(C) of Barlow Twins, +provided that 𝑞 = 2. On the other hand, when 𝑏 = 𝑑, we +recover 𝑅(𝑑) +sum(C) = 𝑅sum(C) in Eq. (6). Thus, this group- +ing formulation gives a generalization of Barlow Twins, +with parameter 𝑏 controlling the trade-off between compu- +tational efficiency and the degree of relaxed regularization. +Empirically, performance can be slightly improved by the +use of a feature group of moderate size, with no substantial +degradation observed in training time and memory usage; +see Sec. 5. +Note that the permutation and grouping of features are +compatible and can be combined. +4.5. Regularizer Based on Sums of Feature Covari- +ances +We used 𝑅sum to define a regularizer based on cross- +correlation, similarly to Barlow Twins. It can also be used +to define a VICReg-style regularizer based on covariance, +simply by replacing 𝑅off with 𝑅sum in Eq. (3), and passing +correlation matrices K(A) or K(B) instead of C(A, B) as +argument. Fast computation is also possible with FFT, and +the grouping version is also straightforward; these are de- +scribed in Supplementary Material. +4.6. Summary of the Proposed Models +The loss functions of the proposed models are summa- +rized below. For Barlow Twins–like cross-correlation regu- +larization, the loss function is +𝐿 = +∑︁ +𝑖 +(1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅(C(A, B)), +(14) +and for VICReg-like covariance regularization, we use +𝐿 = 𝛼 +𝑛 +∑︁ +𝑖 +∥a(𝑖) − b(𝑖) ∥2 +2 ++ 𝜇 +𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) ++ 𝜈 +𝑑 (𝑅(K(A)) + 𝑅(K(B))) , +(15) +where we set 𝑅 = 𝑅sum if feature grouping is not used, or +𝑅 = 𝑅(𝑏) +sum if feature grouping with block size 𝑏 is in effect. +Setting 𝑅 = 𝑅off recovers the original Barlow Twins and +VICReg. +5. Experiments +We empirically evaluate the effect of the proposed reg- +ularizers. To be precise, we train SSL models using the +loss functions of Eqs. (14) and (15) and compare their per- +formance with Barlow Twins and VICReg in downstream +tasks. +Training time and memory consumption are also +5 + +Table 1. Linear evaluation accuracy (%) on ImageNet-100 with +𝑑 = 2048. Bold numbers indicate the best performance within +each family (cross-correlation regularization, covariance regular- +ization, or other SSL models). † indicates the results quoted from +the solo-learn [23] GitHub repository as of December 28, 2022, +and those with ‡ are quoted from [29]. +Model +Top-1 +Top-5 +Barlow Twins† +80.16 +95.14 +Barlow Twins +80.12 +95.24 +proposed (BT-like; no group) +79.94 +94.76 +proposed (BT-like; 𝑏 = 128) +81.02 +95.24 +VICReg† +79.40 +95.02 +VICReg +79.30 +94.30 +proposed (VICReg-like; no group) +79.20 +94.96 +proposed (VICReg-like; 𝑏 = 128) +80.04 +94.98 +W-MSE† [10] +69.06 +91.22 +Zero-FCL‡ [29] +79.32 +94.94 +Zero-CL‡ [29] +79.26 +94.98 +NNCLR† [9] +80.16 +95.30 +BYOL† [12] +80.32 +94.94 +MoCo V3† [7] +80.36 +94.96 +evaluated. +For our models, feature permutation is per- +formed on every batch iteration, except for the ablation +study. +In the following, we briefly present the tasks and data +sets used in the experiments. See Appendix D for the com- +plete experimental setup including the values of the hyper- +parameters, as well as the results of additional experiments. +5.1. Tasks and Datasets +The models are pretrained with images in the ImageNet +dataset [8] or its subset, ImageNet-100 [22], depending on +the experiment. ResNet-50 [14] is used as the backbone for +ImageNet, and ResNet-18 for ImageNet-100. +To evaluate downstream semi-supervised learning per- +formance, we follow the standard linear evaluation proto- +col: After a backbone network is pretrained, we train a lin- +ear classifier on top of the frozen backbone using labeled +data from the ImageNet or ImageNet-100 training set. The +resulting classifier is then evaluated by the top-1 and top-5 +accuracies on the respective validation sets. +For transfer learning evaluation, we apply the pretrained +models to an object detection task on Pascal VOC07+12 +[11]. Following [1, 13, 28], we use the trainval splits of +VOC2007 and VOC2012 for training and the test split of +VOC2007 for testing. We fine-tune Faster R-CNN [19] with +R50-C4. The models are evaluated in three types of aver- +age precision: AP, AP50, and AP75 where AP𝑥 means that +IoU threshold is 𝑥 %. We report the average scores over five +trials. +Table 2. Linear evaluation accuracy (%) on ImageNet; highest +accuracy over 100 epochs of linear head training. 𝑑 = 8192 for +the proposed model, Barlow Twins, and VICReg. †: quoted from +the original papers of the respective methods; ‡: quoted from the +MoCo V3 GitHub repository. +Model +Epochs +Top-1 +Top-5 +Barlow Twins† +1000 +73.2 +91.0 +Barlow Twins +1000 +72.4 +90.6 +proposed (BT-like; no group) +1000 +73.0 +91.2 +proposed (BT-like; 𝑏 = 128) +1000 +73.2 +91.3 +VICReg† +1000 +73.2 +91.1 +VICReg +1000 +72.6 +90.9 +proposed (VICReg-like; no group) +1000 +72.8 +91.1 +W-MSE 4† [10] +400 +72.6 +— +Zero-CL† [29] +400 +72.6 +90.5 +SimCLR† [4] +1000 +69.3 +89.0 +NNCLR† [9] +1000 +75.4 +92.3 +BYOL† [12] +1000 +74.3 +91.6 +MoCo V3‡ [7] +1000 +74.6 +— +Table 3. The results of transfer learning on object detection on +VOC07+12. †: quoted from the original papers; ‡: quoted from +[13]. +Model +AP50 +AP +AP75 +Supervised‡ +81.3 +53.5 +58.8 +Barlow Twins† +82.6 +56.8 +63.4 +proposed (BT-like; no group) +82.5 +55.0 +61.1 +VICReg† +82.4 +— +— +proposed (VICReg-like; no group) +82.3 +56.1 +62.1 +5.2. Results and Discussion +Linear evaluation on ImageNet-100. +Tab. 1 shows the +results. We see that the accuracy of the proposed methods +is comparable with all the existing methods in the table, in- +cluding Barlow Twins and VICReg, with or without feature +grouping. +Linear evaluation on ImageNet. +Tab. 2 shows the re- +sults. Due to the cost of this large-scale experiment, we +do not evaluate feature grouping with the VICReg-like reg- +ularization. The proposed models perform slightly worse +than NNCLR, BYOL, and MoCo V3, but are comparable to +Barlow Twins and VICReg. +Transfer learning evaluation on Pascal VOC object de- +tection. +Tab. 3 show the results of transfer learning, where +the models are pretrained on ImageNet and evaluated on the +Pascal VOC object detection dataset. Again, the proposed +models show competitive performance with Barlow Twins +and VICReg. +6 + +Table 4. Linear evaluation accuracy (%) and the total training time on ImageNet with ResNet-50 backbone (𝑑 = 8192). The per GPU batch +size is 128. Barlow Twins∗ indicates the results quoted from the original paper [28, Figure 2]. +#GPUs (Batch size) +Model +Top-1 +Top-5 +Total training time +8 (1024) +Barlow Twins∗ +73.2 +91.0 +— +Barlow Twins +72.4 +90.7 +6 days 14 hours 0 minutes +proposed (Barlow Twins–like; no grouping) +73.0 +91.3 +5 days 14 hours 58 minutes +4 (512) +Barlow Twins∗ +— +— +— +Barlow Twins +72.1 +90.2 +12 days 19 hours 30 minutes +proposed (Barlow Twins–like; no grouping) +72.8 +91.2 +10 days 21 hours 6 minutes +Training time over 1000 epochs on ImageNet. +Tab. 4 +shows the training time over 1000 epochs on ImageNet. We +tested two situations: training using 8 GPUs and 4 GPUs. +In either situation, we set the per GPU batch size to 128, +which makes the effective batch size of 1024 for 8 GPUs +and 512 for 4 GPUs. As the table shows, the accuracy of the +proposed method is comparable with that of Barlow Twins, +with a noticeable reduction in training time.3 For more pre- +cise evaluation of the speed of the proposed method, see +the next experiment and the additional results in Appen- +dices E.3 and E.4. +Dimensionality of embeddings and computational cost. +Fig. 2 shows the elapsed time and the peak GPU mem- +ory allocation over ten epochs on ImageNet-100, with +varying dimensionality of projected embeddings: +𝑑 +∈ +{2048, 4096, 8192, 16384}. At 𝑑 = 2048, improvement is +only moderate both in terms of time and space. This is be- +cause the computation in the backbone accounts for most +of the computational cost when 𝑑 is small. However, as the +dimensionality is increased, loss computation takes more +time and space, resulting in large performance gaps: At +𝑑 = 8192, the proposed method (without grouping) is 2.2 +times as fast as Barlow Twins; and at 𝑑 = 16384, it is 4.0 +times as fast. In both 𝑑 = 8192 and 16384, memory con- +sumption is reduced by more than half. See Appendix E.3 +for the speed and memory consumption with the ResNet-50 +backbone. +Effectiveness of feature permutation. +Tab. 5 shows the +effect of feature permutation on ImageNet-100, at 𝑑 = +2048. Whether grouping is used or not, the accuracy drops +significantly without permutation, which suggests that fea- +ture permutation is essential for our regularizer to be effec- +tive. As shown in the column “Time” in Tab. 5, the cost of +permutation is negligible, even though it was performed as +3This experiment was carried out on a commercial cloud platform, +which limits a session to a maximum of three days. To finish training +Barlow Twins with 8 GPUs for 1000 epochs, we needed three sessions, +and the proposed model two. The timing reported in Tab. 4 is the total +run time of these sessions that includes the time for reinitialization at the +beginning of each session. +Table 5. The effect of feature permutation: top-1 and top-5 accu- +racy (%) and training time per 10 epochs (second) on ImageNet- +100. +(a) Cross-correlation regularization with 𝑅sum(C(A, B)) +Grouping +Permutation +Top-1 +Top-5 +Time +no +no +59.64 +85.20 +1646.2 +yes +79.94 +94.76 +1668.7 +𝑏 = 128 +no +73.58 +93.36 +1697.5 +yes +81.02 +95.24 +1709.6 +(b) Covariance regularization with 𝑅sum(K(A)), 𝑅sum(K(B)) +Grouping +Permutation +Top-1 +Top-5 +Time +no +no +57.42 +84.26 +1692.2 +yes +79.20 +94.96 +1718.0 +𝑏 = 128 +no +66.26 +89.68 +1802.1 +yes +80.04 +94.98 +1813.3 +frequently as every batch iteration. See also Appendix E.1 +for an additional result in which we quantitatively evaluate +the degree of decorrelation. +Impact of block size in feature grouping. +To evaluate +the effect of feature grouping on ImageNet-100, we fix the +dimension of embeddings at 𝑑 = 2048, and change block +size 𝑏 ∈ {2, 4, 8, ..., 2048}. The block size 𝑏 = 𝑑 = 2048 +corresponds to no feature grouping. Figure 3 shows the re- +sult. We see that unless 𝑏 is extremely small (i.e., 8 or less), +there is no significantly increase in the training time or GPU +memory usage. Setting 𝑏 to a moderate size, e.g., 𝑏 = 128, +improves the performance. +6. Conclusion +We have proposed non-contrastive SSL models with a +new decorrelating regularizer. By exploiting circular con- +volution and FFT, these models require only 𝑂(𝑛𝑑 log 𝑑) +time to calculate the loss for 𝑑-dimensional embeddings of +𝑛 samples, which improves on the 𝑂(𝑛𝑑2) time needed by +the existing models, Barlow Twins and VICReg. Memory +consumption is also reduced, giving more freedom to use +7 + +Cross-correlation regularization family +Covariance regularization family +Time +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +Training time (second) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +Training time (second) +VICReg +proposed (no grouping) +proposed (b = 128) +Space +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +25000 +Peak memory (MB) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +25000 +Peak memory (MB) +VICReg +proposed (no grouping) +proposed (b = 128) +Figure 2. Training time and memory usage on ImageNet-100 with ResNet-18. Upper row: elapsed time per 10 epochs. Lower row: peak +GPU allocated memory (MB). +2 +8 +32 +128 +512 +2048 +Blcok size +79.5 +80.0 +80.5 +81.0 +Accracy (%) +cross-correlation +covariance +(a) Top-1 accuracy +2 +8 +32 +128 +512 +2048 +Blcok size +2000 +4000 +6000 +8000 +10000 +12000 +14000 +Training time (second) +cross-correlation +covariance +(b) Elapsed training time (second) +2 +8 +32 +128 +512 +2048 +Blcok size +2000 +2500 +3000 +3500 +4000 +4500 +Peak memory (MB) +cross-correlation +covariance +(c) Peak GPU memory allocated (MB) +Figure 3. The impact of the block size on ImageNet-100. The 𝑥-axis indicates the block size 𝑏. +a larger batch size, which usually enables better models to +be learned. We also introduced a feature permutation tech- +nique to use with our regularizer and demonstrated its ef- +fectiveness in alleviating the problematic local minima that +can develop with our approach. In empirical evaluations, +training our models is indeed faster with smaller memory +footprints, whereas downstream performance is competi- +tive. Finally, our grouping version of the regularizer gener- +alizes Barlow Twins and VICReg, as they can be regarded +as special cases with specific hyperparameter values. +Acknowledgments +This work was partially supported by the New En- +ergy and Industrial Technology Development Organization +(NEDO) and JSPS Kakenhi Grant 19H04173. +References +[1] Adrien Bardes, Jean Ponce, and Yann LeCun. +VI- +CReg: +Variance-invariance-covariance regularization for +self-supervised learning. In ICLR, 2022. 1, 2, 6, 12, 13 +[2] Lukas Biewald. Experiment tracking with Weights and Bi- +ases. +Software available from https://www.wandb.com/, +2020. 11 +[3] A. Borsellino and T. Poggio. Convolution and correlation +algebras. Kybernetik, 13(2):113–122, 1973. 2 +[4] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Ge- +offrey Hinton. 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Barlow Twins: Self-supervised learning via redun- +dancy reduction. In ICML, pages 12310–12320, 2021. 1, 2, +6, 7, 12, 13 +[29] Shaofeng Zhang, Feng Zhu, Junchi Yan, Rui Zhao, and Xi- +aokang Yang. Zero-CL: Instance and feature decorrelation +for negative-free symmetric contrastive learning. In ICLR, +2022. 1, 2, 6 +9 + +A. Derivation of Eq. (8) +Let 𝑑-dimensional vectors x = [𝑥0 · · · 𝑥𝑑−1]T, y = [𝑦0 · · · 𝑦𝑑−1]T. We first show inv(x) ∗ y = �𝑑−1 +𝑗=0 𝑥 𝑗𝑦(𝑖+ 𝑗) mod 𝑑. +[inv(x) ∗ y]𝑖 = +𝑑−1 +∑︁ +𝑗=0 +[inv(x)] 𝑗 𝑦(𝑖−𝑗) mod 𝑑 += +𝑑−1 +∑︁ +𝑗=0 +𝑥(𝑑− 𝑗) mod 𝑑 𝑦(𝑖−𝑗) mod 𝑑 +∵ [inv(x)] 𝑗 = 𝑥(𝑑−𝑗) mod 𝑑 += +𝑑−1 +∑︁ +𝑗′=0 +𝑥 𝑗′ 𝑦(𝑖−(𝑑−𝑗′)) mod 𝑑 +∵ substituting 𝑗 ′ = (𝑑 − 𝑗) mod 𝑑 += +𝑑−1 +∑︁ +𝑗′=0 +𝑥 𝑗′ 𝑦(𝑖+ 𝑗′) mod 𝑑 +∵ (𝑎 − 𝑑) mod 𝑑 = 𝑎 mod 𝑑 += +𝑑−1 +∑︁ +𝑗=0 +� +xyT� +𝑗,(𝑖+ 𝑗) mod 𝑑 . +∵ renaming variable 𝑗 ′ → 𝑗 +In the literature (e.g., [18, 21]), inv(x) ∗ y is called the circular correlation of x and y, and the above equation is usually +presented as its definition. Setting x = a(𝑘) and y = b(𝑘), we obtain Eq. (8) in Sec. 4.2. +B. Regularizers Based on Sums of Feature Covariances +As mentioned in Sec. 4.5, if we substitute 𝑅sum for 𝑅off in the loss function of VICReg given in Eq. (3), we obtain +regularization based on the covariance matrices K(A), K(B) of individual views. Specifically, the resulting regularizer for +K(A) is: +𝑅sum(K(A)) = +𝑑−1 +∑︁ +𝑖=1 +∥[sumvec(K(A))]𝑖∥𝑞 +𝑞, +(16) +where hyperparameter 𝑞 ∈ {1, 2}. The regularizer for K(B) has the same form and is omitted. +Similarly to when 𝑅sum is applied to cross-correlation matrix C(A, B), 𝑅sum(K(A)) can be efficiently computed by FFT. +Here, we describe how it can be done. For brevity, assume that A is centered; i.e., all features have mean 0 in A. In this case, +its covariance matrix is K(A) = (1/(𝑛 − 1)) �𝑛 +𝑘=1 a(𝑘)a(𝑘)T. +Noting that F(inv(a(𝑘)) = F(a(𝑘)) and the convolution theorem F(x ∗ y) = F(x) ◦ F(y), we have +sumvec(K(A)) = +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +inv(a(𝑘)) ∗ a(𝑘) += +1 +𝑛 − 1 +𝑛 +∑︁ +𝑘=1 +inv(a(𝑘))∗a(𝑘) +������������������������������������������������������ +F−1� +F(a(𝑘)) ◦ F(a(𝑘)) +� += +1 +𝑛 − 1 F−1 +� 𝑛 +∑︁ +𝑘=1 +F(a(𝑘)) ◦ F(a(𝑘)) +� +, +(17) +The grouping version is also straightforward. Partitioning K(A) into block submatrices of size 𝑏 × 𝑏, i.e., K(𝐴) = [K𝑖 𝑗] +(𝑖, 𝑗 = 1, . . . , ⌈𝑑/𝑏⌉), where K𝑖 𝑗 ∈ R𝑏×𝑏, and applying 𝑅(𝑏) +sum defined in Eq. (13) to it, we obtain +𝑅(𝑏) +sum(K(A)) = +⌈𝑑/𝑏⌉ +∑︁ +𝑖=1 +𝑏−1 +∑︁ +ℓ=1 +∥[sumvec(K𝑖𝑖)]ℓ∥𝑞 +𝑞 + +⌈𝑑/𝑏⌉ +∑︁ +𝑖, 𝑗=1 +𝑖≠𝑗 +𝑏−1 +∑︁ +ℓ=0 +��[sumvec(K𝑖 𝑗)]ℓ +��𝑞 +𝑞 . +(18) +where block size 𝑏 is the hyperparameter that controls the granularity of the summary computation. When 𝑏 = 𝑑, i.e., the +block size is (𝑏/𝑑) × (𝑏/𝑑) = 1 × 1, the regularizer 𝑅(𝑏) +sum(K(A)) reduces to 𝑅off(K(A)) of VICReg. +10 + +Table 6. Complexity of loss computation. The space complexity includes 𝑂(𝑛𝑑) memory needed to store input embeddings. Grouping = +𝑏 indicates 𝑏 being the size of the group (i.e., block size). +Regularizer +Grouping +Time +Space +Barlow Twins +— +𝑂(𝑛𝑑2) +𝑂(𝑛𝑑 + 𝑑2) +VICReg +— +𝑂(𝑛𝑑2) +𝑂(𝑛𝑑 + 𝑑2) +proposed (𝑅sum) +no +𝑂(𝑛𝑑 log 𝑑) +𝑂(𝑛𝑑) +proposed (𝑅(𝑏) +sum) +𝑏 +𝑂((𝑛𝑑2/𝑏) log 𝑏) +𝑂(𝑛𝑑) +Table 7. Loss functions and regularizers in the proposed method (with and without grouping), Barlow Twins, and VICReg. Grouping = 𝑏 +indicates 𝑏 being the size of the group (i.e., block size). +(a) Cross-correlation–regularization with Barlow Twins–like loss function 𝐿 = � +𝑖 (1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅(C(A, B)) +Method +Grouping +Regularizer function 𝑅 +Barlow Twins +— +𝑅off +proposed +no +𝑅sum +proposed +𝑏 +𝑅(𝑏) +sum +(b) Covariance regularization with VICReg-like loss function 𝐿 = 𝛼 +𝑛 +� +𝑖 ∥a(𝑖) − b(𝑖) ∥2 +2 + 𝜇 +𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 +𝑑 (𝑅(K(A)) + 𝑅(K(B))) +Method +Grouping +Regularizer function 𝑅 +VICReg +— +𝑅off +proposed +no +𝑅sum +proposed +𝑏 +𝑅(𝑏) +sum +C. Summary of Computational Complexity +Tab. 6 summarizes the computational complexity of the regularizers discussed in this paper. As the table shows, the +proposed regularizers are faster and cheaper than the Barlow Twins and VICReg in terms of time and space complexity. +D. Detailed Experimental Setups +All the experiments in Sec. 5 were conducted on commercial Linux servers with CUDA v11.6.2 and cuDNN v8. We +implemented our model using PyTorch v1.12.0 [17] and solo-learn v1.0.5 [23], a library of self-supervised methods for +visual representation learning built on top of PyTorch and PyTorch Lightning4 v1.6.4. We also used NVIDIA DALI, a library +for data loading and pre-processing to accelerate deep learning applications5. For object detection, detectron2 [26] was used. +To manage the experiments, we used Weights & Biases, a machine learning platform for the tracking and visualization of +experiments [2]. As PyTorch v1.12.0 only provides experimental support for half precision FFT6, we trained every model +with 32-bit precision, including Barlow Twins and VICReg. +D.1. Compared Methods +In each comparison, the proposed method and the two baselines, Barlow Twins and VICReg, used an identical network +architecture, with the exception of the training loss. The loss functions of the baselines are given by Eqs. (1) and (3), which +are repeated below as Eqs. (19) and (20) for convenience. Let A = {a(𝑖)}𝑚 +𝑖=1, B = {b(𝑖)}𝑚 +𝑖=1 be the embeddings of the two +views, with 𝑖 = 1, . . . , 𝑚 indicating the original sample indices, K(A), K(B) ∈ R𝑑×𝑑 are their respective covariance matrices, +4https://www.pytorchlightning.ai/ +5https://github.com/NVIDIA/DALI +6https://pytorch.org/blog/pytorch-1.12-released/#beta-complex32-and-complex-convolutions-in-pytorch +11 + +and C(A, B) ∈ R𝑑×𝑑 is the cross-correlation matrices between 𝐴 and 𝐵. +𝐿BT = +𝑑−1 +∑︁ +𝑖=0 +(1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅off(C(A, B)), +(19) +𝐿VIC = 𝛼 +𝑛 +𝑚 +∑︁ +𝑖=1 +∥a(𝑖) − b(𝑖) ∥2 +2 + 𝜇 +𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 +𝑑 (𝑅off(K(A)) + 𝑅off(K(B))) , +(20) +where hyperparameters 𝛼, 𝜇, 𝜈, 𝜆 ≥ 0 determine the importance of individual terms, and the regularization functions are +given by +𝑅off(M) = +𝑑−1 +∑︁ +𝑖=0 +𝑑−1 +∑︁ +𝑗=0 +𝑗≠𝑖 +[M]2 +𝑖 𝑗, +𝑅var(M) = +𝑑−1 +∑︁ +𝑖=0 +max(0, 𝛾 − +√︁ +[M]𝑖𝑖). +For the proposed method, we replace all occurrences of 𝑅off in Eqs. (19) and (20) with either 𝑅sum (Eq. (6)) or 𝑅(𝑏) +sum (Eq. (13)) +depending on whether grouping is used. These functions are repeated below. +𝑅sum(M) = +𝑑−1 +∑︁ +𝑖=1 +∥[sumvec(M)]𝑖∥𝑞 +𝑞, +(21) +𝑅(𝑏) +sum(M) = +⌈𝑑/𝑏⌉ +∑︁ +𝑖=1 +𝑏−1 +∑︁ +ℓ=1 +∥[sumvec(M𝑖𝑖)]ℓ∥𝑞 +𝑞 + +⌈𝑑/𝑏⌉ +∑︁ +𝑖, 𝑗=1 +𝑖≠𝑗 +𝑏−1 +∑︁ +ℓ=0 +∥[sumvec(M𝑖 𝑗)]ℓ∥𝑞 +𝑞, +(22) +where M = [M𝑖 𝑗] is a block matrix with block elements M𝑖 𝑗 ∈ R𝑏×𝑏, 𝑖, 𝑗 = 1, . . . , ⌈𝑑/𝑏⌉. +Tab. 7 summarizes the regularizers and loss functions for Barlow Twins, VICReg, and the proposed models. +D.2. Data Augmentation +Following the Barlow Twins paper [28], we used non-symmetric parameters for Barlow Twins-like objectives. +For +VICReg-like objectives in ImageNet-100 experiments, we used the symmetrized augmentation pipeline reported in VICReg +paper [1, Appendix C.1]. Following a comment in the VICReg GitHub repository7, we used the non-symmetric augmentation +parameters (the same parameters as in Barlow Twins) for ImageNet experiments. +D.3. Hyperparameters +ImageNet-100. +For ImageNet-100 experiments, we followed the optimization procedure described in [23]. +We used +stochastic gradient descent (SGD) with the LARS optimizer [27] for model training for 400 epochs. We used linear warm-up +with cosine annealing decay for the learning rate scheduler. We set the batch size to 256 (per GPU batch size is 32). We +searched for the loss scaling value and the importance coefficients for the proposed regularizers (𝜈 in Eq. (3) and 𝜆 in Eq. (1)) +by grid search. In addition to these parameters, we further tuned the block size and 𝑞 in our regularizers (Eqs. (6) and (13)). +In linear evaluation, we optimized linear classifiers with SGD for 100 epochs. In training for ImageNet-100, we used the +hyperparameters provided by the solo-learn library. +Tab. 8 summarizes the hyperparameters for ImageNet-100 experiments. +ImageNet. +For ImageNet experiments, we followed the optimization procedure described in [1, 28]. We used SGD with +the LARS optimizer for 1000 epochs and linear warm-up with cosine annealing decay. We set the batch size to 1024 (per +GPU batch size is 128) and used a learning rate of 0.25 by reference to the Barlow Twins GitHub repository8. We searched +for 𝜆 and 𝑞 for the proposed regularizers by grid search. +7https://github.com/facebookresearch/vicreg/issues/3 +8https://github.com/facebookresearch/barlowtwins/issues/7 +12 + +Table 8. The hyperparameters for ImageNet-100 experiments that were used for training models. The hyperparameters for Barlow Twins +and VICReg were set to the values reported by [23]. For the proposed methods, we found values by grid search. +(a) Cross-correlation–regularization with Barlow Twins–like loss function +method +grouping +loss scale +𝑞 +𝜆 +Barlow Twins +— +0.1 +— +0.0051 +proposed +no +0.125 +2 +2−10 +proposed +𝑏 = 128 +0.125 +2 +2−10 +(b) Covariance regularization with VICReg-like loss function +method +grouping +loss scale +𝑞 +𝜈 +VICReg +— +— +— +1.0 +proposed +no +0.25 +1 +1.0 +proposed +𝑏 = 128 +0.25 +1 +2.0 +(c) All other hyperparameters were set to the values reported by [23]. +learning rate +weight decay +batch size +warmup epochs +𝛼 +𝜇 +0.3 +10−4 +256 +10 +25.0 +25.0 +(d) Hyperparameters for linear evaluation on ImageNet-100. +pretrained model +learning rate +steps for learning rate decay +weight decay +batch size +Barlow Twins / proposed (BT-like) +0.1 +[60, 80] +0 +256 +VICReg / proposed (VICReg-like) +0.3 +[60, 80] +0 +512 +In the linear evaluation on ImageNet, the linear head was trained for 100 epochs with SGD and cosine learning rate decay. +We tuned the learning rate and batch size for the proposed methods. For Barlow Twins and VICReg, the learning rate and +batch size are set to the values reported in the original papers [1,28]. +In object detection, we trained a Faster R-CNN with a C-4 backbone for 24K iterations. The backbone is initialized with +the pretrained ResNet-50 backbone. Following [1, 13, 28], we set the batch size to 16 (per GPU batch size is 2) and used a +step learning rate decay (divided by 10 at 18K and 22K iterations) with a linear warmup (slope of 0.333 for 1K iterations). +We tuned the learning rate and the region proposal network loss weight for the proposed methods. +Tab. 9 summarizes the hyperparameters for ImageNet experiments. +D.4. Evaluation of Training Time and Memory Consumption +To discuss empirical complexity, we measured the elapsed time and peak GPU memory allocation over ten epochs. We +conducted three trials and reported the average time and memory allocation. To avoid communication overhead between +GPUs, we evaluated the results of single GPU training (not distributed data parallel training). We used a PyTorch Lightning +profiler9 to record training time10 and a function in PyTorch11 to monitor memory occupied by tensors12. The batch size was +set to 32 for ImageNet-100 and 128 for ImageNet as in pretraining settings (per GPU batch size is 32 and 128). The number +of workers (the argument “num_workers” in the solo-learn library used for implementation) is set to 32 for ImageNet-100 +and 4 for ImageNet. +As mentioned in the footnote of Sec. 5.2, our experiment was performed on a commercial cloud platform that terminates a +session after three days. To finish training Barlow Twins and VICReg with 8 GPUs for 1000 epochs on ImageNet, we needed +three sessions, and the proposed model only two (see Tab. 4). The timing reported in Tab. 4 is the run time of these sessions +that includes the time for initialization at the beginning of each session. This initialization takes only 3–5 seconds at each +9https://pytorch-lightning.readthedocs.io/en/stable/api/pytorch_lightning.profilers.SimpleProfiler.html +10We use the value of the “Total time (s)” in the “run_training_epoch” line as the training time. +11https://pytorch.org/docs/stable/generated/torch.cuda.memory_summary.html +12We use the value of “Peak Usage” in the “Allocated memory” line as the peak GPU memory allocation. +13 + +Table 9. The hyperparameters for ImageNet experiments that were used for training models. +(a) Cross-correlation–regularization with Barlow Twins–like +loss function +method +grouping +loss scale +𝑞 +𝜆 +Barlow Twins +— +0.024 +— +0.0051 +proposed +no +0.024 +2 +2−11 +proposed +𝑏 = 128 +0.024 +2 +2−11 +(b) Covariance regularization with VICReg-like loss function +method +grouping +loss scale +𝑞 +𝛼 +𝜇 +𝜈 +VICReg +— +— +— +25.0 +25.0 +1.0 +proposed +no +— +1 +2.5 +2.5 +0.1 +(c) Hyperparameters for optimization. +learning rate +weight decay +batch size +warmup epochs +0.25 +10−6 +1024 +10 +(d) Hyperparameters for linear evaluation on ImageNet. +pretrained model +learning rate +learning rate decay +weight decay +batch size +Barlow Twins +0.3 +cosine decay +10−6 +256 +VICReg +0.02 +cosine decay +10−6 +256 +proposed (BT-like) +0.125 +cosine decay +10−6 +2048 +proposed (VICReg-like) +0.125 +cosine decay +10−6 +256 +(e) Hyperparameters for object detection on VOC07+12. +pretrained model +learning rate +region proposal network loss weigh +proposed (BT-like) +0.125 +0.03125 +proposed (VICReg-like) +0.125 +0.125 +session and does not affect the trend observed in the table. Note that in addition to this initalization, data copy takes about 15 +minutes at the start of a session, but this has already been excluded from the timing in Tab. 4. +D.5. Computational Resources +We used a cloud computing platform for the experiments. In the main experiments, we trained models with eight Nvidia +A100-SXM4 GPUs on this platform. We used a single Nvidia A100 GPU to evaluate empirical complexity, except where +noted. +D.6. License of the Assets +PyTorch has a BSD-style license13. Solo-learn has an MIT license14. PyTorch Lightning is licensed under the Apache +License 2.015. The ImageNet16 dataset is publicly available and frequently used as the benchmark datasets. The category list +of ImageNet-100 is also publicly available17. +13https://github.com/pytorch/pytorch/blob/master/LICENSE +14https://github.com/vturrisi/solo-learn/blob/main/LICENSE +15https://github.com/Lightning-AI/lightning/blob/master/LICENSE +16https://www.image-net.org/ +17https://github.com/HobbitLong/CMC/blob/master/imagenet100.txt +14 + +Table 10. The regularizers of Barlow Twins and VICReg (Eq. (2)) applied to the covariance/cross-correlation matrices of the proposed +models after they are pretrained. Each loss value is normalized to make it a mean over off-diagonal elements in the matrices. The column +“Diff” indicates the difference to the loss of the baselines. +(a) (Barlow Twins–like) cross-correlation regularization family +Normalized Barlow Twins +Model +Grouping +Permutation +loss (Eq. (23)) +Diff +Barlow Twins +— +— +0.005 +0 +proposed +no +no +0.564 +0.559 +yes +0.010 +0.005 +𝑏 = 128 +no +0.049 +0.044 +yes +0.009 +0.004 +(b) (VICReg-like) covariance regularization family +Normalized VICReg +Model +Grouping +Permutation +loss (Eq. (24)) +Diff +VICReg +— +— +0.002 +0 +proposed +no +no +1.999 +1.997 +yes +0.011 +0.009 +𝑏 = 128 +no +0.379 +0.377 +yes +0.007 +0.005 +E. Results of Additional Experiments +E.1. Effect of Feature Permutation on Decorrelation +In Sec. 4.3, we claimed that the weakness of our regularizer can be overcome with the feature permutation trick, which +we then verified in an ablation study (Tab. 5) by comparing the accuracy of the trained models with and without permutation. +Here, we further quantify the degree to which the embeddings learned by our method is decorrelated. To this end, we use the +values of the original regularizers of Barlow Twins’ and VICReg’s as the yardstick, computing these values with the learned +embeddings by our proposed models. Specifically, after the proposed models are trained, we compute the covariance matrices +K(A), K(B) (of two views) or cross-correlation matrix C(A, B) from the embeddings of the images in the training set. We +then compute the values of Barlow Twins’ and VICReg’s regularizers applied to the embeddings output by our proposed +models on the images in the training set: +𝑅off(C(A, B)) +𝑑(𝑑 − 1) +(normalized Barlow Twins loss), +(23) +𝑅off(K(A)) + 𝑅off(K(B)) +2𝑑(𝑑 − 1) +(normalized VICReg loss). +(24) +The normalization factor 𝑑(𝑑 − 1) is simply to make the resulting values the means over 𝑑(𝑑 − 1) off-diagonal elements +of K(A), K(B), and C(A, B). Since VICReg has two regularization terms (i.e., one each for K(A) and K(B)), its loss is +further divided by 2 in Eq. (24). The results on ImageNet-100 (𝑑=2048) are shown in Tab. 10. We can observe that feature +permutation promotes decorrelation from the point of view of the baseline loss functions. +E.2. Impact of Hyperparameter 𝑞 +We investigate the effect of hyperparameter 𝑞 in our regularizers (Eqs. (6) and (13)). Tab. 11 shows the results with +𝑞 ∈ {1, 2} on ImageNet-100 (𝑑 = 2048). The results indicate that 𝑞 = 1 works better than 𝑞 = 2 in VICReg-like covariance +regularizers. Conversely, 𝑞 = 2 works well in Barlow Twins–like cross-correlation regularizers. +E.3. Training Cost with ResNet-50 Backbone +Figure 4 shows the training cost with ResNet-50 on ImageNet. We were unable to run Barlow Twins and VICReg at +𝑑 = 16384, and also the grouped version of the proposed models with block size 𝑏 = 128. As in the results of ImageNet-100, +we can observe that the proposed regularizers are more efficient than the existing regularizers. +15 + +Table 11. The accuracy with 𝑞 ∈ {1, 2} on ImageNet-100. +Model +Grouping +𝑞 +Top-1 +Top-5 +proposed (Barlow Twins–like) +no +1 +75.94 +94.28 +2 +79.94 +94.76 +𝑏 = 128 +1 +76.44 +94.46 +2 +81.02 +95.24 +proposed (VICReg-like) +no +1 +79.20 +94.96 +2 +57.98 +84.56 +𝑏 = 128 +1 +80.04 +94.98 +2 +71.78 +92.54 +Cross-correlation regularization family +Covariance regularization family +Time +2048 +4096 +8192 +16384 +Dimensionality +0 +10000 +20000 +30000 +40000 +Training time (second) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +10000 +20000 +30000 +40000 +Training time (second) +VICReg +proposed (no grouping) +proposed (b = 128) +Space +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +25000 +30000 +Peak memory (MB) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +5000 +10000 +15000 +20000 +25000 +30000 +Peak memory (MB) +VICReg +proposed (no grouping) +proposed (b = 128) +Figure 4. Training time and memory usage on ImageNet with ResNet-50. Upper row: elapsed time per 10 epochs. Lower row: peak GPU +allocated memory (MB). +Cross-correlation regularization family +Covariance regularization family +Time +2048 +4096 +8192 +16384 +Dimensionality +0 +500 +1000 +1500 +2000 +2500 +3000 +Training time (second) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +500 +1000 +1500 +2000 +2500 +3000 +Training time (second) +VICReg +proposed (no grouping) +proposed (b = 128) +Figure 5. The elapsed DDP training time on ImageNet-100 with ResNet-18. +E.4. Training Time with Distributed Data Parallel +In Sec. 5 and Appendix E.3, we measured training time with a single GPU setting to avoid GPU communication latency. +Here we evaluate the timing of distributed data parallel (DDP) training, when multiple GPUs are available. Fig. 5 shows +16 + +Cross-correlation regularization family +Covariance regularization family +Time +2048 +4096 +8192 +16384 +Dimensionality +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +Training time (second) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2048 +4096 +8192 +16384 +Dimensionality +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +Training time (second) +VICReg +proposed (no grouping) +proposed (b = 128) +Figure 6. The elapsed DDP training time on ImageNet with ResNet-50. +Cross-correlation regularization family +Covariance regularization family +Time +2 nodes +4 nodes +0 +5000 +10000 +15000 +20000 +Training time (second) +Barlow Twins +proposed (no grouping) +proposed (b = 128) +2 nodes +4 nodes +0 +5000 +10000 +15000 +20000 +25000 +Training time (second) +VICReg +proposed (no grouping) +proposed (b = 128) +Figure 7. The elapsed multi-node DDP training time on ImageNet with ResNet-50 (𝑑 = 16384). +the elapsed time of DDP training for ten epochs on eight A100 GPUs. With DDP, the cost of communication between +GPUs emerges as an additional factor determining the total computational time, and hence the relative merit of our method +in reducing loss computation time is expected to diminish. Although this is certainly true, our method is still effective, +improving the computation time by a factor of more than 2.2 (= 945.5/428.7) for VICReg and 2.0 (= 740.6/366.4) for +Barlow Twins when 𝑑 = 8192 and a factor of 4.4 (= 2833.3/647.1) and 3.1 (= 1943.7/622.7) when 𝑑 = 16384. +Fig. 6 shows the elapsed time on ImageNet with ResNet-50. As in ImageNet-100 with ResNet-18 (Fig. 5), our method +improves speed, but the margin is smaller: 1.4 (= 6658/4859) for VICReg and 1.2 (= 5658/4869) for Barlow Twins when +𝑑 = 8192. +In the ImageNet experiments (Figs. 4 and 6), all models triggered an out-of-GPU-memory error when 𝑑 = 16384. To +evaluate the training time with 𝑑 = 16384, here we train the models using the multi-node DDP training. We evaluated two +situations: training using 2 nodes and 4 nodes. In either situation, we set the effective batch size to 1024. Figure 7 shows the +results. Barlow Twins and VICReg still ran out of memory under 2 nodes, but the proposed models (with or without grouping) +worked in this situation thanks to their efficient memory usage. With 4 nodes, all models were trained successfully, and the +proposed models are slightly faster than Barlow Twins and VICReg. However, in this setting, there is no point in training our +models using 4 nodes, when they are trainable on 2 nodes. And if we compare the speed of our models trained on 2 nodes +with Barlow Twins and VICReg (which failed to be trained on 2 nodes) on 4 nodes, the advantage of our models becomes +more pronounced. +F. Code +Listings 1 and 2 show Python-based implementations for covariance and cross-correlation regularizers (without feature +grouping). As explained in Sec. 4, the summary vectors can be efficiently calculated with FFT (see Listing 3). +In the computation of the proposed regularizers, we do not conduct collective operations, such as all-reduce and gather +functions. +17 + +1# Z1, Z2: projected image embeddings (n x d) +2# q: a hyperprameter for L_q^q norm +3 +4def xsum_regularizer(Z1, Z2, G, q): +5 +# pre-process: centering and normalization +6 +Z1 = batch_normalization(Z1) +7 +Z2 = batch_normalization(Z2) +8 +9 +# feature permutation +10 +idx = torch.randperm(Z1.shape[1]) +11 +Z1 = Z1[:, idx] +12 +Z2 = Z2[:, idx] +13 +14 +# summary vector +15 +sumvec = cal_sumvec(Z1, Z2, 0) / n +16 +17 +# loss for off-diagonal elements +18 +if q == 1: +19 +loss = torch.sum(sumvec[1:].abs()) +20 +elif q == 2: +21 +loss = torch.sum(sumvec[1:].pow(2)) +22 +23 +return loss +Listing 1. Computing Barlow Twins–style cross-correlation regularizer +1# Z: projected image embeddings ([n: batch size] x [d: embedding dimension]) +2# q: a hyperparameter for L_q^q norm +3 +4def covsum_regularizer(Z, blck_size, q): +5 +# pre-process: centering +6 +Z = +Z - Z.mean(dim=0) +7 +8 +# feature permutation +9 +idx = torch.randperm(Z.shape[1]) +10 +Z = Z[:, idx] +11 +12 +# summary vector +13 +sumvec = cal_sumvec(Z, Z, 0) / (n - 1) +14 +15 +# loss for off-diagonal elements +16 +if q == 1: +17 +loss = torch.sum(sumvec[1:].abs()) +18 +elif q == 2: +19 +loss = torch.sum(sumvec[1:].pow(2)) +20 +21 +return loss +Listing 2. Computing VICReg-style covariance regularizer +1def cal_sumvec(z1, z2, dim): +2 +fz1 = fft.rfft(z1) +3 +fz2 = fft.rfft(z2) +4 +fz1_conj = fz1.conj() +5 +fz_prod = fz1_conj * fz2 +6 +fc = fz_prod.sum(dim=dim) +7 +sumvec= fft.irfft(fc) +8 +return sumvec +Listing 3. Summary vector computation +18 + diff --git a/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/load_file.txt b/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f6bf31dfee619f9c574e617233fd3e9e52b2bd2 --- /dev/null +++ b/YtAzT4oBgHgl3EQfmv2c/content/tmp_files/load_file.txt @@ -0,0 +1,1078 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf,len=1077 +page_content='Learning Decorrelated Representations Efficiently Using Fast Fourier Transform Yutaro Shigeto* Masashi Shimbo∗ Yuya Yoshikawa Akikazu Takeuchi {shigeto,shimbo,yoshikawa,takeuchi}@stair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='center STAIR Lab, Chiba Institute of Technology, Narashino, Chiba, Japan Abstract Barlow Twins and VICReg are self-supervised represen- tation learning models that use regularizers to decorrelate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Although they work as well as conventional repre- sentation learning models, their training can be computa- tionally demanding if the dimension of projected represen- tations is high;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' as these regularizers are defined in terms of individual elements of a cross-correlation or covariance matrix, computing the loss for 𝑑-dimensional projected rep- resentations of 𝑛 samples takes 𝑂(𝑛𝑑2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In this paper, we propose a relaxed version of decorrelating regulariz- ers that can be computed in 𝑂(𝑛𝑑 log 𝑑) time by the fast Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We also propose an inexpensive trick to mitigate the undesirable local minima that develop with the relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Models learning representations using the pro- posed regularizers show comparable accuracy to existing models in downstream tasks, whereas the training requires less memory and is faster when 𝑑 is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Introduction Unsupervised or self-supervised learning (SSL) of repre- sentations [1,4–6,10,12,13,24,25,28,29] has become an in- tegral part of deep learning applications in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In SSL, a network is first pretrained on a “self-supervised” pretext task for which labeled data can be readily obtained on a large scale without human supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Lower layers of the pretrained network are then reused for downstream tasks, with the expectation that these layers produce generic representations of the input that are also useful in down- stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Most of the SSL models for visual representations em- ploy a multi-view, Siamese network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' First, an input image is converted by different random transforma- tions that do not alter its original semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' These two augmented examples are then fed to a neural network (or in some cases, two different networks) that consists of a Equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' backbone network cascased with a small projection net- work (usually a multi-layer perceptron), to produce “twin” projected embeddings of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Finally, the network weights are trained so that the two embeddings (also called “positive pair”) are similar, reflecting the fact that they rep- resent the same original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' After training, the projec- tion network is discarded, and only the backbone network is reused for downstream tasks as the encoder of input images that produces their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' One major issue in SSL is the representation collapse, or the presence of meaningless solutions such that all ex- amples are projected to a single vector embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Con- trastive approaches [4,5,13,24,25] eliminate such solutions by a loss term to repel embeddings produced from different original images, or “negative pairs.” Because considering all possible negative pairs is infeasible, negative sampling is usually performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Some recent work has explored non-contrastive SSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Among these, Barlow Twins [28] and VICReg [1] use loss functions to penarlize the features of embeddings with a small variance, which in turn discourages the col- lapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' They further introduce regularization terms to decor- relate features, by promoting off-diagonals of the correla- tion/covariance matrices to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Although these mod- els perform as well as contrastive models, their regularizers are computationally demanding for high-dimensional em- beddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' since the regulaizers are defined in terms of indi- vidual elements in covariance or cross-correlation matrices, they require 𝑂(𝑛𝑑2) time to compute, where 𝑛 is the num- ber of samples in a batch, and 𝑑 is the dimensionality of the projected embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is unfortunate, as improved performance is reported for both Barlow Twins and VICReg as 𝑑 is increased [1,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In this paper, we address the inefficiency of Barlow Twins and VICReg mentioned above with a re- laxed version of decorrelating regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This regulariza- tion does not require the calculation of the correlation/co- variance matrices explicitly and can be computed in time 𝑂(𝑛𝑑 log 𝑑) by means of circular convolution and the fast 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='01569v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='LG] 4 Jan 2023 Fourier transform (FFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Undesirable local minima that can develop with the use of the relaxed regularizer can be over- come by feature permutation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We show that the resulting method achieves competitive performance with Barlow Twins and VICReg on downstream tasks, with substantially less computation time when 𝑑 is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The proposed method also reduces memory consumption, which allows an increased batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We use the 0-based indexing for vector and ma- trix components unless stated otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' thus, for a vector x ∈ R𝑑, the component index ranges from 0 to 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We denote by [x]𝑖 the 𝑖th component of vector x, and [M]𝑖 𝑗 denotes the (𝑖, 𝑗)-element of matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For a complex vec- tor c, c denotes its componentwise complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For vectors x and y, x ◦ y denotes their componentwise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Related Work Contrastive SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Contrastive representation learning uses positive and negative pairs of augmented samples [4,5, 7,9,13,22,24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The commonly used InfoNCE loss [24] consists of an alignment term, which maximizes similarity between positive pairs, and a uniformity term, which mini- mizes similarity between negative pairs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' SimCLR [4] is one of the state-of-the-art methods in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' How- ever, to obtain good representations, SimCLR needs a large number of negative pairs [4], or, in other words, a large batch size 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This can be a computational bottleneck, as the loss computation of SimCLR takes 𝑂(𝑛2𝑑) time where 𝑑 is the dimensionality of the projected embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Non-contrastive SSL by Asymmetric Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Recently, researchers have started exploring the possibility of non-contrastive approaches to self-supervised represen- tation learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', those that do not use negative pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To overcome the representation collapse, models such as BYOL [12] and SimSiam [6] employ asymmetric network architectures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', by suppressing gradient updates and/or using the moving average of network parameters for one view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' These methods are heuristically motivated, as they do not explicitly penalize the collapse but work well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Non-contrastive SSL by Decorrelating Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins [28] was the first method to introduce a loss function that explicitly penalizes collapsed embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' It uses a regularizer based on cross-correlation matrices over two views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' VICReg [1] also introduced two regularizers, but they are defined in terms of covariance matrices of indi- vidual views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We review these methods in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 3 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Non-contrastive SSL by Whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Some authors [10, 15] used whitening to explicitly decorrelate features during training, without resorting to regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' [29] whitened both the feature and sample covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Because the whitening procedures used in these approaches require the computation of all the eigenvalues of the covariance ma- trices, a training epoch takes time 𝑂(𝑑3) (or 𝑂(𝑑3 + 𝑛3) with [29]), which can be problematic with large 𝑑 or 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Use of Convolution in Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Convolution is the basic building block of convolution neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' CNNs take (linear) convolution of input vec- tors with small learnable filter kernels to extract local fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In contrast, we use convolution to compute summary statistics of the covariance/cross-correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Al- though FFT reduces the asymptotic complexity of convolu- tion computation, it is seldom used with CNNs, because the size of kernels is typically too small to warrant speed-up by FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In other areas of machine learning, circular convolu- tion and its non-commutative analogue, circular correla- tion, have been applied to implement associative memory [3, 18, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The idea has recently been revived for knowl- edge graph embeddings [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' SSL Models Using Decorrelating Regulariz- ers In this section, Barlow Twins [28] and VICReg [1] are reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Given a batch of 𝑛 original training examples, these models apply two different transformations chosen randomly to each example and input the transformed ex- amples into a neural network to obtain twin embeddings (views) of the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Let a(𝑘), b(𝑘) ∈ R𝑑 denote the twin embeddings thus obtained for the 𝑘th example, and let A = {a(𝑘)}𝑛 𝑘=1, B = {b(𝑘)}𝑛 𝑘=1 be the sets of embeddings for individual views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The network is then trained to minimize the loss function specific to each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins optimizes the loss function defined in terms of the cross-correlation matrix C(A, B) ∈ R𝑑×𝑑 between views A and B: 𝐿BT = 𝑑−1 ∑︁ 𝑖=0 (1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅off(C(A, B)), (1) where hyperparameter 𝜆 ≥ 0 controls the strength of reg- ularization, and 𝑅off : R𝑑×𝑑 → R is a regularizer function defined as 𝑅off(M) = 𝑑−1 ∑︁ 𝑖=0 𝑑−1 ∑︁ 𝑗=0 𝑗≠𝑖 [M]2 𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (2) 2 The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (1) is minimized when the corre- sponding features of two views are fully correlated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', [C(A, B)]𝑖𝑖 = 1 for 𝑖 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This term can be efficiently computed in time 𝑂(𝑛𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizer 𝑅off(C(A, B)) in the second term is for feature decorrela- tion, as off-diagonal elements in C(A, B) are pushed toward zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This regularization term can be a computational bur- den when 𝑑 is large, as it takes 𝑂(𝑛𝑑2) time to compute, thanks to the 𝑑 × 𝑑 matrix C(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Let K(A), K(B) ∈ R𝑑×𝑑 be the covariance ma- trices of A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In VICReg, the loss function is defined as 𝐿VIC = 𝛼 𝑛 𝑛 ∑︁ 𝑘=1 ∥a(𝑘) − b(𝑘) ∥2 2 + 𝜇 𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 𝑑 (𝑅off(K(A)) + 𝑅off(K(B))) , (3) where 𝛼, 𝜇, 𝜈 ≥ 0 are hyperparameters to control the impor- tance of individual terms, and 𝑅var is the regularizer defined as 𝑅var(M) = 𝑑−1 ∑︁ 𝑖=0 max(0, 𝛾 − √︁ [M]𝑖𝑖), (4) with the target standard deviation 𝛾 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Function 𝑅off is the same regularizer as used in Barlow Twins (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (2)), but applied to K(A) and K(B) instead of C(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (3) brings two views of the same ex- ample closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizer 𝑅var penalizes collapsed embed- dings with zero variances, whereas 𝑅off promotes the diver- sity of features by encouraging the covariance of features to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The time complexity of caclculating 𝑅off(K(A)) and 𝑅off(K(B)) is 𝑂(𝑛𝑑2), just like 𝑅off(C(A, B)) in Barlow Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Proposed Method We propose a weaker but efficiently computable alterna- tive to the regularizer function 𝑅off used by Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In the following, we present our regularizer in terms of cross-correlation matrix C(A, B), similarly to Bar- low Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' However, if applied to K(A) and K(B) instead, this regulizer can be used as a drop-in replacement for 𝑅off in the VICReg’s loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizer Based on Sums of Cross-correla- tion Recall that the loss of Barlow Twins is based on the cross-correlation matrix C = C(A, B) of two views A = {a(𝑘)}𝑛 𝑘=1 and B = {b(𝑘)}𝑛 𝑘=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For brevity, assume that both 𝑐00 𝑐01 𝑐02 𝑐10 𝑐11 𝑐12 𝑐20 𝑐21 𝑐22 𝑣0 𝑣1 𝑣2 ���� ���� 𝑣0 = 𝑐00 + 𝑐11 + 𝑐22 𝑣1 = 𝑐01 + 𝑐12 + 𝑐20 𝑣2 = 𝑐02 + 𝑐10 + 𝑐21 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' A 3×3 cross-correlation matrix C = � 𝑐𝑖 𝑗 � (𝑖, 𝑗 = 0, 1, 2) and sumvec(C) = [𝑣0 𝑣1 𝑣2]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝐴 and 𝐵 are standardized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Then, the cross-correlation ma- trix is simply given by C = (1/(𝑛 − 1)) �𝑛 𝑘=1 a(𝑘)b(𝑘)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Our regularizer is defined in terms of a 𝑑-dimensional “summary” vector of the 𝑑 × 𝑑 cross-correlation matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This vector, denoted here by sumvec(C), is given compo- nentwise by [sumvec(C)]𝑖 = 𝑑−1 ∑︁ 𝑗=0 [C] 𝑗,(𝑖+ 𝑗) mod 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5) Note that the component indices are 0-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The 0th com- ponent [sumvec(C)]0 is the trace of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Each of the remain- ing 𝑑 − 1 components corresponds to a sum of 𝑑 different off-diagonal elements of C, with no single element appear- ing in two distinct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Thus, every element in C appears exactly once in the summations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The calculation of a summary vector for a 3 × 3 covariance matrix is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Now, we define a regularizer in terms of all but the 0th component of sumvec(C): 𝑅sum(C) = 𝑑−1 ∑︁ 𝑖=1 ∥[sumvec(C)]𝑖∥𝑞 𝑞, (6) where hyperparameter 𝑞 ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This function 𝑅sum can be used as a drop-in replacement for 𝑅off in Barlow Twins’ loss function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The 0th component [sumvec(C)]0 is excluded from the summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6), because it is equal to the sum of the diagonal elements of C, which are irrele- vant to feature decorrelation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' they do not appear in Barlow Twin’s regularizer 𝑅off(C), either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The regularizer 𝑅sum is weaker than 𝑅off in that it im- poses constraints on the components of the summary vec- tor, or the sums of 𝑑 elements of C, whereas 𝑅off constrains individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Indeed, 𝑅sum(C) is a lower bound of 𝑅off(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' However, as we discuss in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2, 𝑅sum allows faster computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Furthermore, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3, we provide a simple technique to mitigate the weakness of our regular- izer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Efficient Computation Computing sumvec(C) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5) requires cross- correlation matrix C(A, B), whose calculation incurs the 3 same computational inefficiency as the regularizer in Bar- low Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Fortunately, sumvec(C) can be calculated di- rectly from the vectors in A and B without their cross- correlation matrix calculated explicitly, by means of FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To this end, we first need the definitions of involution and circular convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The involution [20] (also called flipping [21]) inv(x) of a vector x ∈ R𝑑 is the vector obtained by reversing the or- der of its 1st (not the 0th) to (𝑑 − 1)st components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', [inv(x)]𝑖 = [x] (𝑑−𝑖) mod 𝑑 for 𝑖 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For vectors x, y ∈ R𝑑, their circular convolution x ∗ y is a 𝑑-dimensional vector with components [x ∗ y]𝑖 = 𝑑−1 ∑︁ 𝑗=0 � x yT� 𝑗,(𝑖−𝑗) mod 𝑑 , (7) Due to this definition, circular convolution is known as the “compressed outer product.” Now, for each twin representations a(𝑘) ∈ A and b(𝑘) ∈ B (𝑘 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑛), let us consider vector inv(a(𝑘)) ∗ b(𝑘) ∈ R𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 Noting the indices altered by involution, we see that this vector is given componentwise by � inv(a(𝑘)) ∗ b(𝑘)� 𝑖 = 𝑑−1 ∑︁ 𝑗=0 � a(𝑘)b(𝑘)T� 𝑗,(𝑖+ 𝑗) mod 𝑑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (8) Substituting C = (1/(𝑛 − 1)) �𝑛 𝑘=1 a(𝑘)b(𝑘)T into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (8), we have [sumvec(C)]𝑖 = 𝑑−1 ∑︁ 𝑗=0 C ������������������������������������������������ � 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 a(𝑘)b(𝑘)T � 𝑗,(𝑖+ 𝑗) mod 𝑑 = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 𝑑−1 ∑︁ 𝑗=0 � a(𝑘)b(𝑘)T� 𝑗,(𝑖+ 𝑗) mod 𝑑 = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 � inv(a(𝑘)) ∗ b(𝑘)� 𝑖 , (9) or, as a vector, sumvec(C) = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 inv(a(𝑘)) ∗ b(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (10) Now, let F and F−1 denote the (discrete) Fourier and the inverse Fourier transforms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Noting that F(inv(x)) = F(x) for any x ∈ R𝑑 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', [21], Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2) and using the celebrated convolution 1The vector inv(x) ∗ y is known as the circular (cross-)correlation of x and y [18, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We opt not to use this term in this paper to avoid confusion with the cross-correlation of random vectors, which is used in Barlow Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' theorem F(x ∗ y) = F(x) ◦ F(y), we have inv(a(𝑘)) ∗ b(𝑘) = F−1� F(a(𝑘)) ◦ F(b(𝑘)) � (11) where ◦ denotes componentwise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (11) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (10), we obtain sumvec(C) = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 inv(a(𝑘))∗b(𝑘) ������������������������������������������������������ F−1� F(a(𝑘)) ◦ F(b(𝑘)) � = 1 𝑛 − 1 F−1 � 𝑛 ∑︁ 𝑘=1 F(a(𝑘)) ◦ F(b(𝑘)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (12) Using this equation, we can calculate sumvec(C) directly from the representation vectors in A and B, bypassing the cumbersome calculation of C: First compute the Fourier transform of all reprenstations a(𝑘) and b(𝑘), and simply apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Since the (inverse) Fourier transform of a 𝑑-dimensional vector can be done in time 𝑂(𝑑 log 𝑑) by the FFT algorithm, and the computation of complex conjugates and component products, and the sum of the vectors of 𝑛 takes 𝑂(𝑛𝑑) time, the overall time to obtain sumvec(C) is 𝑂(𝑛𝑑 log 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The time to calculate 𝑅sum(C) is also 𝑂(𝑛𝑑 log 𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is a substantial improvement over Barlow Twin’s 𝑂(𝑛𝑑2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The space requirement is 𝑂(𝑛𝑑), which is optimal if we consider the same 𝑂(𝑛𝑑) space needed to store input vectors A and B as part of the space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In contrast, Barlow Twins needs extra 𝑂(𝑑2) space to store C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Feature Permutation to Mitigate Undesirable Local Minima As seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5), the components of sumvec(C) are the sums of 𝑑 elements in C, and the proposed regu- larizer 𝑅sum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6)) encourages these sums to be close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is weaker than Barlow Twins’ regularizer 𝑅off (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (2)), which pushes individual elements of C towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Indeed, 𝑅sum(C) can be close to zero even if individ- ual elements in C are not;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', the summands in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (5) can cancel each other, since they can be either positive or neg- ative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As a result, undesirable local minima develop in the parameter space, making our regularizer ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Here, we propose a simple trick to eliminate these local minima: Randomly permute feature indices during train- ing, so that the combination of features appearing in a sum in sumvec(C) changes frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To see why this works, consider minimizing 𝑅sum(C), regarding the elements of C as independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' It is easy to see that the minimum is attained by the solutions to a homogeneous system of lin- 4 ear equations: [sumvec(C)]𝑖 �������������������������������������� 𝑑−1 ∑︁ 𝑗=0 [C] 𝑗,(𝑖+ 𝑗) mod 𝑑 = 0, for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is an underdetermined system, with only 𝑑 − 1 equa- tions but with 𝑑(𝑑 − 1) unknowns, namely, [C] 𝑗ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑗, ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑑 − 1, 𝑗 ≠ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is why nontrivial solutions arise such that [C] 𝑗ℓ ≠ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', those in which summands with opposite signs cancel each other in an equation and which are undesirable for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Now, by repeatedly permuting the feature indices and minimizing the loss, we effectively introduce more and more equations to the system, since permutation can pro- duce different sets of linear equations over the unknowns, and these new constraints eventually make non-trivial solu- tions inadmissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For ease of implementation, we permute feature indices randomly during training, instead of generating all permu- tations systematically at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Note that the permuted fea- ture indices need not be identical across mini-batches, even within a single epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' indeed, in the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5, we use a different random permutation of features in every mini-batch in every epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Feature Grouping to Control the Degree of Re- laxation Instead of computing a summary vector for an entire cross-correlation matrix C, we can compute summaries at a more fine-grained level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Specifically, we partition 𝑑 fea- tures into groups of size 𝑏 each2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This partitioning induces in C a total of ⌈𝑑/𝑏⌉2 block submatrices of size 𝑏 × 𝑏, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', C = [C𝑖 𝑗] (𝑖, 𝑗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , ⌈𝑑/𝑏⌉) with submatrices C𝑖 𝑗 ∈ R𝑏×𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We then define the regularizer by 𝑅(𝑏) sum(C) = ⌈𝑑/𝑏⌉ ∑︁ 𝑖=1 𝑏−1 ∑︁ ℓ=1 ∥[sumvec(C𝑖𝑖)]ℓ∥𝑞 𝑞 + ⌈𝑑/𝑏⌉ ∑︁ 𝑖, 𝑗=1 𝑖≠ 𝑗 𝑏−1 ∑︁ ℓ=0 ��[sumvec(C𝑖 𝑗)]ℓ ��𝑞 𝑞 , (13) As before, sumvec(C𝑖 𝑗) can be computed without ex- plicitly computing C𝑖 𝑗 by means of involution, circu- lar convolution (of subvectors of embeddings), and the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Calculating a single sumvec(C𝑖 𝑗) takes 𝑂(𝑛𝑏 log 𝑏) time using FFT, and since there are ⌈𝑑/𝑏⌉2 blocks, the total time needed to compute 𝑅(𝑏) sum is 𝑂((𝑛𝑑2/𝑏) log 𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 2If 𝑑 is not divisible by 𝑏, pad dummy features that are constantly 0 in the last group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The block size hyperparameter 𝑏 controls the granular- ity of the summary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In particular, when 𝑏 = 1, the regularizer 𝑅(1) sum(C) reduces 𝑅off(C) of Barlow Twins, provided that 𝑞 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' On the other hand, when 𝑏 = 𝑑, we recover 𝑅(𝑑) sum(C) = 𝑅sum(C) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Thus, this group- ing formulation gives a generalization of Barlow Twins, with parameter 𝑏 controlling the trade-off between compu- tational efficiency and the degree of relaxed regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Empirically, performance can be slightly improved by the use of a feature group of moderate size, with no substantial degradation observed in training time and memory usage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Note that the permutation and grouping of features are compatible and can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizer Based on Sums of Feature Covari- ances We used 𝑅sum to define a regularizer based on cross- correlation, similarly to Barlow Twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' It can also be used to define a VICReg-style regularizer based on covariance, simply by replacing 𝑅off with 𝑅sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (3), and passing correlation matrices K(A) or K(B) instead of C(A, B) as argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Fast computation is also possible with FFT, and the grouping version is also straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' these are de- scribed in Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Summary of the Proposed Models The loss functions of the proposed models are summa- rized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For Barlow Twins–like cross-correlation regu- larization, the loss function is 𝐿 = ∑︁ 𝑖 (1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅(C(A, B)), (14) and for VICReg-like covariance regularization, we use 𝐿 = 𝛼 𝑛 ∑︁ 𝑖 ∥a(𝑖) − b(𝑖) ∥2 2 + 𝜇 𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 𝑑 (𝑅(K(A)) + 𝑅(K(B))) , (15) where we set 𝑅 = 𝑅sum if feature grouping is not used, or 𝑅 = 𝑅(𝑏) sum if feature grouping with block size 𝑏 is in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Setting 𝑅 = 𝑅off recovers the original Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Experiments We empirically evaluate the effect of the proposed reg- ularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To be precise, we train SSL models using the loss functions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (14) and (15) and compare their per- formance with Barlow Twins and VICReg in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Training time and memory consumption are also 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Linear evaluation accuracy (%) on ImageNet-100 with 𝑑 = 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Bold numbers indicate the best performance within each family (cross-correlation regularization, covariance regular- ization, or other SSL models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' † indicates the results quoted from the solo-learn [23] GitHub repository as of December 28, 2022, and those with ‡ are quoted from [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Model Top-1 Top-5 Barlow Twins† 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='14 Barlow Twins 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='12 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='24 proposed (BT-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='76 proposed (BT-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑏 = 128) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='24 VICReg† 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='40 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='02 VICReg 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30 proposed (VICReg-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='96 proposed (VICReg-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑏 = 128) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='04 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='98 W-MSE† [10] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='06 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='22 Zero-FCL‡ [29] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='32 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 Zero-CL‡ [29] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='26 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='98 NNCLR† [9] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30 BYOL† [12] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='32 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 MoCo V3† [7] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='36 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='96 evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For our models, feature permutation is per- formed on every batch iteration, except for the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In the following, we briefly present the tasks and data sets used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' See Appendix D for the com- plete experimental setup including the values of the hyper- parameters, as well as the results of additional experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tasks and Datasets The models are pretrained with images in the ImageNet dataset [8] or its subset, ImageNet-100 [22], depending on the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' ResNet-50 [14] is used as the backbone for ImageNet, and ResNet-18 for ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To evaluate downstream semi-supervised learning per- formance, we follow the standard linear evaluation proto- col: After a backbone network is pretrained, we train a lin- ear classifier on top of the frozen backbone using labeled data from the ImageNet or ImageNet-100 training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The resulting classifier is then evaluated by the top-1 and top-5 accuracies on the respective validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For transfer learning evaluation, we apply the pretrained models to an object detection task on Pascal VOC07+12 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Following [1, 13, 28], we use the trainval splits of VOC2007 and VOC2012 for training and the test split of VOC2007 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We fine-tune Faster R-CNN [19] with R50-C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The models are evaluated in three types of aver- age precision: AP, AP50, and AP75 where AP𝑥 means that IoU threshold is 𝑥 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We report the average scores over five trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Linear evaluation accuracy (%) on ImageNet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' highest accuracy over 100 epochs of linear head training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑑 = 8192 for the proposed model, Barlow Twins, and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' †: quoted from the original papers of the respective methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' ‡: quoted from the MoCo V3 GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Model Epochs Top-1 Top-5 Barlow Twins† 1000 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 Barlow Twins 1000 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 proposed (BT-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 1000 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 proposed (BT-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑏 = 128) 1000 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 VICReg† 1000 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 VICReg 1000 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='9 proposed (VICReg-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 1000 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 W-MSE 4† [10] 400 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 — Zero-CL† [29] 400 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 SimCLR† [4] 1000 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 NNCLR† [9] 1000 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 BYOL† [12] 1000 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 MoCo V3‡ [7] 1000 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 — Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The results of transfer learning on object detection on VOC07+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' †: quoted from the original papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' ‡: quoted from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Model AP50 AP AP75 Supervised‡ 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8 Barlow Twins† 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 proposed (BT-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 VICReg† 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 — — proposed (VICReg-like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no group) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Results and Discussion Linear evaluation on ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We see that the accuracy of the proposed methods is comparable with all the existing methods in the table, in- cluding Barlow Twins and VICReg, with or without feature grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Linear evaluation on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 2 shows the re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Due to the cost of this large-scale experiment, we do not evaluate feature grouping with the VICReg-like reg- ularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The proposed models perform slightly worse than NNCLR, BYOL, and MoCo V3, but are comparable to Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Transfer learning evaluation on Pascal VOC object de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 3 show the results of transfer learning, where the models are pretrained on ImageNet and evaluated on the Pascal VOC object detection dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Again, the proposed models show competitive performance with Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 6 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Linear evaluation accuracy (%) and the total training time on ImageNet with ResNet-50 backbone (𝑑 = 8192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The per GPU batch size is 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins∗ indicates the results quoted from the original paper [28, Figure 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' #GPUs (Batch size) Model Top-1 Top-5 Total training time 8 (1024) Barlow Twins∗ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 — Barlow Twins 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7 6 days 14 hours 0 minutes proposed (Barlow Twins–like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no grouping) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 5 days 14 hours 58 minutes 4 (512) Barlow Twins∗ — — — Barlow Twins 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 12 days 19 hours 30 minutes proposed (Barlow Twins–like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' no grouping) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 10 days 21 hours 6 minutes Training time over 1000 epochs on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4 shows the training time over 1000 epochs on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We tested two situations: training using 8 GPUs and 4 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In either situation, we set the per GPU batch size to 128, which makes the effective batch size of 1024 for 8 GPUs and 512 for 4 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As the table shows, the accuracy of the proposed method is comparable with that of Barlow Twins, with a noticeable reduction in training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 For more pre- cise evaluation of the speed of the proposed method, see the next experiment and the additional results in Appen- dices E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Dimensionality of embeddings and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 2 shows the elapsed time and the peak GPU mem- ory allocation over ten epochs on ImageNet-100, with varying dimensionality of projected embeddings: 𝑑 ∈ {2048, 4096, 8192, 16384}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' At 𝑑 = 2048, improvement is only moderate both in terms of time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This is be- cause the computation in the backbone accounts for most of the computational cost when 𝑑 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' However, as the dimensionality is increased, loss computation takes more time and space, resulting in large performance gaps: At 𝑑 = 8192, the proposed method (without grouping) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 times as fast as Barlow Twins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' and at 𝑑 = 16384, it is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 times as fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In both 𝑑 = 8192 and 16384, memory con- sumption is reduced by more than half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 for the speed and memory consumption with the ResNet-50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Effectiveness of feature permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5 shows the effect of feature permutation on ImageNet-100, at 𝑑 = 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Whether grouping is used or not, the accuracy drops significantly without permutation, which suggests that fea- ture permutation is essential for our regularizer to be effec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As shown in the column “Time” in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5, the cost of permutation is negligible, even though it was performed as 3This experiment was carried out on a commercial cloud platform, which limits a session to a maximum of three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To finish training Barlow Twins with 8 GPUs for 1000 epochs, we needed three sessions, and the proposed model two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The timing reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4 is the total run time of these sessions that includes the time for reinitialization at the beginning of each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The effect of feature permutation: top-1 and top-5 accu- racy (%) and training time per 10 epochs (second) on ImageNet- 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (a) Cross-correlation regularization with 𝑅sum(C(A, B)) Grouping Permutation Top-1 Top-5 Time no no 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='64 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20 1646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 yes 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='76 1668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7 𝑏 = 128 no 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='58 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='36 1697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 yes 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='24 1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6 (b) Covariance regularization with 𝑅sum(K(A)), 𝑅sum(K(B)) Grouping Permutation Top-1 Top-5 Time no no 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='42 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='26 1692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 yes 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='96 1718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 𝑏 = 128 no 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='26 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='68 1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 yes 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='04 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='98 1813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 frequently as every batch iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' See also Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 for an additional result in which we quantitatively evaluate the degree of decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Impact of block size in feature grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To evaluate the effect of feature grouping on ImageNet-100, we fix the dimension of embeddings at 𝑑 = 2048, and change block size 𝑏 ∈ {2, 4, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', 2048}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The block size 𝑏 = 𝑑 = 2048 corresponds to no feature grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Figure 3 shows the re- sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We see that unless 𝑏 is extremely small (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', 8 or less), there is no significantly increase in the training time or GPU memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Setting 𝑏 to a moderate size, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', 𝑏 = 128, improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Conclusion We have proposed non-contrastive SSL models with a new decorrelating regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' By exploiting circular con- volution and FFT, these models require only 𝑂(𝑛𝑑 log 𝑑) time to calculate the loss for 𝑑-dimensional embeddings of 𝑛 samples, which improves on the 𝑂(𝑛𝑑2) time needed by the existing models, Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Memory consumption is also reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' giving more freedom to use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Cross-correlation regularization family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Covariance regularization family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Training time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Barlow Twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Training time and memory usage on ImageNet-100 with ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Upper row: elapsed time per 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Lower row: peak GPU allocated memory (MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 2 8 32 128 512 2048 Blcok size 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 Accracy (%) cross-correlation covariance (a) Top-1 accuracy 2 8 32 128 512 2048 Blcok size 2000 4000 6000 8000 10000 12000 14000 Training time (second) cross-correlation covariance (b) Elapsed training time (second) 2 8 32 128 512 2048 Blcok size 2000 2500 3000 3500 4000 4500 Peak memory (MB) cross-correlation covariance (c) Peak GPU memory allocated (MB) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The impact of the block size on ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The 𝑥-axis indicates the block size 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' a larger batch size, which usually enables better models to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We also introduced a feature permutation tech- nique to use with our regularizer and demonstrated its ef- fectiveness in alleviating the problematic local minima that can develop with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In empirical evaluations, training our models is indeed faster with smaller memory footprints, whereas downstream performance is competi- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Finally, our grouping version of the regularizer gener- alizes Barlow Twins and VICReg, as they can be regarded as special cases with specific hyperparameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Acknowledgments This work was partially supported by the New En- ergy and Industrial Technology Development Organization (NEDO) and JSPS Kakenhi Grant 19H04173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' References [1] Adrien Bardes, Jean Ponce, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' VI- CReg: Variance-invariance-covariance regularization for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In ICLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1, 2, 6, 12, 13 [2] Lukas Biewald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Experiment tracking with Weights and Bi- ases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Software available from https://www.' metadata={'source': 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self-supervised mod- els are strong semi-supervised learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In NeurIPS, pages 22243–22255, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1, 2 8 [6] Xinlei Chen and Kaiming He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Exploring simple siamese rep- resentation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In CVPR, pages 15750–15758, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1, 2 [7] Xinlei Chen, Saining Xie, and Kaiming He.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='03888, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 12 [28] Jure Zbontar, Li Jing, Ishan Misra, Yann LeCun, and Stéhane Deny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins: Self-supervised learning via redun- dancy reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In ICML, pages 12310–12320, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1, 2, 6, 7, 12, 13 [29] Shaofeng Zhang, Feng Zhu, Junchi Yan, Rui Zhao, and Xi- aokang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Zero-CL: Instance and feature decorrelation for negative-free symmetric contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In ICLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 1, 2, 6 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (8) Let 𝑑-dimensional vectors x = [𝑥0 · · · 𝑥𝑑−1]T, y = [𝑦0 · · · 𝑦𝑑−1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We first show inv(x) ∗ y = �𝑑−1 𝑗=0 𝑥 𝑗𝑦(𝑖+ 𝑗) mod 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' [inv(x) ∗ y]𝑖 = 𝑑−1 ∑︁ 𝑗=0 [inv(x)] 𝑗 𝑦(𝑖−𝑗) mod 𝑑 = 𝑑−1 ∑︁ 𝑗=0 𝑥(𝑑− 𝑗) mod 𝑑 𝑦(𝑖−𝑗) mod 𝑑 ∵ [inv(x)] 𝑗 = 𝑥(𝑑−𝑗) mod 𝑑 = 𝑑−1 ∑︁ 𝑗′=0 𝑥 𝑗′ 𝑦(𝑖−(𝑑−𝑗′)) mod 𝑑 ∵ substituting 𝑗 ′ = (𝑑 − 𝑗) mod 𝑑 = 𝑑−1 ∑︁ 𝑗′=0 𝑥 𝑗′ 𝑦(𝑖+ 𝑗′) mod 𝑑 ∵ (𝑎 − 𝑑) mod 𝑑 = 𝑎 mod 𝑑 = 𝑑−1 ∑︁ 𝑗=0 � xyT� 𝑗,(𝑖+ 𝑗) mod 𝑑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' ∵ renaming variable 𝑗 ′ → 𝑗 In the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', [18, 21]), inv(x) ∗ y is called the circular correlation of x and y, and the above equation is usually presented as its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Setting x = a(𝑘) and y = b(𝑘), we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (8) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizers Based on Sums of Feature Covariances As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5, if we substitute 𝑅sum for 𝑅off in the loss function of VICReg given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (3), we obtain regularization based on the covariance matrices K(A), K(B) of individual views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Specifically, the resulting regularizer for K(A) is: 𝑅sum(K(A)) = 𝑑−1 ∑︁ 𝑖=1 ∥[sumvec(K(A))]𝑖∥𝑞 𝑞, (16) where hyperparameter 𝑞 ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The regularizer for K(B) has the same form and is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Similarly to when 𝑅sum is applied to cross-correlation matrix C(A, B), 𝑅sum(K(A)) can be efficiently computed by FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Here, we describe how it can be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For brevity, assume that A is centered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', all features have mean 0 in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In this case, its covariance matrix is K(A) = (1/(𝑛 − 1)) �𝑛 𝑘=1 a(𝑘)a(𝑘)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Noting that F(inv(a(𝑘)) = F(a(𝑘)) and the convolution theorem F(x ∗ y) = F(x) ◦ F(y), we have sumvec(K(A)) = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 inv(a(𝑘)) ∗ a(𝑘) = 1 𝑛 − 1 𝑛 ∑︁ 𝑘=1 inv(a(𝑘))∗a(𝑘) ������������������������������������������������������ F−1� F(a(𝑘)) ◦ F(a(𝑘)) � = 1 𝑛 − 1 F−1 � 𝑛 ∑︁ 𝑘=1 F(a(𝑘)) ◦ F(a(𝑘)) � , (17) The grouping version is also straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Partitioning K(A) into block submatrices of size 𝑏 × 𝑏, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', K(𝐴) = [K𝑖 𝑗] (𝑖, 𝑗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , ⌈𝑑/𝑏⌉), where K𝑖 𝑗 ∈ R𝑏×𝑏, and applying 𝑅(𝑏) sum defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (13) to it, we obtain 𝑅(𝑏) sum(K(A)) = ⌈𝑑/𝑏⌉ ∑︁ 𝑖=1 𝑏−1 ∑︁ ℓ=1 ∥[sumvec(K𝑖𝑖)]ℓ∥𝑞 𝑞 + ⌈𝑑/𝑏⌉ ∑︁ 𝑖, 𝑗=1 𝑖≠𝑗 𝑏−1 ∑︁ ℓ=0 ��[sumvec(K𝑖 𝑗)]ℓ ��𝑞 𝑞 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (18) where block size 𝑏 is the hyperparameter that controls the granularity of the summary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' When 𝑏 = 𝑑, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', the block size is (𝑏/𝑑) × (𝑏/𝑑) = 1 × 1, the regularizer 𝑅(𝑏) sum(K(A)) reduces to 𝑅off(K(A)) of VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 10 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Complexity of loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The space complexity includes 𝑂(𝑛𝑑) memory needed to store input embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Grouping = 𝑏 indicates 𝑏 being the size of the group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', block size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Regularizer Grouping Time Space Barlow Twins — 𝑂(𝑛𝑑2) 𝑂(𝑛𝑑 + 𝑑2) VICReg — 𝑂(𝑛𝑑2) 𝑂(𝑛𝑑 + 𝑑2) proposed (𝑅sum) no 𝑂(𝑛𝑑 log 𝑑) 𝑂(𝑛𝑑) proposed (𝑅(𝑏) sum) 𝑏 𝑂((𝑛𝑑2/𝑏) log 𝑏) 𝑂(𝑛𝑑) Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Loss functions and regularizers in the proposed method (with and without grouping), Barlow Twins, and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Grouping = 𝑏 indicates 𝑏 being the size of the group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', block size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (a) Cross-correlation–regularization with Barlow Twins–like loss function 𝐿 = � 𝑖 (1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅(C(A, B)) Method Grouping Regularizer function 𝑅 Barlow Twins — 𝑅off proposed no 𝑅sum proposed 𝑏 𝑅(𝑏) sum (b) Covariance regularization with VICReg-like loss function 𝐿 = 𝛼 𝑛 � 𝑖 ∥a(𝑖) − b(𝑖) ∥2 2 + 𝜇 𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 𝑑 (𝑅(K(A)) + 𝑅(K(B))) Method Grouping Regularizer function 𝑅 VICReg — 𝑅off proposed no 𝑅sum proposed 𝑏 𝑅(𝑏) sum C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Summary of Computational Complexity Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 6 summarizes the computational complexity of the regularizers discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As the table shows, the proposed regularizers are faster and cheaper than the Barlow Twins and VICReg in terms of time and space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Detailed Experimental Setups All the experiments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5 were conducted on commercial Linux servers with CUDA v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 and cuDNN v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We implemented our model using PyTorch v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 [17] and solo-learn v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 [23], a library of self-supervised methods for visual representation learning built on top of PyTorch and PyTorch Lightning4 v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We also used NVIDIA DALI, a library for data loading and pre-processing to accelerate deep learning applications5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For object detection, detectron2 [26] was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To manage the experiments, we used Weights & Biases, a machine learning platform for the tracking and visualization of experiments [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As PyTorch v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 only provides experimental support for half precision FFT6, we trained every model with 32-bit precision, including Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Compared Methods In each comparison, the proposed method and the two baselines, Barlow Twins and VICReg, used an identical network architecture, with the exception of the training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The loss functions of the baselines are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (1) and (3), which are repeated below as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (19) and (20) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Let A = {a(𝑖)}𝑚 𝑖=1, B = {b(𝑖)}𝑚 𝑖=1 be the embeddings of the two views, with 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , 𝑚 indicating the original sample indices, K(A), K(B) ∈ R𝑑×𝑑 are their respective covariance matrices, 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='pytorchlightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='ai/ 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/NVIDIA/DALI 6https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='org/blog/pytorch-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='12-released/#beta-complex32-and-complex-convolutions-in-pytorch 11 and C(A, B) ∈ R𝑑×𝑑 is the cross-correlation matrices between 𝐴 and 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝐿BT = 𝑑−1 ∑︁ 𝑖=0 (1 − [C(A, B)]𝑖𝑖)2 + 𝜆𝑅off(C(A, B)), (19) 𝐿VIC = 𝛼 𝑛 𝑚 ∑︁ 𝑖=1 ∥a(𝑖) − b(𝑖) ∥2 2 + 𝜇 𝑑 (𝑅var(K(A)) + 𝑅var(K(B))) + 𝜈 𝑑 (𝑅off(K(A)) + 𝑅off(K(B))) , (20) where hyperparameters 𝛼, 𝜇, 𝜈, 𝜆 ≥ 0 determine the importance of individual terms, and the regularization functions are given by 𝑅off(M) = 𝑑−1 ∑︁ 𝑖=0 𝑑−1 ∑︁ 𝑗=0 𝑗≠𝑖 [M]2 𝑖 𝑗, 𝑅var(M) = 𝑑−1 ∑︁ 𝑖=0 max(0, 𝛾 − √︁ [M]𝑖𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For the proposed method, we replace all occurrences of 𝑅off in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (19) and (20) with either 𝑅sum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6)) or 𝑅(𝑏) sum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (13)) depending on whether grouping is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' These functions are repeated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 𝑅sum(M) = 𝑑−1 ∑︁ 𝑖=1 ∥[sumvec(M)]𝑖∥𝑞 𝑞, (21) 𝑅(𝑏) sum(M) = ⌈𝑑/𝑏⌉ ∑︁ 𝑖=1 𝑏−1 ∑︁ ℓ=1 ∥[sumvec(M𝑖𝑖)]ℓ∥𝑞 𝑞 + ⌈𝑑/𝑏⌉ ∑︁ 𝑖, 𝑗=1 𝑖≠𝑗 𝑏−1 ∑︁ ℓ=0 ∥[sumvec(M𝑖 𝑗)]ℓ∥𝑞 𝑞, (22) where M = [M𝑖 𝑗] is a block matrix with block elements M𝑖 𝑗 ∈ R𝑏×𝑏, 𝑖, 𝑗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' , ⌈𝑑/𝑏⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 7 summarizes the regularizers and loss functions for Barlow Twins, VICReg, and the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Data Augmentation Following the Barlow Twins paper [28], we used non-symmetric parameters for Barlow Twins-like objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For VICReg-like objectives in ImageNet-100 experiments, we used the symmetrized augmentation pipeline reported in VICReg paper [1, Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Following a comment in the VICReg GitHub repository7, we used the non-symmetric augmentation parameters (the same parameters as in Barlow Twins) for ImageNet experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Hyperparameters ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For ImageNet-100 experiments, we followed the optimization procedure described in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We used stochastic gradient descent (SGD) with the LARS optimizer [27] for model training for 400 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We used linear warm-up with cosine annealing decay for the learning rate scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We set the batch size to 256 (per GPU batch size is 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We searched for the loss scaling value and the importance coefficients for the proposed regularizers (𝜈 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (3) and 𝜆 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (1)) by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In addition to these parameters, we further tuned the block size and 𝑞 in our regularizers (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6) and (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In linear evaluation, we optimized linear classifiers with SGD for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In training for ImageNet-100, we used the hyperparameters provided by the solo-learn library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 8 summarizes the hyperparameters for ImageNet-100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For ImageNet experiments, we followed the optimization procedure described in [1, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We used SGD with the LARS optimizer for 1000 epochs and linear warm-up with cosine annealing decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We set the batch size to 1024 (per GPU batch size is 128) and used a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25 by reference to the Barlow Twins GitHub repository8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We searched for 𝜆 and 𝑞 for the proposed regularizers by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/facebookresearch/vicreg/issues/3 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/facebookresearch/barlowtwins/issues/7 12 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The hyperparameters for ImageNet-100 experiments that were used for training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The hyperparameters for Barlow Twins and VICReg were set to the values reported by [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For the proposed methods, we found values by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (a) Cross-correlation–regularization with Barlow Twins–like loss function method grouping loss scale 𝑞 𝜆 Barlow Twins — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0051 proposed no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 2 2−10 proposed 𝑏 = 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 2 2−10 (b) Covariance regularization with VICReg-like loss function method grouping loss scale 𝑞 𝜈 VICReg — — — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 proposed no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 proposed 𝑏 = 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 (c) All other hyperparameters were set to the values reported by [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' learning rate weight decay batch size warmup epochs 𝛼 𝜇 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 10−4 256 10 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 (d) Hyperparameters for linear evaluation on ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' pretrained model learning rate steps for learning rate decay weight decay batch size Barlow Twins / proposed (BT-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 [60, 80] 0 256 VICReg / proposed (VICReg-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 [60, 80] 0 512 In the linear evaluation on ImageNet, the linear head was trained for 100 epochs with SGD and cosine learning rate decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We tuned the learning rate and batch size for the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' For Barlow Twins and VICReg, the learning rate and batch size are set to the values reported in the original papers [1,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In object detection, we trained a Faster R-CNN with a C-4 backbone for 24K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The backbone is initialized with the pretrained ResNet-50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Following [1, 13, 28], we set the batch size to 16 (per GPU batch size is 2) and used a step learning rate decay (divided by 10 at 18K and 22K iterations) with a linear warmup (slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='333 for 1K iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We tuned the learning rate and the region proposal network loss weight for the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 9 summarizes the hyperparameters for ImageNet experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Evaluation of Training Time and Memory Consumption To discuss empirical complexity, we measured the elapsed time and peak GPU memory allocation over ten epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We conducted three trials and reported the average time and memory allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To avoid communication overhead between GPUs, we evaluated the results of single GPU training (not distributed data parallel training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We used a PyTorch Lightning profiler9 to record training time10 and a function in PyTorch11 to monitor memory occupied by tensors12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The batch size was set to 32 for ImageNet-100 and 128 for ImageNet as in pretraining settings (per GPU batch size is 32 and 128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The number of workers (the argument “num_workers” in the solo-learn library used for implementation) is set to 32 for ImageNet-100 and 4 for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As mentioned in the footnote of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2, our experiment was performed on a commercial cloud platform that terminates a session after three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To finish training Barlow Twins and VICReg with 8 GPUs for 1000 epochs on ImageNet, we needed three sessions, and the proposed model only two (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The timing reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4 is the run time of these sessions that includes the time for initialization at the beginning of each session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' This initialization takes only 3–5 seconds at each 9https://pytorch-lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='io/en/stable/api/pytorch_lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='profilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='SimpleProfiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='html 10We use the value of the “Total time (s)” in the “run_training_epoch” line as the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 11https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='org/docs/stable/generated/torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='cuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='memory_summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='html 12We use the value of “Peak Usage” in the “Allocated memory” line as the peak GPU memory allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 13 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The hyperparameters for ImageNet experiments that were used for training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (a) Cross-correlation–regularization with Barlow Twins–like loss function method grouping loss scale 𝑞 𝜆 Barlow Twins — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='024 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0051 proposed no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='024 2 2−11 proposed 𝑏 = 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='024 2 2−11 (b) Covariance regularization with VICReg-like loss function method grouping loss scale 𝑞 𝛼 𝜇 𝜈 VICReg — — — 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 proposed no — 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 (c) Hyperparameters for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' learning rate weight decay batch size warmup epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25 10−6 1024 10 (d) Hyperparameters for linear evaluation on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' pretrained model learning rate learning rate decay weight decay batch size Barlow Twins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3 cosine decay 10−6 256 VICReg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='02 cosine decay 10−6 256 proposed (BT-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 cosine decay 10−6 2048 proposed (VICReg-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 cosine decay 10−6 256 (e) Hyperparameters for object detection on VOC07+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' pretrained model learning rate region proposal network loss weigh proposed (BT-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='03125 proposed (VICReg-like) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='125 session and does not affect the trend observed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Note that in addition to this initalization, data copy takes about 15 minutes at the start of a session, but this has already been excluded from the timing in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Computational Resources We used a cloud computing platform for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In the main experiments, we trained models with eight Nvidia A100-SXM4 GPUs on this platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We used a single Nvidia A100 GPU to evaluate empirical complexity, except where noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' License of the Assets PyTorch has a BSD-style license13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Solo-learn has an MIT license14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' PyTorch Lightning is licensed under the Apache License 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The ImageNet16 dataset is publicly available and frequently used as the benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The category list of ImageNet-100 is also publicly available17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/pytorch/pytorch/blob/master/LICENSE 14https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/vturrisi/solo-learn/blob/main/LICENSE 15https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/Lightning-AI/lightning/blob/master/LICENSE 16https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='image-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='org/ 17https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='com/HobbitLong/CMC/blob/master/imagenet100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='txt 14 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The regularizers of Barlow Twins and VICReg (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (2)) applied to the covariance/cross-correlation matrices of the proposed models after they are pretrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Each loss value is normalized to make it a mean over off-diagonal elements in the matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The column “Diff” indicates the difference to the loss of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (a) (Barlow Twins–like) cross-correlation regularization family Normalized Barlow Twins Model Grouping Permutation loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (23)) Diff Barlow Twins — — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='005 0 proposed no no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='559 yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='005 𝑏 = 128 no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='044 yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='004 (b) (VICReg-like) covariance regularization family Normalized VICReg Model Grouping Permutation loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (24)) Diff VICReg — — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='002 0 proposed no no 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='997 yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='009 𝑏 = 128 no 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='377 yes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='005 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Results of Additional Experiments E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Effect of Feature Permutation on Decorrelation In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3, we claimed that the weakness of our regularizer can be overcome with the feature permutation trick, which we then verified in an ablation study (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5) by comparing the accuracy of the trained models with and without permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Here, we further quantify the degree to which the embeddings learned by our method is decorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To this end, we use the values of the original regularizers of Barlow Twins’ and VICReg’s as the yardstick, computing these values with the learned embeddings by our proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Specifically, after the proposed models are trained, we compute the covariance matrices K(A), K(B) (of two views) or cross-correlation matrix C(A, B) from the embeddings of the images in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We then compute the values of Barlow Twins’ and VICReg’s regularizers applied to the embeddings output by our proposed models on the images in the training set: 𝑅off(C(A, B)) 𝑑(𝑑 − 1) (normalized Barlow Twins loss), (23) 𝑅off(K(A)) + 𝑅off(K(B)) 2𝑑(𝑑 − 1) (normalized VICReg loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (24) The normalization factor 𝑑(𝑑 − 1) is simply to make the resulting values the means over 𝑑(𝑑 − 1) off-diagonal elements of K(A), K(B), and C(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Since VICReg has two regularization terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=', one each for K(A) and K(B)), its loss is further divided by 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The results on ImageNet-100 (𝑑=2048) are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We can observe that feature permutation promotes decorrelation from the point of view of the baseline loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Impact of Hyperparameter 𝑞 We investigate the effect of hyperparameter 𝑞 in our regularizers (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' (6) and (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 11 shows the results with 𝑞 ∈ {1, 2} on ImageNet-100 (𝑑 = 2048).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The results indicate that 𝑞 = 1 works better than 𝑞 = 2 in VICReg-like covariance regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Conversely, 𝑞 = 2 works well in Barlow Twins–like cross-correlation regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Training Cost with ResNet-50 Backbone Figure 4 shows the training cost with ResNet-50 on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We were unable to run Barlow Twins and VICReg at 𝑑 = 16384, and also the grouped version of the proposed models with block size 𝑏 = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As in the results of ImageNet-100, we can observe that the proposed regularizers are more efficient than the existing regularizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 15 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The accuracy with 𝑞 ∈ {1, 2} on ImageNet-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Model Grouping 𝑞 Top-1 Top-5 proposed (Barlow Twins–like) no 1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='28 2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='94 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='76 𝑏 = 128 1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='44 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='46 2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='02 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='24 proposed (VICReg-like) no 1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='96 2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='98 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='56 𝑏 = 128 1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='04 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='98 2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='78 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Cross-correlation regularization family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Covariance regularization family ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Training time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Barlow Twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Training time (second) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='VICReg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Peak memory (MB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Barlow Twins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='8192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='16384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Dimensionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='25000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Peak memory (MB) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='VICReg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (no grouping) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='proposed (b = 128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Training time and memory usage on ImageNet with ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Upper row: elapsed time per 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Lower row: peak GPU allocated memory (MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Cross-correlation regularization family Covariance regularization family Time 2048 4096 8192 16384 Dimensionality 0 500 1000 1500 2000 2500 3000 Training time (second) Barlow Twins proposed (no grouping) proposed (b = 128) 2048 4096 8192 16384 Dimensionality 0 500 1000 1500 2000 2500 3000 Training time (second) VICReg proposed (no grouping) proposed (b = 128) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The elapsed DDP training time on ImageNet-100 with ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Training Time with Distributed Data Parallel In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5 and Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3, we measured training time with a single GPU setting to avoid GPU communication latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Here we evaluate the timing of distributed data parallel (DDP) training, when multiple GPUs are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5 shows 16 Cross-correlation regularization family Covariance regularization family Time 2048 4096 8192 16384 Dimensionality 0 1000 2000 3000 4000 5000 6000 7000 Training time (second) Barlow Twins proposed (no grouping) proposed (b = 128) 2048 4096 8192 16384 Dimensionality 0 1000 2000 3000 4000 5000 6000 7000 Training time (second) VICReg proposed (no grouping) proposed (b = 128) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The elapsed DDP training time on ImageNet with ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Cross-correlation regularization family Covariance regularization family Time 2 nodes 4 nodes 0 5000 10000 15000 20000 Training time (second) Barlow Twins proposed (no grouping) proposed (b = 128) 2 nodes 4 nodes 0 5000 10000 15000 20000 25000 Training time (second) VICReg proposed (no grouping) proposed (b = 128) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' The elapsed multi-node DDP training time on ImageNet with ResNet-50 (𝑑 = 16384).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' the elapsed time of DDP training for ten epochs on eight A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' With DDP, the cost of communication between GPUs emerges as an additional factor determining the total computational time, and hence the relative merit of our method in reducing loss computation time is expected to diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Although this is certainly true, our method is still effective, improving the computation time by a factor of more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 (= 945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='5/428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7) for VICReg and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='0 (= 740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='6/366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4) for Barlow Twins when 𝑑 = 8192 and a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 (= 2833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='3/647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='1 (= 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7/622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='7) when 𝑑 = 16384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 6 shows the elapsed time on ImageNet with ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As in ImageNet-100 with ResNet-18 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 5), our method improves speed, but the margin is smaller: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='4 (= 6658/4859) for VICReg and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='2 (= 5658/4869) for Barlow Twins when 𝑑 = 8192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In the ImageNet experiments (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4 and 6), all models triggered an out-of-GPU-memory error when 𝑑 = 16384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' To evaluate the training time with 𝑑 = 16384, here we train the models using the multi-node DDP training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' We evaluated two situations: training using 2 nodes and 4 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In either situation, we set the effective batch size to 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Figure 7 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Barlow Twins and VICReg still ran out of memory under 2 nodes, but the proposed models (with or without grouping) worked in this situation thanks to their efficient memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' With 4 nodes, all models were trained successfully, and the proposed models are slightly faster than Barlow Twins and VICReg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' However, in this setting, there is no point in training our models using 4 nodes, when they are trainable on 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' And if we compare the speed of our models trained on 2 nodes with Barlow Twins and VICReg (which failed to be trained on 2 nodes) on 4 nodes, the advantage of our models becomes more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Code Listings 1 and 2 show Python-based implementations for covariance and cross-correlation regularizers (without feature grouping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 4, the summary vectors can be efficiently calculated with FFT (see Listing 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' In the computation of the proposed regularizers, we do not conduct collective operations, such as all-reduce and gather functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' 17 1# Z1, Z2: projected image embeddings (n x d) 2# q: a hyperprameter for L_q^q norm 3 4def xsum_regularizer(Z1, Z2, G, q): 5 # pre-process: centering and normalization 6 Z1 = batch_normalization(Z1) 7 Z2 = batch_normalization(Z2) 8 9 # feature permutation 10 idx = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='randperm(Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='shape[1]) 11 Z1 = Z1[:, idx] 12 Z2 = Z2[:, idx] 13 14 # summary vector 15 sumvec = cal_sumvec(Z1, Z2, 0) / n 16 17 # loss for off-diagonal elements 18 if q == 1: 19 loss = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='sum(sumvec[1:].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='abs()) 20 elif q == 2: 21 loss = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='sum(sumvec[1:].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='pow(2)) 22 23 return loss Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Computing Barlow Twins–style cross-correlation regularizer 1# Z: projected image embeddings ([n: batch size] x [d: embedding dimension]) 2# q: a hyperparameter for L_q^q norm 3 4def covsum_regularizer(Z, blck_size, q): 5 # pre-process: centering 6 Z = Z - Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='mean(dim=0) 7 8 # feature permutation 9 idx = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='randperm(Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='shape[1]) 10 Z = Z[:, idx] 11 12 # summary vector 13 sumvec = cal_sumvec(Z, Z, 0) / (n - 1) 14 15 # loss for off-diagonal elements 16 if q == 1: 17 loss = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='sum(sumvec[1:].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='abs()) 18 elif q == 2: 19 loss = torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='sum(sumvec[1:].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='pow(2)) 20 21 return loss Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Computing VICReg-style covariance regularizer 1def cal_sumvec(z1, z2, dim): 2 fz1 = fft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='rfft(z1) 3 fz2 = fft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='rfft(z2) 4 fz1_conj = fz1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='conj() 5 fz_prod = fz1_conj * fz2 6 fc = fz_prod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='sum(dim=dim) 7 sumvec= fft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content='irfft(fc) 8 return sumvec Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} +page_content=' Summary vector computation 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAzT4oBgHgl3EQfmv2c/content/2301.01569v1.pdf'} diff --git a/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/2301.00066v1.pdf.txt b/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/2301.00066v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a2650f361256a994aeb788a7e8262190c956ac4 --- /dev/null +++ b/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/2301.00066v1.pdf.txt @@ -0,0 +1,686 @@ +MEMORY AUGMENTED LOOKUP DICTIONARY BASED LANGUAGE MODELING FOR +AUTOMATIC SPEECH RECOGNITION +Yukun Feng1,∗, Ming Tu2,†, Rui Xia2, Chuanzeng Huang2, Yuxuan Wang2 +Johns Hopkins University1 +Speech and Music Intelligence (SAMI), ByteDance2 +yfeng55@jhu.edu, {mingtu, rui.xia, huangchuanzeng, wangyuxuan.11}@bytedance.com +ABSTRACT +Recent studies have shown that using an external Language Model +(LM) benefits the end-to-end Automatic Speech Recognition (ASR). +However, predicting tokens that appear less frequently in the training +set is still quite challenging. The long-tail prediction problems have +been widely studied in many applications, but only been addressed +by a few studies for ASR and LMs. In this paper, we propose a new +memory augmented lookup dictionary based Transformer architec- +ture for LM. The newly introduced lookup dictionary incorporates +rich contextual information in training set, which is vital to correctly +predict long-tail tokens. With intensive experiments on Chinese and +English data sets, our proposed method is proved to outperform the +baseline Transformer LM by a great margin on both word/character +error rate and tail tokens error rate. This is achieved without impact +on the decoding efficiency. Overall, we demonstrate the effective- +ness of our proposed method in boosting the ASR decoding perfor- +mance, especially for long-tail tokens. +Index Terms— Automatic speech recognition, Language mod- +eling, rare words recognition, long-tail recognition. +1. INTRODUCTION +While a lot of studies have demonstrated the superiority of end-to- +end (E2E) Automatic Speech Recognition (ASR) systems [1,2] and +the effectiveness of incorporating Language Models (LM) into the +E2E ASR systems [3, 4], recognition and prediction of words that +appear only a few or zero times in training data are still big chal- +lenges, especially for E2E ASR systems which are optimized only +on text in the training data. +Some studies have addressed this long-tail problem for E2E +ASR [5–11]. The studies in [6, 9] resort to adding large corpora of +textual data or adjusting the distribution of head and tail words in +LM training to improve the modeling ability of tail words. In [7,8], +the authors propsoed to improve the prediction of tail words with +the help of large-scale pretrained LMs (BERT [12] variants) which +inevitably increases the decoding computational cost. Another line +of research modified the training loss or introduced extra loss terms +to regularize the ASR training, and results showed improved perfor- +mance on rare words [5, 11]. In [10], the authors tried to scale up +the embedding capacity of an RNN LM by incorporating N-gram +context embedding into the embedding layer without sacrificing +decoding efficiency. However, it ignored the frequency information +of words and N-grams and only reply on the input embedding layer +to learn enough contextual information to predict rare words. +∗Work done during internship at ByteDance +†Corresponding author +Since Transformer LMs have shown better performance than +RNN LMs for ASR [13], in this paper we extend the Transformer +LMs with a lookup dictionary that maps the current context to can- +didate tokens that have occurred during training. Inspired by [14] +which focuses on effective training of BERT, we initialize a dictio- +nary by aggregating the N-gram token IDs of the current token as +keys and utilize a multi-vector array as values to enable memoriza- +tion of rich context information. We now consider the dictionary’s +values as the memory of the corresponding N-gram context. Specif- +ically, the contextual memory is updated by the current token’s sub- +sequent token embedding in the training based on how often the +subsequent token occurs in the training corpus. For each key, the +frequency of the subsequent token decides how many vectors in the +corresponding multi-vector value will be updated. We then use an +attention module at the last layer of the transformer blocks to map +the dictionary memory to the contextualized embedding of the cur- +rent token, in which the current context will query the most relevant +vectors from the corresponding multi-vector memory. +We experimented on two Mandarin ASR data sets and improve +8.5% relatively of the Character Error Rate (CER) over the baseline +Transformer LM. Notably, our method show 13% and 12.5% rela- +tive CER reduction on the 1-gram and 2-gram tail tokens. Also, we +achieve Word Error Rate (WER) improvement on the two test sets +of LibriSpeech. The results indicate the success of our method on +improving not only the general ASR decoding but also the predic- +tion of tail tokens for both Mandarin and English. We did intensive +analysis to investigate the benefits of different aspects of our pro- +posed method. Overall, this paper makes several contributions as +the following: +1. We propose a new Transformer based LM for ASR equipped +with a lookup dictionary consisting of multi-vector memory +that builds bonding between the current context and to-be- +predicted candidate tokens. +2. We incorporate the N-gram context information and the to- +ken frequency in training data into the lookup dictionary to +improve the prediction of rare words. +3. Our proposed LM significantly outperforms the baseline LM +in both Mandarin and English ASR while keeping the same +inference efficiency. +2. PROPOSED APPROACH +In Figure 1, we show the diagram of the Transformer LM equipped +with our proposed memory augmented lookup dictionary. +Each +module will be introduced separately in the following subsections. +arXiv:2301.00066v1 [cs.CL] 30 Dec 2022 + +E0 +TM +TM +TM +TM +TM +TM +C0 +C1 +C2 +E1 +E2 +W0 +W1 +W2 +Attention +M +Di +Dn +[ ID(W0) + ID(W1) ] mod U = i +Output +M +Dictionary +Selection +Update +U +Fig. 1. Overview of the proposed memory augmented lookup dictionary based Transformer LM. W k represents the input tokens, Ek is for +the token embedding,”TM” means Transformer blocks in auto-regressive manner. Ck is the contextualized embedding corresponding to the +input tokens. Input and output embedding weights are shared in the LM. +2.1. Dictionary Construction and Indexing +We initialize the dictionary as DDD ∈ RU×M×demb, where demb is the +embedding size and U is the dictionary size. Instead of using one +vector as value for each key as in [14], we scale up the dictionary +size by introducing an extra hyper-parameter M to form a key-value +pair as (i, Di) where Di ∈ RM×demb. For the kth token in the input +sequence, the corresponding dictionary index i is mapped through a +modular hash with U, defined as: +i = ID(Tokenk) mod U +(1) +where ID() refers to the vocabulary id of the input token. We believe +with multiple vectors stored for each entry, much richer contextual +information could be memorized compared than the single vector +counterpart. To consider more context in dictionary indexing, we +also extend Equation 1 to N-gram case as in [10], where the dictio- +nary index i is calculated as follows +i = ( +k +� +n=k−N+1 +ID(Tokenn)) mod U +(2) +where N indicates the number of token IDs to aggregate. For ex- +ample, if N = 2, we sum up the IDs of the current token and its +previous token before the modulo operation. +To trade off information redundancy and memory capacity, col- +lision is allowed when doing hashing. U, N and M can be adjusted +appropriately. We assume this approach could utilize the dictionary +memory more efficiently. We will show the influence of changing +the three hyper-parameters to the performance in results part. +2.2. Dictionary Update +As the kth token is mapped to the dictionary memory Di through +Equation 1, each memory vector dm +i +is updated by the embedding +of the embedding of current token’s next token ek+1, which can be +formulated as: +� +dm +i = +� +dm +i ∗ α + ek+1 ∗ (1 − α) +if Xk+1 = 1 +dm +i +if Xk+1 = 0 +(3) +where α is a smoothing hyper-parameter that indicates how much in- +formation comes from ek+1, and we set it as 0.5 for all later experi- +ments. We define a Bernoulli Variable Xk+1 ∼ Bern(Pk+1), which +decides how many vectors will be updated in the matrix Di. Pk+1 +indicates the update ratio, computed by the normalized occurrence +of the k + 1th token in training data: +Pk+1 = +1 +log (Count of Tokenk+1). +(4) +In this case, embeddings of low frequency tokens are able to con- +tribute more to the corresponding memory compared to high fre- +quency tokens. Token frequencies are calcualted with training text +and corresponding text tokenizer. We will further discuss the effect +of the update ratio in Table 3. +2.3. Context Selection +We use an attention module to relate the output representation of +the current token to the corresponding dictionary memory. Attention +performs as a mapping function for the input query (Q) and key- +value (K-V) pairs, as +Attention(Q, K, V) = Softmax( QKT +√demb +)V. +(5) +As we share the input and output embedding weight in the Trans- +former LM, and the dictionary memory stores the candidate tokens’ +embedding during training, we assume the attention module could +help select useful information from the memory given the output +representation of the current token. For the kth token, we define the +contextualized token embedding from the Transformer model as ck, +and its corresponding dictionary memory is defined as Di, where i +is the hashing index from Eq 1 or 2. The new output representation +� +ck is computed as: +� +ck = Attention(ck, Di, Di), +(6) +which will then be used to calculate the output token distribution. + +Model +Input +Search +Overall +Tail-1 +Tail-2 +Overall +Tail-1 +Tail-2 +CER/SER +CER +CER +CER/SER +CER +CER +Conformer +6.19 / 44.91 +15.87 +13.82 +13.51 / 45.38 +22.38 +21.46 +with Language Model ++ LM +5.55 / 41.80 +13.46 +11.86 +9.07 / 30.11 +13.86 +13.78 ++ LM [10] +5.42 / 40.37 +12.50 +11.18 +8.91 / 29.79 +13.50 +13.44 ++ LM [14] +5.35 / 40.19 +12.48 +10.97 +9.02 / 30.23 +13.80 +13.68 ++ Ours +5.09 / 38.86 +11.46 +10.30 +8.29 / 27.60 +12.34 +12.13 ++8.3% ++14.9% ++13.2% ++8.6% ++11.0% ++12.0% ++ LML +4.83 / 37.29 +10.94 +9.67 +8.24 / 27.21 +12.26 +12.27 ++ OursL +4.73 / 36.90 +10.54 +9.54 +7.80 / 26.20 +11.49 +11.41 ++2.1% ++3.7% ++1.3% ++5.4% ++6.3% ++7.0% +Table 1. Evaluation of CER and SER on two internal Chinese ASR test sets. “Input” and “Search” refer to voice input and voice search +domain test sets respectively. L refers to the LM with 1024 embedding size. +2.4. Training and Inference +During training, the context selection operation was done before the +dictionary update for the reason that the update information in Eq +3 for the current input sentence will not affect the current context +selection. To stabilize training, we also disable the dictionary up- +date for the first 1000 training steps to warmup the newly initialized +embedding to a good distribution. During inference, the dictionary +update is also disabled to avoid any information leakage for auto- +regressive prediction. With the trained memory augmented Trans- +former LM, we apply shallow fusion to integrate the LM to ASR de- +coding with weight λsf. Also, Internal Language Model Estimation +(ILME) [15] is adopted to suppress the internal LM of the E2E ASR +and advocate the contribution of the external LM, which has been +proved to be quite effective especially there is domain mismatch be- +tween textual distribution of ASR and LM training data. The weight +of ILME is noted by λi. We also tried LM rescoring over the N-best +output of beam search, and the weight of rescoring is noted by λres. +3. EXPERIMENT +3.1. Datasets +We adopt the LibriSpeech [16] dataset to evaluate the ASR perfor- +mance in English. We use the standard 960 hours data for training +and the ”clean” and ”other” test sets for evaluation. The correspond- +ing LM is trained on PG-19 [17], an 11GB in-domain text corpus +consisting of books extracted from Project Gutenberg. To match the +averaged sentence length in LibriSpeech, we process the PG-19 into +a sentence-level corpus. We use the unigram tokenizer [18] with vo- +cabulary size of 5000 from ESPnet [19] for both ASR and LM train- +ing. Also, we evaluate our method on two internal Chinese video +datasets. We have a 10k hours annotated audio dataset for general +ASR training and two test sets: one is voice input domain (5103 +utterances) and the other is voice search domain (6424 utterances), +which are two different domains compared to the ASR training set. +As for the LM training, we have a 60GB text corpus for the voice +input domain and 2GB corpus for the voice search domain. We pro- +cess the Chinese text at the character level with a vocabulary size of +11k (with both Chinese characters and English subword tokens). +Besides evaluating the overall performance on the above men- +tioned test sets, we also assess the ASR metrics on tail tokens. Tail +tokens are defined as the tokens whose accumulated frequency in +the training corpus is lower than a threshold, which we set as 5%, +i.e. the frequency ratio of head and tail tokens is 95:5. Both 1-gram +(Tail-1) and 2-gram (Tail-2) tail tokens are extracted from test sets at +character-level for Chinese. For English teset ses, we only extracted +1-gram word-level tail tokens. +3.2. Experimental settings +We train both Chinese and English ASR models with a LAS [2] ar- +chitecture, for which we use a 12-layer Conformer [20] encoder and +6-layer Transformer decoder for Librispeech (as in ESPnet), and a +18-layer Conformer encoder and 4-layer Transformer decoder for +the 10k hours Chinese dataset. +For LibriSpeech, we configure the LM as a 16-layer Transformer +blocks with 1024 embedding size (as in ESPnet). It is trained on +PG-19 for sentence-level language modeling with a dropout rate of +0.3 and an effective token number of 524288 in each update. Adam +with betas of (0.9, 0.98), and weight decay of 0.01 is used for the +optimization with 10k warmup steps. For the proposed look-up dic- +tionary, we use 2-gram for dictionary hashing (as in Eq. 2); U is set +to 5k; M is set to 64. The LM for Chinese datasets consists of 4 +layer Transformer blocks with the embedding size of 384 and 1024 +for small and large configuration respectively 1. For look-up dictio- +nary, U is set to 10k and other hyper-parameters are the same with +the Librispeech settings. +For ASR inference in this paper, we set λsf={0.15, 0.4, 0.4}, +λres={0.0, 0.0, 0.1}, λi={0.0, 0.2, 0.2} for {”LibriSpeech”, ”Input, +”Search”} respectively, which give the best performance. A beam +size of 60 is used for the LibriSpeech and 10 for Chinese sets. We +use Word Error Rate (WER) as ASR metric for LibriSpeech test sets, +and Character Error Rate (CER) and Sentence Error Rate (SER) for +Chinese test sets. For both test sets, we also calculate the tail token +error rate by only counting errors on tail tokens and ignoring errors +on other tokens within the same testing utterances. +3.3. Results +We compare our model with the original Transformer LM, as well as +two other baselines: N-gram augmented embedding for LM training +in [10] and single-vector memory for BERT pretraining in [14]. In +Table 1, while the original LM helps the ASR model achieve lower +CER and SER, our method shows significant improvement over it +and the two baseline methods. We achieve 8.3% and 8.6% CER +improvement on the general ”Input” and ”Search” test sets over the +Transformer LM, and the CER improvement of tail tokens are even +1We avoided the LibriSpeech settings because of impact on the decoding +efficiency. + +Model +Clean +Other +Overall +Tail-1 +Overall +Tail-1 +Conformer +3.12% +11.92% +6.23% +24.52% ++ LM +3.08% +10.93% +5.81% +23.30% ++ Ours +3.01% +10.57% +5.73% +22.93% +Table 2. Evaluation of WER on the LibriSpeech test sets. +Fig. 2. Change of the overall CER (%) on the ”Search” test set with +different dictionary size and different N-gram settings. +higher: 13% on 1gram and 12.5% on 2gram tail tokens. As we in- +crease the hidden size of the LM from 384 to 1024, the performance +gain is not as much as the small LMs, but our method still outper- +forms the LM by 3.7% on overall CER and 4.6% on tail tokens CER. +In Table 2, our proposed method also shows consistent improvement +on the two LibriSpeech test sets. The improvement on tail word error +rate is more significantly compared to the overall WER improvement +as on Chinese test sets. +4. ANALYSIS +In this section, we analyze how the different hyper-parameters, in- +cluding dictionary size U, N in N-gram for hashing, memory up- +date ratio and memory size of each entry M, affect the performance. +All experiments are conducted on the Chinese ”Search” test set, and +the Transformer LM model with the proposed memory augmented +lookup dictionary has 4 layers and 384 hidden size. +In Figure 2, we show the change of the overall CER (y axis) +with the increase of dictionary size in different N-gram settings. It is +clear that for each N-gram setting, increasing the dictionary size will +boost the performance, and 2-gram achieves the best performance. +Since the degree of collision elevates with bigger U and N, larger +N means more collision; thus 4-gram performs even worse than 1 +gram case when the dictionary size is not large. Considering the +extra space taken by large dictionary size, wo choose the 2-gram +with 10k dictionary size. +In Table 3, we show the performance of both the overall CER +and CER on tail tokens under different memory update settings. The +ratios ”0.2”, ”0.5” and ”0.8” indicate we set a fixed probability for all +tokens when sampling the Bernoulli variable Xk+1 in Eq. 3 to up- +date the memory, while the ”freq” means we use Eq. 4 to decide the +Pk+1 for different tokens depending on their frequency in training +set. The results demonstrate that large update ratio tends to improve +the performance and our proposed frequency-based memory update +strategy marginally beat other options. +Figure 3 analyzes if a large memory size M would help the se- +lection and the overall performance. We use the Information Gain +(IG) which is computed by the difference in the attention entropy +(as in Eq. 5 and 6) between a randomly initialized dictionary and +a well-trained one. The entropy indicates how well the dictionary +Ratio +Overall +Tail-1 +Tail-2 +0.2 +8.37% +12.47% +12.37% +0.5 +8.35% +12.38% +12.24% +0.8 +8.31% +12.34% +12.16% +freq +8.29% +12.34% +12.13% +Table 3. Change of the overall and tail tokens CER under different +memory update options. +16 +32 +64 +128 +Memory Size +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Information Gain / Gradients +8.25 +8.30 +8.35 +8.40 +8.45 +CER +IG +Gradients +CER +Fig. 3. Overall CER, Gradients, and Information Gain (IG) change +on the ”Search” test set with the increase of memory size M +maps the information to the contextualized embedding of the cur- +rent token � +ck [21]. The results show the IG is highly correlated with +M. Besides, we adopt the Gradient Attribution test [22, 23] to ad- +dress the dictionary memory’s contribution further. It computes the +normalized gradient of the model variables to reflect its contribution +to the output prediction. It shows the gradients are also consistent +with the previous finding that a larger memory would receive more +gradients, indicating a greater contribution to the model prediction. +However, considering the small relative gain and high computational +cost when we increase the memory size from 64 to 128, we set the +memory size as 64 in our experiments. +Finally, we want to discuss how the proposed memory aug- +mented lookup dictionary will affect the model size and inference +speed. +During inference, compared to the baseline Transformer +LM, the additional computation of our method is only the dictionary +indexing (Eq. 2) and context selection (Eq. 6). For lookup dictio- +nary, the indexing operation requires O(1) time cost. The context +selection also performs as a constant time cost as O(M), where M +is the memory size of the dictionary. We evaluate the Real Time +Factor (RTF) on the ”Search” test set on a NVIDIA A100 GPU with +beam size batch size equals to 1. The RTF is 0.124 for ASR model +only, and 0.195 and 0.198 for the baseline Transformer LM and our +proposed LM, respectively. We notice that such additional opera- +tions almost do not affect the decoding speed in practice though the +model size increases by introducing the lookup dictionary. +5. CONCLUSIONS +In this paper, we propose a memory augmented lookup dictionary +based Transformer LM to improve the language modeling in ASR, +especially for long tail tokens. We have improved the baseline Trans- +former LM in terms of overall ASR metrics and the tail words er- +ror rate in both Chinese and English test sets. We also analyze our +method under different hyper-parameter settings. Overall, the results +prove the superiority of the method over the baseline Transformer +LM without sacrificing inference speed. Future work includes more +experiments on English data sets, especially in domain mismatch +condition. We are also interested in applying the method to general +language modeling tasks. + +8.50 +1gram +3gram +2gram +4gram +Overall CER +8.45 +8.40 +8.35 +8.30 +8.25 +2k +5k +10k +20k +Dictionary Size6. REFERENCES +[1] A. Graves, “Sequence transduction with recurrent neural net- +works,” arXiv preprint arXiv:1211.3711, 2012. +[2] W. Chan, N. Jaitly, Q. V. Le, and O. Vinyals, “Listen, attend +and spell,” arXiv preprint arXiv:1508.01211, 2015. +[3] S. Toshniwal, A. Kannan, C.-C. Chiu, Y. Wu, T. N. Sainath, +and K. Livescu, “A comparison of techniques for language +model integration in encoder-decoder speech recognition,” +in 2018 IEEE spoken language technology workshop (SLT). +IEEE, 2018, pp. 369–375. +[4] A. Kannan, Y. Wu, P. Nguyen, T. N. Sainath, Z. Chen, and +R. 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[Online]. +Available: https://arxiv.org/abs/2202.07856 + diff --git a/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/load_file.txt b/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a300018500572331e161de95692552ecc8d5176b --- /dev/null +++ b/_dAyT4oBgHgl3EQfRfZA/content/tmp_files/load_file.txt @@ -0,0 +1,538 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf,len=537 +page_content='MEMORY AUGMENTED LOOKUP DICTIONARY BASED LANGUAGE MODELING FOR AUTOMATIC SPEECH RECOGNITION Yukun Feng1,∗, Ming Tu2,†, Rui Xia2, Chuanzeng Huang2, Yuxuan Wang2 Johns Hopkins University1 Speech and Music Intelligence (SAMI), ByteDance2 yfeng55@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='edu, {mingtu, rui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='xia, huangchuanzeng, wangyuxuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='11}@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='com ABSTRACT Recent studies have shown that using an external Language Model (LM) benefits the end-to-end Automatic Speech Recognition (ASR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' However, predicting tokens that appear less frequently in the training set is still quite challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The long-tail prediction problems have been widely studied in many applications, but only been addressed by a few studies for ASR and LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In this paper, we propose a new memory augmented lookup dictionary based Transformer architec- ture for LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The newly introduced lookup dictionary incorporates rich contextual information in training set, which is vital to correctly predict long-tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' With intensive experiments on Chinese and English data sets, our proposed method is proved to outperform the baseline Transformer LM by a great margin on both word/character error rate and tail tokens error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' This is achieved without impact on the decoding efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Overall, we demonstrate the effective- ness of our proposed method in boosting the ASR decoding perfor- mance, especially for long-tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Index Terms— Automatic speech recognition, Language mod- eling, rare words recognition, long-tail recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' INTRODUCTION While a lot of studies have demonstrated the superiority of end-to- end (E2E) Automatic Speech Recognition (ASR) systems [1,2] and the effectiveness of incorporating Language Models (LM) into the E2E ASR systems [3, 4], recognition and prediction of words that appear only a few or zero times in training data are still big chal- lenges, especially for E2E ASR systems which are optimized only on text in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Some studies have addressed this long-tail problem for E2E ASR [5–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The studies in [6, 9] resort to adding large corpora of textual data or adjusting the distribution of head and tail words in LM training to improve the modeling ability of tail words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In [7,8], the authors propsoed to improve the prediction of tail words with the help of large-scale pretrained LMs (BERT [12] variants) which inevitably increases the decoding computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Another line of research modified the training loss or introduced extra loss terms to regularize the ASR training, and results showed improved perfor- mance on rare words [5, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In [10], the authors tried to scale up the embedding capacity of an RNN LM by incorporating N-gram context embedding into the embedding layer without sacrificing decoding efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' However, it ignored the frequency information of words and N-grams and only reply on the input embedding layer to learn enough contextual information to predict rare words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' ∗Work done during internship at ByteDance †Corresponding author Since Transformer LMs have shown better performance than RNN LMs for ASR [13], in this paper we extend the Transformer LMs with a lookup dictionary that maps the current context to can- didate tokens that have occurred during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Inspired by [14] which focuses on effective training of BERT, we initialize a dictio- nary by aggregating the N-gram token IDs of the current token as keys and utilize a multi-vector array as values to enable memoriza- tion of rich context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We now consider the dictionary’s values as the memory of the corresponding N-gram context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Specif- ically, the contextual memory is updated by the current token’s sub- sequent token embedding in the training based on how often the subsequent token occurs in the training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For each key, the frequency of the subsequent token decides how many vectors in the corresponding multi-vector value will be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We then use an attention module at the last layer of the transformer blocks to map the dictionary memory to the contextualized embedding of the cur- rent token, in which the current context will query the most relevant vectors from the corresponding multi-vector memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We experimented on two Mandarin ASR data sets and improve 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5% relatively of the Character Error Rate (CER) over the baseline Transformer LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Notably, our method show 13% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5% rela- tive CER reduction on the 1-gram and 2-gram tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Also, we achieve Word Error Rate (WER) improvement on the two test sets of LibriSpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The results indicate the success of our method on improving not only the general ASR decoding but also the predic- tion of tail tokens for both Mandarin and English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We did intensive analysis to investigate the benefits of different aspects of our pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Overall, this paper makes several contributions as the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We propose a new Transformer based LM for ASR equipped with a lookup dictionary consisting of multi-vector memory that builds bonding between the current context and to-be- predicted candidate tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We incorporate the N-gram context information and the to- ken frequency in training data into the lookup dictionary to improve the prediction of rare words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Our proposed LM significantly outperforms the baseline LM in both Mandarin and English ASR while keeping the same inference efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' PROPOSED APPROACH In Figure 1, we show the diagram of the Transformer LM equipped with our proposed memory augmented lookup dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Each module will be introduced separately in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='00066v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='CL] 30 Dec 2022 E0 TM TM TM TM TM TM C0 C1 C2 E1 E2 W0 W1 W2 Attention M Di Dn [ ID(W0) + ID(W1) ] mod U = i Output M Dictionary Selection Update U Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Overview of the proposed memory augmented lookup dictionary based Transformer LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' W k represents the input tokens, Ek is for the token embedding,”TM” means Transformer blocks in auto-regressive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Ck is the contextualized embedding corresponding to the input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Input and output embedding weights are shared in the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Dictionary Construction and Indexing We initialize the dictionary as DDD ∈ RU×M×demb, where demb is the embedding size and U is the dictionary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Instead of using one vector as value for each key as in [14], we scale up the dictionary size by introducing an extra hyper-parameter M to form a key-value pair as (i, Di) where Di ∈ RM×demb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For the kth token in the input sequence, the corresponding dictionary index i is mapped through a modular hash with U, defined as: i = ID(Tokenk) mod U (1) where ID() refers to the vocabulary id of the input token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We believe with multiple vectors stored for each entry, much richer contextual information could be memorized compared than the single vector counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' To consider more context in dictionary indexing, we also extend Equation 1 to N-gram case as in [10], where the dictio- nary index i is calculated as follows i = ( k � n=k−N+1 ID(Tokenn)) mod U (2) where N indicates the number of token IDs to aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For ex- ample, if N = 2, we sum up the IDs of the current token and its previous token before the modulo operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' To trade off information redundancy and memory capacity, col- lision is allowed when doing hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' U, N and M can be adjusted appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We assume this approach could utilize the dictionary memory more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We will show the influence of changing the three hyper-parameters to the performance in results part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Dictionary Update As the kth token is mapped to the dictionary memory Di through Equation 1, each memory vector dm i is updated by the embedding of the embedding of current token’s next token ek+1, which can be formulated as: � dm i = � dm i ∗ α + ek+1 ∗ (1 − α) if Xk+1 = 1 dm i if Xk+1 = 0 (3) where α is a smoothing hyper-parameter that indicates how much in- formation comes from ek+1, and we set it as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5 for all later experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We define a Bernoulli Variable Xk+1 ∼ Bern(Pk+1), which decides how many vectors will be updated in the matrix Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Pk+1 indicates the update ratio, computed by the normalized occurrence of the k + 1th token in training data: Pk+1 = 1 log (Count of Tokenk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' (4) In this case, embeddings of low frequency tokens are able to con- tribute more to the corresponding memory compared to high fre- quency tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Token frequencies are calcualted with training text and corresponding text tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We will further discuss the effect of the update ratio in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Context Selection We use an attention module to relate the output representation of the current token to the corresponding dictionary memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Attention performs as a mapping function for the input query (Q) and key- value (K-V) pairs, as Attention(Q, K, V) = Softmax( QKT √demb )V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' (5) As we share the input and output embedding weight in the Trans- former LM, and the dictionary memory stores the candidate tokens’ embedding during training, we assume the attention module could help select useful information from the memory given the output representation of the current token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For the kth token, we define the contextualized token embedding from the Transformer model as ck, and its corresponding dictionary memory is defined as Di, where i is the hashing index from Eq 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The new output representation � ck is computed as: � ck = Attention(ck, Di, Di), (6) which will then be used to calculate the output token distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Model Input Search Overall Tail-1 Tail-2 Overall Tail-1 Tail-2 CER/SER CER CER CER/SER CER CER Conformer 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='19 / 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='91 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='87 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='82 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='51 / 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='38 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='38 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='46 with Language Model + LM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='55 / 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='80 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='86 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='07 / 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='11 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='86 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='78 + LM [10] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='42 / 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='37 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='91 / 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='79 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='44 + LM [14] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='35 / 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='19 12.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='41 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='1% +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='7% +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3% +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='4% +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3% +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0% Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Evaluation of CER and SER on two internal Chinese ASR test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' “Input” and “Search” refer to voice input and voice search domain test sets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' L refers to the LM with 1024 embedding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Training and Inference During training, the context selection operation was done before the dictionary update for the reason that the update information in Eq 3 for the current input sentence will not affect the current context selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' To stabilize training, we also disable the dictionary up- date for the first 1000 training steps to warmup the newly initialized embedding to a good distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' During inference, the dictionary update is also disabled to avoid any information leakage for auto- regressive prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' With the trained memory augmented Trans- former LM, we apply shallow fusion to integrate the LM to ASR de- coding with weight λsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Also, Internal Language Model Estimation (ILME) [15] is adopted to suppress the internal LM of the E2E ASR and advocate the contribution of the external LM, which has been proved to be quite effective especially there is domain mismatch be- tween textual distribution of ASR and LM training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The weight of ILME is noted by λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We also tried LM rescoring over the N-best output of beam search, and the weight of rescoring is noted by λres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' EXPERIMENT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Datasets We adopt the LibriSpeech [16] dataset to evaluate the ASR perfor- mance in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We use the standard 960 hours data for training and the ”clean” and ”other” test sets for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The correspond- ing LM is trained on PG-19 [17], an 11GB in-domain text corpus consisting of books extracted from Project Gutenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' To match the averaged sentence length in LibriSpeech, we process the PG-19 into a sentence-level corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We use the unigram tokenizer [18] with vo- cabulary size of 5000 from ESPnet [19] for both ASR and LM train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Also, we evaluate our method on two internal Chinese video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We have a 10k hours annotated audio dataset for general ASR training and two test sets: one is voice input domain (5103 utterances) and the other is voice search domain (6424 utterances), which are two different domains compared to the ASR training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' As for the LM training, we have a 60GB text corpus for the voice input domain and 2GB corpus for the voice search domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We pro- cess the Chinese text at the character level with a vocabulary size of 11k (with both Chinese characters and English subword tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Besides evaluating the overall performance on the above men- tioned test sets, we also assess the ASR metrics on tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Tail tokens are defined as the tokens whose accumulated frequency in the training corpus is lower than a threshold, which we set as 5%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' the frequency ratio of head and tail tokens is 95:5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Both 1-gram (Tail-1) and 2-gram (Tail-2) tail tokens are extracted from test sets at character-level for Chinese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For English teset ses, we only extracted 1-gram word-level tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Experimental settings We train both Chinese and English ASR models with a LAS [2] ar- chitecture, for which we use a 12-layer Conformer [20] encoder and 6-layer Transformer decoder for Librispeech (as in ESPnet), and a 18-layer Conformer encoder and 4-layer Transformer decoder for the 10k hours Chinese dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For LibriSpeech, we configure the LM as a 16-layer Transformer blocks with 1024 embedding size (as in ESPnet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' It is trained on PG-19 for sentence-level language modeling with a dropout rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3 and an effective token number of 524288 in each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Adam with betas of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='98), and weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='01 is used for the optimization with 10k warmup steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For the proposed look-up dic- tionary, we use 2-gram for dictionary hashing (as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' U is set to 5k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' M is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The LM for Chinese datasets consists of 4 layer Transformer blocks with the embedding size of 384 and 1024 for small and large configuration respectively 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For look-up dictio- nary, U is set to 10k and other hyper-parameters are the same with the Librispeech settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For ASR inference in this paper, we set λsf={0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='4}, λres={0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='1}, λi={0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2} for {”LibriSpeech”, ”Input, ”Search”} respectively, which give the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' A beam size of 60 is used for the LibriSpeech and 10 for Chinese sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We use Word Error Rate (WER) as ASR metric for LibriSpeech test sets, and Character Error Rate (CER) and Sentence Error Rate (SER) for Chinese test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For both test sets, we also calculate the tail token error rate by only counting errors on tail tokens and ignoring errors on other tokens within the same testing utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Results We compare our model with the original Transformer LM, as well as two other baselines: N-gram augmented embedding for LM training in [10] and single-vector memory for BERT pretraining in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In Table 1, while the original LM helps the ASR model achieve lower CER and SER, our method shows significant improvement over it and the two baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We achieve 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='3% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='6% CER improvement on the general ”Input” and ”Search” test sets over the Transformer LM, and the CER improvement of tail tokens are even 1We avoided the LibriSpeech settings because of impact on the decoding efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Model Clean Other Overall Tail-1 Overall Tail-1 Conformer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='12% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='92% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='23% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='52% + LM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='08% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='93% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='81% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='30% + Ours 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='01% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='57% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='73% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='93% Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Evaluation of WER on the LibriSpeech test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Change of the overall CER (%) on the ”Search” test set with different dictionary size and different N-gram settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' higher: 13% on 1gram and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5% on 2gram tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' As we in- crease the hidden size of the LM from 384 to 1024, the performance gain is not as much as the small LMs, but our method still outper- forms the LM by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='7% on overall CER and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='6% on tail tokens CER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In Table 2, our proposed method also shows consistent improvement on the two LibriSpeech test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The improvement on tail word error rate is more significantly compared to the overall WER improvement as on Chinese test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' ANALYSIS In this section, we analyze how the different hyper-parameters, in- cluding dictionary size U, N in N-gram for hashing, memory up- date ratio and memory size of each entry M, affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' All experiments are conducted on the Chinese ”Search” test set, and the Transformer LM model with the proposed memory augmented lookup dictionary has 4 layers and 384 hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In Figure 2, we show the change of the overall CER (y axis) with the increase of dictionary size in different N-gram settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' It is clear that for each N-gram setting, increasing the dictionary size will boost the performance, and 2-gram achieves the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Since the degree of collision elevates with bigger U and N, larger N means more collision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' thus 4-gram performs even worse than 1 gram case when the dictionary size is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Considering the extra space taken by large dictionary size, wo choose the 2-gram with 10k dictionary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' In Table 3, we show the performance of both the overall CER and CER on tail tokens under different memory update settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The ratios ”0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2”, ”0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5” and ”0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='8” indicate we set a fixed probability for all tokens when sampling the Bernoulli variable Xk+1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3 to up- date the memory, while the ”freq” means we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 4 to decide the Pk+1 for different tokens depending on their frequency in training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The results demonstrate that large update ratio tends to improve the performance and our proposed frequency-based memory update strategy marginally beat other options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Figure 3 analyzes if a large memory size M would help the se- lection and the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We use the Information Gain (IG) which is computed by the difference in the attention entropy (as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 5 and 6) between a randomly initialized dictionary and a well-trained one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The entropy indicates how well the dictionary Ratio Overall Tail-1 Tail-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='37% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='47% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='37% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='35% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='38% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='24% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='31% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='34% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='16% freq 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='29% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='34% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='13% Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Change of the overall and tail tokens CER under different memory update options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 16 32 64 128 Memory Size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='0 Information Gain / Gradients 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='45 CER IG Gradients CER Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Overall CER, Gradients, and Information Gain (IG) change on the ”Search” test set with the increase of memory size M maps the information to the contextualized embedding of the cur- rent token � ck [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The results show the IG is highly correlated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Besides, we adopt the Gradient Attribution test [22, 23] to ad- dress the dictionary memory’s contribution further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' It computes the normalized gradient of the model variables to reflect its contribution to the output prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' It shows the gradients are also consistent with the previous finding that a larger memory would receive more gradients, indicating a greater contribution to the model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' However, considering the small relative gain and high computational cost when we increase the memory size from 64 to 128, we set the memory size as 64 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Finally, we want to discuss how the proposed memory aug- mented lookup dictionary will affect the model size and inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' During inference, compared to the baseline Transformer LM, the additional computation of our method is only the dictionary indexing (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 2) and context selection (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' For lookup dictio- nary, the indexing operation requires O(1) time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The context selection also performs as a constant time cost as O(M), where M is the memory size of the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We evaluate the Real Time Factor (RTF) on the ”Search” test set on a NVIDIA A100 GPU with beam size batch size equals to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' The RTF is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='124 for ASR model only, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='195 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='198 for the baseline Transformer LM and our proposed LM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We notice that such additional opera- tions almost do not affect the decoding speed in practice though the model size increases by introducing the lookup dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' CONCLUSIONS In this paper, we propose a memory augmented lookup dictionary based Transformer LM to improve the language modeling in ASR, especially for long tail tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We have improved the baseline Trans- former LM in terms of overall ASR metrics and the tail words er- ror rate in both Chinese and English test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We also analyze our method under different hyper-parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Overall, the results prove the superiority of the method over the baseline Transformer LM without sacrificing inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Future work includes more experiments on English data sets, especially in domain mismatch condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' We are also interested in applying the method to general language modeling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='50 1gram 3gram 2gram 4gram Overall CER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='25 2k 5k 10k 20k Dictionary Size6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content=' Graves, “Sequence transduction with recurrent neural net- works,” arXiv preprint arXiv:1211.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='org/abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} +page_content='07856' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfRfZA/content/2301.00066v1.pdf'} diff --git a/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/2301.05002v1.pdf.txt b/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/2301.05002v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b050a927e07decb861561550f7886cb7437e118b --- /dev/null +++ b/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/2301.05002v1.pdf.txt @@ -0,0 +1,1236 @@ +arXiv:2301.05002v1 [math.OC] 12 Jan 2023 +Convergence Analysis of the Proximal +Gradient Method in the Presence of +the Kurdyka–Łojasiewicz Property +without Global Lipschitz Assumptions +Xiaoxi Jia∗ +Christian Kanzow† +Patrick Mehlitz‡ +January 13, 2023 +Abstract. +We consider a composite optimization problem where the sum of a con- +tinuously differentiable and a merely lower semicontinuous function has to be minimized. +The proximal gradient algorithm is the classical method for solving such a problem numer- +ically. The corresponding global convergence and local rate-of-convergence theory typically +assumes, besides some technical conditions, that the smooth function has a globally Lip- +schitz continuous gradient and that the objective function satisfies the Kurdyka–Łojasiewicz +property. Though this global Lipschitz assumption is satisfied in several applications where +the objective function is, e.g., quadratic, this requirement is very restrictive in the non- +quadratic case. Some recent contributions therefore try to overcome this global Lipschitz +condition by replacing it with a local one, but, to the best of our knowledge, they still +require some extra condition in order to obtain the desired global and rate-of-convergence +results. The aim of this paper is to show that the local Lipschitz assumption together with +the Kurdyka–Łojasiewicz property is sufficient to recover these convergence results. +Keywords. Non-Lipschitz Optimization, Nonsmooth Optimization, Proximal Gradient +Method, Kurdyka–Łojasiewicz Property, Rate-of-Convergence +AMS subject classifications. 49J52, 90C30 +1 +Introduction +In this paper, we are concerned with problems from composite optimization where the +sum of a continuously differentiable function f and a merely lower semicontinuous +function φ has to be minimized. Problems of this type appear quite frequently in +∗University +of +Würzburg, +Institute +of +Mathematics, +97074 +Würzburg, +Germany, +xiaoxi.jia@mathematik.uni-wuerzburg.de +†University +of +Würzburg, +Institute +of +Mathematics, +97074 +Würzburg, +Germany, +kanzow@mathematik.uni-wuerzburg.de, ORCID: 0000-0003-2897-2509 +‡Brandenburgische Technische Universität Cottbus-Senftenberg, Institute of Mathematics, 03046 +Cottbus, Germany, mehlitz@b-tu.de, ORCID: 0000-0002-9355-850X +1 + +many practically relevant areas like, e.g., machine learning, data compression, matrix +completion, and image processing, see [10,18,19,24,30,33], where, typically, f models +a tracking-type term while φ is used to promote sparse structures in the solutions. +For an algorithmic treatment of such problems, it seems a nearby idea to exploit +the composite form, i.e., differentiability of f on the one hand and additional struc- +tural properties of the function φ on the other hand (typically, the nonsmoothness +encapsulated within φ is of specific type in all aforementioned applications). More +precisely, the so-called proximal mapping of the function φ has to be available, which +is typically the case in the aforementioned practically relevant scenarios. The idea +behind the definition of proximal mappings is to interrelate the search for minimizers +(or at least stationary points) with a fixed-point problem, and to apply a fixed-point +iteration to the proximal mapping in order to tackle the minimization of the under- +lying function. Combining the available oracles for f and φ in order to construct an +algorithm to minimize f + φ led to the development of so-called proximal gradient +methods which date back to [25]. It is worth to note that proximal gradient algo- +rithms can be interpreted as so-called forward-backward splitting methods which are +far older, see [17,37] for their origins and [6] for a modern view. Popular instances +of proximal gradient methods are the iterative shrinkage/threshold algorithm (ISTA) +and its accelerated version (FISTA = fast ISTA), see [8], where φ has to be convex. +The monograph [7] presents a nice overview of existing results addressing proximal +gradient methods where the nonsmooth part enjoys convexity. +It has been pointed out in the seminal works [4,13] that the convergence theory +for proximal gradient methods can be extended to situations where the nonsmooth +part φ is merely lower semicontinuous and not necessarily convex. In both afore- +mentioned papers, the analysis, which covers both (global) convergence and rate- +of-convergence results, requires a so-called descent lemma as well as the celebrated +Kurdyka–Łojasiewicz property, originating from [29,31,32]. The majority of available +convergence results regarding proximal gradient methods seems to indicate that the +price we have to pay for allowing φ to be nonsmooth is that the gradient ∇f of the +smooth part has to be globally Lipschitz continuous. This requirement, which holds +naturally when f is a (convex) quadratic function (as indicated above, this happens +to be the case in many standard applications from image processing and data sci- +ence), turns out to be rather restrictive in the non-quadratic situation which also is +of practical interest, see Examples 3.6 and 3.7 below. +Let us review some contributions where the authors try to get rid of this global Lip- +schitz assumption. First, we would like to mention [5] where composite optimization +problems with convex functions f and φ are considered without postulating global +Lipschitzness of ∇f. It is shown that local Lipschitz continuity of ∇f is enough to +obtain rate-of-convergence results for the iterates generated by a Bregman-type prox- +imal gradient method. However, the authors of [5] require the additional assumption +that there is a constant L > 0 such that Lh − f is convex, where h is a convex func- +tion which defines the Bregman distance (let us mention that h equals the squared +Euclidean norm in our setting). This convexity-type condition is satisfied in a couple +of practically relevant situations. The approach of [5] was generalized to the noncon- +vex setting in [14] using, once again, a local Lipschitz assumption on ∇f, as well as +2 + +the slightly stronger assumption (in order to deal with the nonconvexity) that there +exist a constant L > 0 and a convex function h such that both Lh − f and Lh + f +are convex. Let us emphasize that this constant L plays a central role in the de- +sign of the corresponding proximal-type methods. More precisely, it is used explicitly +for the determination of the stepsizes. In the recent paper [21], global convergence +results are proven under a local Lipschitz assumption on ∇f (without postulating +any of the convexity-type conditions from above), but the authors assume (a priori) +boundedness of iterates and stepsizes. +The present paper is based on [28] where the authors show global convergence +results for proximal gradient methods in the sense that every accumulation point is +shown to be a suitable stationary point of the composite optimization problem. The +analysis in [28] is based on the local Lipschitz continuity of ∇f, and does not require +the iterates to be bounded. An extension of this work, using a nonmonotone line +search, is given in [22]. In contrast to most existing papers on proximal gradient +methods, however, convergence of the entire sequence is not addressed in [22, 28]. +Hence, no associated rate-of-convergence results could be given ([22] presents some +standard worst-case rate-of-convergence results addressing the difference of two sub- +sequent iterates along convergent subsequences). The aim of this paper is to fill this +gap. More precisely, we show that the entire sequence generated by the proximal +gradient method converges to a limit with a suitable rate, provided that this point +satisfies the aforementioned Kurdyka–Łojasiewicz property. The underlying conver- +gence theory is still based on a merely local Lipschitz assumption on ∇f, neither +its global Lipschitzness nor the (a priori) boundedness of the iterates and stepsizes +is presumed. To this end, we stress that our analysis is not based on any kind of +descent lemma, which is in contrast to the contributions [5,14] mentioned above. +The paper is organized as follows: In Section 2, we formally introduce the model +problem of interest and provide some necessary notation as well as background ma- +terial from generalized differentiation. The proximal gradient method together with +the global convergence properties known from [28] are stated in Section 3. The con- +vergence and rate-of-convergence analysis is then given in Section 4. We close with +some final remarks in Section 5. +2 +Problem Setting and Preliminaries +2.1 +Problem Setting +Throughout the paper, we investigate the numerical treatment of the composite op- +timization problem +min +x +ψ(x) := f(x) + φ(x), +x ∈ X, +(P) +where f : X → R is continuously differentiable, φ: X → R := R ∪ {∞} is lower semi- +continuous (possibly infinite-valued and nondifferentiable), and X denotes a Euclidean +space, i.e., a real and finite-dimensional Hilbert space. Since we do not want to deal +with trivial situations, we assume that there exist points in X where the value of φ is +3 + +finite. Let us underline that X is chosen to be Euclidean because this allows to cover +applications from matrix analysis like low-rank optimization or matrix completion. +In order to minimize the function ψ: X → R in (P), we will exploit its composite +structure which allows for gradient steps with respect to the continuously differen- +tiable function f on the one hand and so-called proximal steps with respect to φ on +the other hand, i.e., we rely on a splitting approach. Throughout the last decades, +experiments on numerous practically relevant optimization problems have shown that +splitting methods are superior to the direct applications of standard methods from +nonsmooth optimization to the function ψ. +2.2 +Basic Notation +Throughout the paper, the Euclidean space X will be equipped with the inner product +⟨·, ·⟩: X × X → R and the associated norm ∥·∥. Given a set A ⊂ X and an element +x ∈ X, we use A+x := x+A := {x}+A := {x+a | a ∈ A} for brevity. Furthermore, +dist(x, A) := inf{∥y − x∥ | y ∈ A} +denotes the distance of the point x to the set A with dist(x, ∅) := ∞. For given ε > 0, +Bε(x) := {y ∈ X | ∥y − x∥ ≤ ε} denotes the closed ε-ball around x. +The continuous linear operator f ′(x): X → R denotes the derivative of the con- +tinuously differentiable function f : X → R at x ∈ X, and we will make use of +∇f(x) := f ′(x)∗1 where f ′(x)∗: R → X is the adjoint of f ′(x). This way, ∇f is +a mapping from X to X. +We further say that a sequence {xk} ⊂ X converges Q-linearly to x∗ ∈ X if there +is a constant c ∈ (0, 1) such that the inequality +∥xk+1 − x∗∥ ≤ c∥xk − x∗∥ +holds for all sufficiently large k ∈ N. Furthermore, {xk} is said to converge R-linearly +to x∗ if we have +lim sup +k→∞ +∥xk − x∗∥1/k < 1. +Note that this R-linear convergence holds if there exist constants ω > 0 and µ ∈ (0, 1) +such that ∥xk − x∗∥ ≤ ωµk holds for all sufficiently large k ∈ N, i.e., if the expression +∥xk − x∗∥ is dominated by a Q-linearly convergent null sequence. +2.3 +Generalized Differentiation +The following concepts are standard in variational analysis, and we refer the interested +reader to the monographs [34,39] for more details. +Let us fix a merely lower semicontinuous function ϑ: X → R and pick x ∈ dom ϑ +where dom ϑ := {x ∈ X | ϑ(x) < ∞} denotes the domain of ϑ. Then the set +�∂ϑ(x) := +� +η ∈ X +���� lim inf +y→x, y̸=x +ϑ(y) − ϑ(x) − ⟨η, y − x⟩ +∥y − x∥ +≥ 0 +� +4 + +is called the regular (or Fréchet) subdifferential of ϑ at x. Furthermore, the set +∂ϑ(x) := +� +η ∈ X +����� +∃{xk}, {ηk} ⊂ X: +xk → x, ϑ(xk) → ϑ(x), ηk → η, ηk ∈ �∂ϑ(xk) ∀k ∈ N +� +is well known as the limiting (or Mordukhovich) subdifferential of ϑ at x. Clearly, we +always have �∂ϑ(x) ⊂ ∂ϑ(x) by construction of these sets. Whenever ϑ is a convex +function, equality holds, and both subdifferentials coincide with the subdifferential of +convex analysis, i.e., +�∂ϑ(x) = ∂ϑ(x) = {η ∈ X | ∀y ∈ dom ϑ: ϑ(y) ≥ ϑ(x) + ⟨η, y − x⟩} +is valid in this situation. By definition of the regular subdifferential, it is clear that +whenever x∗ ∈ dom ϑ is a local minimizer of ϑ, then 0 ∈ �∂ϑ(x∗) hold. The latter +fact is known as Fermat’s rule, see [34, Proposition 1.30(i)]. +Thus, the inclusion +0 ∈ ∂ϑ(x∗) is a necessary optimality condition for x∗ being a local minimizer of ϑ +as well. Note that, for ϑ being convex, this necessary optimality condition is also +sufficient for (global) minimality of x∗ for ϑ. +Let us now apply this to the special case where ϑ := ψ is the sum of the contin- +uously differentiable function f and a merely lower semicontinuous function φ, as it +happens to be the case when investigating (P). Whenever x ∈ dom φ is fixed, the +sum rule +∂(f + φ)(x) = ∇f(x) + ∂φ(x) +(2.1) +holds due to the assumed continuous differentiability of f, see [34, Proposition 1.30(ii)]. +Application of Fermat’s rule therefore shows that the optimality condition +0 ∈ ∇f(x∗) + ∂φ(x∗) +holds at any local minimizer x∗ ∈ dom φ of the composite optimization problem (P). +Any point x∗ ∈ dom φ satisfying this necessary optimality condition will be called an +M-stationary point of (P) due to the appearance of the limiting (or Mordukhovich) +subdifferential. +We next introduce the famous Kurdyka–Łojasiewicz property that was already +mentioned in Section 1 and which plays a central role in our subsequent convergence +analysis. The version of this property stated below is a generalization of the classical +Kurdyka–Łojasiewicz inequality for nonsmooth functions as introduced in [3,11,12] +and afterwards used in the local convergence analysis of several nonsmooth optimiza- +tion methods, see [2,4,13,15,16,35,36] for a couple of examples. +Definition 2.1. Let g : X → R be lower semicontinuous. We say that g has the KL +property, where KL abbreviates Kurdyka–Łojasiewicz, at x∗ ∈ {x ∈ X | ∂g(x) ̸= ∅} if +there exist a constant η > 0, a neighborhood U ⊂ X of x∗, and a continuous concave +function χ: [0, η] → [0, ∞) which is continuously differentiable on (0, η) and satisfies +χ(0) = 0 as well as χ′(t) > 0 for all t ∈ (0, η) such that the so-called KL inequality +χ′� +g(x) − g(x∗) +� +dist +� +0, ∂g(x) +� +≥ 1 +holds for all x ∈ U ∩ +� +x ∈ X | g(x∗) < g(x) < g(x∗) + η +� +. The function χ from above +is referred to as the desingularization function. +5 + +We note that there exist classes of functions where the KL property holds with +the corresponding desingularization function given by χ(t) := ctκ for κ ∈ (0, 1] and +some constant c > 0, where the parameter κ is called the KL exponent, see [12,29]. +3 +A Proximal Gradient Method and its Global Con- +vergence Properties +This section begins with a formal description of a proximal gradient method for +the composite optimization problem (P), and then summarizes the associated global +convergence properties established in [28]. Note that our proximal gradient method +uses a line search which is important to get global convergence properties without a +global Lipschitz assumption. We start with a precise statement of the algorithm. +Algorithm 3.1 (Proximal Gradient Method). +Require: τ > 1, 0 < γmin ≤ γmax < ∞, δ ∈ (0, 1), x0 ∈ dom φ +1: Set k := 0. +2: while A suitable termination criterion is violated at iteration k do +3: +Choose γ0 +k ∈ [γmin, γmax]. +4: +For i = 0, 1, 2, . . ., compute a solution xk,i of +min +x +f(xk) + ⟨∇f(xk), x − xk⟩ + γk,i +2 ∥x − xk∥2 + φ(x), +x ∈ X +(3.1) +with γk,i := τ iγ0 +k, until the acceptance criterion +ψ(xk,i) ≤ ψ(xk) − δγk,i +2 ∥xk,i − xk∥2 +(3.2) +holds. +5: +Denote by ik := i the terminal value, and set γk := γk,ik and xk+1 := xk,ik. +6: +Set k ← k + 1. +7: end while +8: return xk +Our convergence analysis requires some technical assumptions as well as a local +Lipschitz condition on the gradient of the continuously differentiable function f. +Assumption 3.2. +(a) The function ψ is bounded from below on dom φ. +(b) The function φ is bounded from below by an affine function. +(c) The function ∇f : X → X is locally Lipschitz continuous. +Keeping in mind that our goal is to minimize the function ψ in (P), Assumption 3.2 (a) +is reasonable. Furthermore, Assumption 3.2 (b) is employed to guarantee existence +of solutions for the appearing subproblems (3.1). To be precise, Assumption 3.2 (b) +implies that the objective function of the subproblem (3.1) is, for fixed k, i ∈ N, +6 + +coercive, and therefore always attains a global minimizer xk,i (which does not need +to be unique). Finally, the local Lipschitz condition for ∇f from Assumption 3.2 (c) +will play a crucial role especially in Section 4 where we consider situations where a +sequence generated by Algorithm 3.1 converges as a whole and give associated rate- +of-convergence results. +In the following, we recall the central global convergence properties of Algorithm 3.1 +whose proofs can be found in [28, Section 3]. Note that, throughout our analysis of +Algorithm 3.1, we implicitly assume that this method generates an infinite sequence. +For a discussion of a practical termination criterion, we refer to [28, Remark 3.1] for +more details. +First, we recall that the stepsize rule in Step 4 of Algorithm 3.1 is always finite if +the current iterate is not already stationary. Hence, the overall method is well-defined. +Lemma 3.3. Consider a fixed iteration k ∈ N of Algorithm 3.1, assume that xk is +not an M-stationary point of (P), and suppose that Assumption 3.2 (b) holds. Then +the inner loop in Step 4 of Algorithm 3.1 is finite, i.e., we have γk = γk,ik for some +finite index ik ∈ {0, 1, 2, . . .}. +The following result summarizes some of the properties of Algorithm 3.1 that will +later be used in Section 4. +Proposition 3.4. Let Assumption 3.2 (a) and (b) hold, and let {xk} be a sequence +generated by Algorithm 3.1. Then the following statements hold: +(a) ∥xk+1 − xk∥ → 0 as k → ∞, +(b) for any convergent subsequence {xk}K, γk∥xk+1 − xk∥ →K 0 holds as k →K ∞, +(c) if, additionally, Assumption 3.2 (c) is valid, then for any convergent subse- +quence {xk}K, {γk}K is bounded. +Finally, we restate the main global convergence result for Algorithm 3.1, see again +[28, Section 3] for the corresponding details. +Theorem 3.5. Let Assumption 3.2 be satisfied. Then each accumulation point of a +sequence {xk} generated by Algorithm 3.1 is an M-stationary point of (P). +Note that [28, Theorem 3.1] shows that a result like Theorem 3.5 also holds with- +out any Lipschitz condition regarding ∇f, but it then requires a slightly stronger +condition for the nonsmooth function φ, namely the continuity of φ on its domain +(this condition holds, e.g., if φ is the indicator function of a constraint set). Our +analysis in Section 4, however, requires the local Lipschitz condition for the gradient +∇f, so we decided to treat it as a standing assumption. +We close this section by mentioning two classes of examples where the standard +global Lipschitz assumption on the gradient of f is typically violated, whereas a local +Lipschitz condition is often satisfied. +Example 3.6. (Augmented Lagrangian Methods) +Consider the constrained optimization problem +min +x +f(x) + φ(x) +s.t. +c(x) ∈ C, +7 + +where f : X → R and φ: X → R are as in (P). In addition, we have some constraints +defined by a continuously differentiable function c: X → Y, where Y is another Eu- +clidean space, and a nonempty, closed, and convex set C ⊂ Y. +Given a current iterate xk ∈ X and a corresponding Lagrange multiplier estimate +λk ∈ Y, augmented Lagrangian techniques then compute the next iterate xk+1 by +solving (approximately) the subproblem +min +x f(x) + φ(x) + ρk +2 dist2 +� +c(x) + λk +ρk +, C +� +, +x ∈ X +for some penalty parameter ρk > 0. +Since the squared distance function y �→ +dist2(y, C) is continuously differentiable by convexity of C, see [6, Corollary 12.30], +this subproblem has exactly the structure of the composite optimization problem +(P) and can therefore, in principle, be solved by a proximal gradient method, see +[20,23,26,27] for suitable realizations of this approach. +Assuming that the gradient of the smooth part of this objective function (with +respect to the variable x) is globally Lipschitz continuous, however, is pretty strong +is this setting and, basically, requires the constraint function c to be linear and the +set C to be polyhedral, whereas local Lipschitzness of this gradient holds under mild +conditions on the smoothness of f and c. +The following example makes use of conjugate functions, see [6, Definition 3.1]. +Since, within this paper, they only occur in this particular application, we refrain +from stating their precise definitions and properties, and refer the interested reader +to the excellent monographs [6,7,39] for more details. +Example 3.7. (Dual Proximal Gradient Methods) +Consider the (primal) optimization problem +min +x +g(x) + h(Ax), +x ∈ X +(3.3) +where both functions g : X → R and h: Y → R are lower semicontinuous and convex +while possessing nonempty domains, and A: X → Y is a linear operator. Above, Y +is another Euclidean space. Note that none of the functions g or h is assumed to be +(continuously) differentiable. +The (Fenchel) dual problem of (3.3) is given by +min +y +g∗(A∗y) + h∗(−y), +y ∈ Y +(3.4) +with the two conjugate functions g∗: X → R and h∗ : Y → R being lower semicontinu- +ous and convex, and A∗ : Y → X being the adjoint of A. Under suitable assumptions, +the pair (3.3), (3.4) enjoys strong duality, i.e., the optimal objective function values +of these problems coincide, see [38], which motivates to solve (3.4) instead of (3.3) in +some applications where the conjugate functions are explicitly available. +Assuming, in addition, that g is uniformly convex, it is known that g∗ is real- +valued everywhere and continuously differentiable with a globally Lipschitz continuous +8 + +gradient, see [39, Proposition 12.60]. Consequently, as promoted in [9], a standard +proximal gradient algorithm can be applied to the dual problem (3.4). On the other +hand, if g is only strictly convex, then the domain of g∗ is, in general, no longer the +entire space, but g∗ can still be shown to be continuously differentiable on the interior +of its domain. Its gradient, however, is no longer guaranteed to be globally Lipschitz +continuous on the domain. +4 +Convergence Analysis in the Presence of the KL +Property +The aim of this section is to show convergence of the entire sequence {xk} generated by +Algorithm 3.1 provided that there exists an accumulation point x∗ which, in addition, +satisfies the KL property, and to present associated rate-of-convergence results. The +proofs of these results are based on a local Lipschitz assumption on ∇f only, without +the a priori assumption that the whole sequence {xk} is bounded. Based on some +recent contributions in the area of proximal gradient and related first-order methods, +it seems reasonable to expect such a result to hold. For example, [13,35] consider a +whole class of first-order methods and investigate their (essentially local) convergence +showing, in particular, that the entire sequence {xk} generated by their methods stays +within a certain neighborhood of a solution provided that the KL property holds at +this solution. +Their approach is not directly applicable to our situation since, on the one hand, +we do not use the a priori assumption that our iterates are bounded, and, on the other +hand, because the adaption of the methods considered in [13, 35] to the proximal +gradient setting would result in an algorithm with a constant stepsize. +However, +having an accumulation point of Algorithm 3.1 satisfying the KL property, we know +from the local Lipschitz assumption on ∇f that a respective global Lipschitz condition +holds in a suitable neighborhood of this point, which then can be used to verify +that the stepsizes computed by Algorithm 3.1 remain bounded. This – more or less +heuristic – idea fortifies us to believe that one can also get convergence and rate- +of-convergence results under the KL property in the presence of Assumption 3.2 (c). +The following analysis is a careful mathematical realization of this somewhat vague +idea. +We begin with a result which shows that, locally around an accumulation point +of the sequence {xk}, the associated stepsizes γk remain bounded. This observation +and its proof are related to [28, Corollary 3.1]. Note that this statement is essentially +different from the boundedness of stepsizes along convergent subsequences of iterates +which is inherent in the presence of Assumption 3.2, see Proposition 3.4 (c). +Lemma 4.1. Let Assumption 3.2 hold, let {xk} be any sequence generated by Algorithm 3.1, +and let x∗ be an accumulation point of this sequence. Then, for any ρ > 0, there is a +constant ¯γρ > 0 (usually depending on ρ) such that γk ≤ ¯γρ holds for all k ∈ N such +that xk ∈ Bρ(x∗). +9 + +Proof. First, recall from Lemma 3.3 that the stepsize γk is well-defined for each k ∈ N. +Let ρ > 0 be fixed, and recall that the assumed local Lipschitz continuity of ∇f +implies that this gradient mapping is (globally) Lipschitz continuous on the compact +set B2ρ(x∗) (note that we took 2ρ as the radius of this ball here). Let us denote the +corresponding Lipschitz constant by L2ρ. Since x∗ is an accumulation point of the +sequence {xk}, there are infinitely many iterates of this sequence belonging to Bρ(x∗). +Now, assume, by contradiction, that there is a subsequence {γk}K with xk ∈ +Bρ(x∗) for all k ∈ K such that {γk}K is unbounded. Without loss of generality, we +may assume that γk →K ∞, that the subsequence of iterates {xk}K converges to +some point ¯x (not necessarily equal to x∗), and that, for each k ∈ K, the acceptance +criterion (3.2) is violated in the first iteration of the inner loop. Then, for the trial +stepsize ˆγk := γk/τ = τ ik−1γ0 +k, we also have ˆγk →K ∞, whereas the corresponding +trial vector ˆxk := xk,ik−1 does not satisfy the acceptance criterion from (3.2), i.e., we +have +ψ(ˆxk) > ψ(xk) − δ ˆγk +2 ∥ˆxk − xk∥2 +∀k ∈ K. +(4.1) +On the other hand, since ˆxk solves the corresponding subproblem (3.1) with ˆγk in +place of γk,i, we have +⟨∇f(xk), ˆxk − xk⟩ + ˆγk +2 ∥ˆxk − xk∥2 + φ(ˆxk) − φ(xk) ≤ 0. +(4.2) +We claim that this, in particular, implies ˆxk →K ¯x. In fact, using (4.2), the Cauchy- +Schwarz inequality, and the fact that {ψ(xk)} is monotonically decreasing by con- +struction of Algorithm 3.1, we obtain +ˆγk +2 ∥ˆxk − xk∥2 ≤ ∥∇f(xk)∥∥ˆxk − xk∥ + φ(xk) − φ(ˆxk) += ∥∇f(xk)∥∥ˆxk − xk∥ + ψ(xk) − f(xk) − φ(ˆxk) +≤ ∥∇f(xk)∥∥ˆxk − xk∥ + ψ(x0) − f(xk) − φ(ˆxk). +Since f is continuously differentiable and −φ is bounded from above by an affine +function in view of Assumption 3.2 (b), the above estimate implies ∥ˆxk − xk∥ →K 0. +In fact, if {∥ˆxk − xk∥}K would be unbounded, then the left-hand side would grow +more rapidly than the right-hand side, and if {∥ˆxk − xk∥}K would be bounded, but +staying away, at least on a subsequence, from zero by a positive number, the right- +hand side would be bounded, whereas the left-hand side would be unbounded on +the corresponding subsequence. Consequently, we have ∥ˆxk − xk∥ →K 0, and since +xk →K ¯x, this implies ˆxk →K ¯x. In particular, since ¯x ∈ Bρ(x∗), this implies that, +for all sufficiently large k ∈ K, we have both xk ∈ B2ρ(x∗) and ˆxk ∈ B2ρ(x∗). +Let us fix some k ∈ K. Using the mean-value theorem yields the existence of a +point ξk on the line segment connecting xk with ˆxk such that +ψ(ˆxk) − ψ(xk) = f(ˆxk) + φ(ˆxk) − f(xk) − φ(xk) += ⟨∇f(ξk), ˆxk − xk⟩ + φ(ˆxk) − φ(xk). +10 + +Substituting the resulting expression for φ(ˆxk) − φ(xk) into (4.2) yields +⟨∇f(xk) − ∇f(ξk), ˆxk − xk⟩ + ˆγk +2 ∥ˆxk − xk∥2 + ψ(ˆxk) − ψ(xk) ≤ 0. +(4.3) +Exploiting (4.1), we therefore obtain +ˆγk +2 ∥ˆxk − xk∥2 ≤ −⟨∇f(xk) − ∇f(ξk), ˆxk − xk⟩ + ψ(xk) − ψ(ˆxk) +≤ ∥∇f(xk) − ∇f(ξk)∥∥ˆxk − xk∥ + δ ˆγk +2 ∥ˆxk − xk∥2 +which can be rewritten as +(1 − δ)ˆγk +2 ∥ˆxk − xk∥ ≤ ∥∇f(xk) − ∇f(ξk)∥. +Since ξk in an element from the line connecting xk and ˆxk, it follows that ξk ∈ B2ρ(x∗) +for all k ∈ K sufficiently large. Hence, the Lipschitz continuity of ∇f on this ball +yields +(1 − δ)ˆγk +2 ∥ˆxk − xk∥ ≤ L2ρ∥xk − ξk∥ ≤ L2ρ∥xk − ˆxk∥ +for all sufficiently large k ∈ K. Since ˆxk ̸= xk in view of (4.1), this implies that +{ˆγk}K is bounded which, in turn, yields the boundedness of the subsequence {γk}K, +contradicting our assumption. This completes the proof. +We next show that the entire sequence {ψ(xk)} converges to ψ(x∗), where x∗ is an +arbitrary accumulation point of a sequence {xk} generated by Algorithm 3.1. Note +that this result is not completely obvious since ψ is only lower semicontinuous but +not continuous in general. Indeed, this property results from the construction of the +iterates xk+1 of Algorithm 3.1. +Lemma 4.2. Let Assumption 3.2 be satisfied, and let x∗ be an accumulation point +of a sequence {xk} generated by Algorithm 3.1. Then the entire sequence {ψ(xk)} +converges to ψ(x∗). +Proof. Let {xk}K be a subsequence converging to x∗. By means of Proposition 3.4 (a), +we also have xk+1 →K x∗. Since ψ is lower semicontinuous, we then obtain +ψ(x∗) ≤ lim inf +k→K∞ ψ(xk+1). +(4.4) +On the other hand, by construction, the entire sequence {ψ(xk)} is monotonically +decreasing. Since it is also bounded from below by ψ(x∗) as a consequence of (4.4), it +follows that the whole sequence {ψ(xk)} converges. It remains to show that its limit +is equal to (the lower bound) ψ(x∗). +To this end, we first note that xk+1 solves the subproblem (3.1) with stepsize γk +in place of γk,i. Hence, we have +⟨∇f(xk), xk+1 − xk⟩ + γk +2 ∥xk+1 − xk∥2 + φ(xk+1) +≤ ⟨∇f(xk), x∗ − xk⟩ + γk +2 ∥x∗ − xk∥2 + φ(x∗) +11 + +for each k ∈ N. Taking the upper limit as k →K ∞, and using the continuity of ∇f +as well as Proposition 3.4, we obtain +lim sup +k→K∞ +φ(xk+1) ≤ φ(x∗). +Combining this with (4.4) and using the continuity of f yields ψ(xk+1) →K ψ(x∗). +Since {ψ(xk)} converges, the assertion follows. +All results stated so far are independent of the KL property. The remaining part of +our analysis, however, is heavily based on the assumption that our objective function +ψ satisfies the KL property at a given accumulation point x∗ of a sequence {xk} +generated by Algorithm 3.1. In particular, let η > 0 be the corresponding constant +from the definition of the associated desingularization function χ. Furthermore, we +will assume that Assumption 3.2 is valid. In view of Proposition 3.4, we can find a +sufficiently large index ˆk ∈ N such that +sup +k≥ˆk +∥xk+1 − xk∥ ≤ η. +(4.5) +We then define +ρ := η + 1 +2 +(4.6) +as well as the compact set +Cρ := Bρ(x∗) ∩ Lψ(x0), +(4.7) +where Lψ(x0) := {x ∈ X | ψ(x) ≤ ψ(x0)} is the sublevel set of ψ with respect to x0, +the starting point exploited in Algorithm 3.1. By monotonicity of {ψ(xk)}, we have +{xk} ⊂ Lψ(x0). Finally, throughout the section, let Lρ > 0 be a (global) Lipschitz +constant of ∇f on Cρ from (4.7). Finally, in view of Lemma 4.1, we have +γk ≤ ¯γρ +∀xk ∈ Cρ +(4.8) +with some suitable upper bound ¯γρ > 0 (depending on our choice of ρ from (4.6)). +Using this notation, we can formulate the following result. +Lemma 4.3. Let Assumption 3.2 hold, and let {xk} be any sequence generated by +Algorithm 3.1. Suppose that {xk}K is a subsequence converging to some limit point +x∗, and that ψ has the KL property at x∗ with desingularization function χ. Then +there is a sufficiently large constant k0 ∈ K such that the corresponding constant +α := ∥xk0 − x∗∥ + +� +8 +� +ψ(xk0) − ψ(x∗) +� +δγmin ++ 2 +� +¯γρ + Lρ +� +δγmin +χ +� +ψ(xk0) − ψ(x∗) +� +(4.9) +satisfies α < 1 +2, where ρ > 0 and ¯γρ > 0 are the constants defined in (4.6) and (4.8), +respectively, while Lρ > 0 is a Lipschitz constant of ∇f on Cρ from (4.7), and δ > 0 +as well as γmin > 0 are the parameters from Algorithm 3.1. +12 + +Proof. The statement follows from the fact that each summand on the right-hand +side of (4.9) can be made arbitrarily small. This is clear for the first one since the +subsequence {xk}K converges to x∗. This is also true for the second summand as a +consequence of Lemma 4.2. Finally, the third one can be made arbitrarily small since +we have ψ(xk) → ψ(x∗) by Lemma 4.2, taking into account that the desingularization +function χ is continuous at the origin. Hence, the statement follows by taking an index +k0 ∈ K sufficiently large. +We next state another technical result. +Lemma 4.4. Let Assumption 3.2 hold, and let {xk} be any sequence generated by +Algorithm 3.1. Suppose that {xk}K is a subsequence converging to some limit point +x∗, and that ψ has the KL property at x∗ with desingularization function χ. Then +dist +� +0, ∂ψ(xk+1) +� +≤ +� +¯γρ + Lρ +� +∥xk+1 − xk∥ +holds for all sufficiently large k ∈ N such that xk ∈ Bα(x∗), where α < 1 +2 denotes the +constant from (4.9), ¯γρ > 0 is the constant from (4.8), and Lρ > 0 is the Lipschitz +constant of ∇f on Cρ from (4.7). +Proof. For any k ∈ N, since xk+1 is a solution of (3.1), we obtain +0 ∈ ∇f(xk) + γk(xk+1 − xk) + ∂φ(xk+1) +from the corresponding M-stationary condition. This implies +γk(xk − xk+1) + ∇f(xk+1) − ∇f(xk) ∈ ∇f(xk+1) + ∂φ(xk+1) = ∂ψ(xk+1) +(4.10) +for all k ∈ N, where we used the sum rule (2.1) for the limiting subdifferential. +Now, take an arbitrary index k ∈ N sufficiently large such that xk ∈ Bα(x∗) and +k ≥ ˆk, where ˆk is the index from (4.5). In view of (4.6) and Lemma 4.3, we have +α ≤ ρ. Therefore, Lemma 4.1 shows that +γk ≤ ¯γρ. +(4.11) +Moreover, using (4.5), (4.6), and Lemma 4.3, we get +∥xk+1 − x∗∥ ≤ ∥xk+1 − xk∥ + ∥xk − x∗∥ ≤ η + α ≤ ρ. +Hence, xk, xk+1 ∈ Cρ holds with the compact set Cρ from (4.7). Therefore, we have +��∇f(xk+1) − ∇f(xk) +�� ≤ Lρ∥xk+1 − xk∥ +by definition of Lρ. Together with (4.10) and (4.11), we thus obtain +dist +� +0, ∂ψ(xk+1) +� +≤ +��γk(xk − xk+1) + ∇f(xk+1) − ∇f(xk) +�� +≤ γk∥xk+1 − xk∥ + Lρ∥xk+1 − xk∥ +≤ (¯γρ + Lρ +� +∥xk+1 − xk∥ +for all k ∈ N satisfying k ≥ ˆk and xk ∈ Bα(x∗). +13 + +The following result shows that the entire sequence {xk}, generated by Algorithm 3.1, +already converges to one of its accumulation points x∗ provided that the objective +function ψ satisfies the KL property at this point. The proof combines our previous +results with a technique used in [13]. +Theorem 4.5. Let Assumption 3.2 hold, and let {xk} be any sequence generated by +Algorithm 3.1. Suppose that {xk}K is a subsequence converging to some limit point +x∗, and that ψ has the KL property at x∗. Then the entire sequence {xk} converges +to x∗. +Proof. In view of Lemma 4.2, we know that the whole sequence {ψ(xk)} is monoton- +ically decreasing and converging to ψ(x∗). This implies that ψ(xk) ≥ ψ(x∗) holds for +all k ∈ N. +Now, suppose we have ψ(xk) = ψ(x∗) for some index k ∈ N. Then, by monotonic- +ity, we also get ψ(xk+1) = ψ(x∗). Consequently, we obtain from (3.2) that +0 ≤ δγmin +2 +∥xk+1 − xk∥2 ≤ ψ(xk) − ψ(xk+1) = 0 +and, thus, xk+1 = xk. Since, by assumption, the subsequence {xk}K converges to +x∗, this implies that xk = x∗ for all k ∈ N sufficiently large. In particular, we have +convergence of the entire (eventually constant) sequence {xk} to x∗ in this situation. +For the remainder of this proof, we can therefore assume that ψ(xk) > ψ(x∗) holds +for all k ∈ N. We then let α ∈ (0, 1/2) be the constant from (4.9), and k0 ∈ K be +the corresponding iteration index which is used in the definition of α, see Lemma 4.3. +We then have 0 < ψ(xk) − ψ(x∗) ≤ ψ(xk0) − ψ(x∗) for all k ≥ k0. Without loss of +generality, we may also assume that k0 ≥ ˆk (the latter being the index defined by +(4.5)) and that k0 is sufficiently large to satisfy +ψ(xk0) < ψ(x∗) + η. +(4.12) +Let χ: [0, η] → [0, ∞) be the desingularization function which comes along with the +validity of the KL property at x∗. Due to χ(0) = 0 and χ′(t) > 0 for all t ∈ (0, η), we +obtain +χ +� +ψ(xk) − ψ(x∗) +� +≥ 0 +∀k ≥ k0. +(4.13) +We now claim that the following two statements hold for all k ≥ k0: +(a) xk ∈ Bα(x∗), +(b) ∥xk0 − x∗∥ + �k +i=k0 ∥xi+1 − xi∥ ≤ α, which is equivalent to +k +� +i=k0 +∥xi+1−xi∥ ≤ +� +8 +� +ψ(xk0) − ψ(x∗) +� +δγmin ++ 2 +� +¯γρ + Lρ +� +δγmin +χ +� +ψ(xk0)−ψ(x∗) +� +. (4.14) +We verify these two statements jointly by induction. For k = k0, statement (a) holds +simply by the definition of α in (4.9). Furthermore, the acceptance criterion (3.2) +together with the monotonicity of {ψ(xk)} implies +∥xk0+1 − xk0∥ ≤ +� +2 +� +ψ(xk0) − ψ(xk0+1) +� +δγmin +≤ +� +2 +� +ψ(xk0) − ψ(x∗) +� +δγmin +. +(4.15) +14 + +In particular, this shows that (4.14) holds for k = k0. Suppose that both statements +hold for some k ≥ k0. Using the triangle inequality, the induction hypothesis, and +the definition of α, we obtain +∥xk+1 − x∗∥ ≤ +k +� +i=k0 +∥xi+1 − xi∥ + ∥xk0 − x∗∥ ≤ α, +i.e., statement (a) holds for k + 1 in place of k. The verification of the induction step +for (b) is more involved. +To this end, first note that (4.12) implies +ψ(x∗) < ψ(xi) < ψ(x∗) + η +∀i ≥ k0. +(4.16) +Since ψ has the KL property at x∗, we have +χ′� +ψ(xi) − ψ(x∗) +� +dist +� +0, ∂ψ(xi) +� +≥ 1 +∀i ≥ k0. +(4.17) +Since xi ∈ Bα(x∗) for all i ∈ {k0, k0 + 1, . . . , k} by our induction hypothesis, we can +apply Lemma 4.4 and obtain (after a simple index shift) +dist +� +0, ∂ψ(xi) +� +≤ +� +¯γρ + Lρ +� +∥xi − xi−1∥ +∀i ∈ {k0 + 1, k0 + 2, . . . , k + 1}. +In view of (4.17), we therefore obtain +χ′� +ψ(xi) − ψ(x∗) +� +≥ +1 +� +¯γρ + Lρ +� +∥xi − xi−1∥ +∀i ∈ {k0 + 1, k0 + 2, . . . , k + 1}. (4.18) +To simplify some of the subsequent formulas, we follow [13] and introduce the short- +hand notation +∆i,j := χ +� +ψ(xi) − ψ(x∗) +� +− χ +� +ψ(xj) − ψ(x∗) +� +for i, j ∈ N. The assumed concavity of χ then implies +∆i,i+1 ≥ χ′� +ψ(xi) − ψ(x∗) +�� +ψ(xi) − ψ(xi+1) +� +. +(4.19) +Using (4.18), (4.19), and the acceptance criterion (3.2), we therefore get +∆i,i+1 ≥ χ′� +ψ(xi) − ψ(x∗) +�� +ψ(xi) − ψ(xi+1) +� +≥ +ψ(xi) − ψ(xi+1) +(¯γρ + Lρ)∥xi − xi−1∥ ≥ +δγmin +2(¯γρ + Lρ) +∥xi+1 − xi∥2 +∥xi − xi−1∥ = β∥xi+1 − xi∥2 +∥xi − xi−1∥ +for all i ∈ {k0+1, k0+2, . . . , k+1}, where we used the constant β := +δγmin +2(¯γρ+Lρ). Noting +that a + b ≥ 2 +√ +ab holds for all real numbers a, b ≥ 0, we therefore obtain +1 +β∆i,i+1 + ∥xi − xi−1∥ ≥ 2 +� 1 +β∆i,i+1∥xi − xi−1∥ ≥ 2∥xi+1 − xi∥ +15 + +for all i ∈ {k0 + 1, k0 + 2, . . . , k + 1}. Summation yields +2 +k+1 +� +i=k0+1 +∥xi+1 − xi∥ ≤ +k+1 +� +i=k0+1 +∥xi − xi−1∥ + 1 +β +k+1 +� +i=k0+1 +∆i,i+1 += +k +� +i=k0+1 +∥xi+1 − xi∥ + ∥xk0+1 − xk0∥ + 1 +β ∆k0+1,k+2 +≤ +k+1 +� +i=k0+1 +∥xi+1 − xi∥ + ∥xk0+1 − xk0∥ + 1 +β∆k0+1,k+2. +Subtracting the first summand from the right-hand side, exploiting the estimate +(4.15), and using the nonnegativity as well as monotonicity of the desingularization +function χ, we obtain +k+1 +� +i=k0+1 +∥xi+1 − xi∥ ≤ +� +2 +� +ψ(xk0) − ψ(x∗) +� +δγmin ++ 1 +β χ +� +ψ(xk0) − ψ(x∗) +� +. +Adding the term ∥xk0+1 − xk0∥ to both sides and using (4.15) once again, we get +k+1 +� +i=k0 +∥xi+1 − xi∥ ≤ +� +8 +� +ψ(xk0) − ψ(x∗) +� +δγmin ++ 1 +β χ +� +ψ(xk0) − ψ(x∗) +� +. +Hence, statement (b) holds for k + 1 in place of k, and this completes the induction. +In particular, it follows from (a) that xk ∈ Bα(x∗) for all k ≥ k0. Taking k → ∞ +in (4.14) therefore shows that {xk} is a Cauchy sequence and, thus, convergent. Since +we already know that x∗ is an accumulation point, it follows that the entire sequence +{xk} converges to x∗. +We finally state our rate-of-convergence result for one particular class of desin- +gularization functions. The result holds for a more general class of such functions, +and we comment on this after the proof. To keep the notation simple and since this +result, having in mind the previous ones, is more or less a standard observation, we +decided to state this rate-of-convergence result in the following way. +Theorem 4.6. Let Assumption 3.2 hold, and let {xk} be any sequence generated by +Algorithm 3.1. Suppose that {xk}K is a subsequence converging to some limit point +x∗, and that ψ has the KL property at x∗. Then the entire sequence {xk} converges +to x∗, and if the corresponding desingularization function has the form χ(t) = ct1/2 +for some c > 0, the following statements hold: +(a) the sequence {ψ(xk)} converges Q-linearly to ψ(x∗), +(b) the sequence {xk} converges R-linearly to x∗. +Proof. In view of Theorem 4.5, we only need to verify the quantitative statements (a) +and (b) of the theorem. +16 + +As noted at the beginning of the proof of Theorem 4.5, we may assume, without +loss of generality, that ψ(xk) > ψ(x∗) holds for all k ∈ N. In view of Lemma 4.2, we +then have +xk ∈ Bα(x∗) ∩ +� +x ∈ dom φ | ψ(x∗) < ψ(x) < ψ(x∗) + η +� +for all k ∈ N sufficiently large, where α > 0 is the constant from (4.9) and η > 0 +denotes the constant from the definition of the desingularization function χ. Since ψ +satisfies the KL property at x∗ with χ(t) = ct1/2, we have +1 ≤ χ′� +ψ(xk+1) − ψ(x∗) +� +dist +� +0, ∂ψ(xk+1) +� += c +2 +� +ψ(xk+1) − ψ(x∗) +�−1/2 dist +� +0, ∂ψ(xk+1) +� +for all sufficiently large k ∈ N. Taking into account Lemma 4.4, this yields +1 ≤ c(¯γρ + Lρ) +2 +� +ψ(xk+1) − ψ(x∗) +�−1/2∥xk+1 − xk∥ +for all k ∈ N sufficiently large, where ¯γρ > 0 is the constant from (4.8) and Lρ > 0 +is the global Lipschitz constant of ∇f on Cρ from (4.7). Rearranging this expression +yields +∥xk+1 − xk∥ ≥ +2 +c(¯γρ + Lρ) +� +ψ(xk+1) − ψ(x∗) +�1/2. +(4.20) +On the other hand, by the acceptance criterion (3.2) and γk ≥ γmin, we have +ψ(xk+1) − ψ(xk) ≤ −δγmin +2 ∥xk+1 − xk∥2. +(4.21) +Combining (4.20) and (4.21), we obtain +� +ψ(xk+1) − ψ(x∗) +� +− +� +ψ(xk) − ψ(x∗) +� += ψ(xk+1) − ψ(xk) +≤ −δγmin +2 ∥xk+1 − xk∥2 +≤ − +2δγmin +c2(¯γρ + Lρ)2 +� +ψ(xk+1) − ψ(x∗) +� += −σ +� +ψ(xk+1) − ψ(x∗) +� +for all k ∈ N sufficiently large, where we used the constant σ := +2δγmin +c2(¯γρ+Lρ)2 for brevity. +Rearranging these terms yields +ψ(xk+1) − ψ(x∗) ≤ +1 +1 + σ +� +ψ(xk) − ψ(x∗) +� +(4.22) +for all k ∈ N large enough, which shows that the sequence {ψ(xk)} converges Q- +linearly to ψ(x∗). +To verify statement (b), observe that the descent test (3.2) and the monotonicity +of the sequence {ψ(xk)} yield +δγmin +2 +∥xk+1 − xk∥2 ≤ ψ(xk) − ψ(xk+1) ≤ ψ(xk) − ψ(x∗) =: ψk, +17 + +and that the sequence {ψk} is Q-linearly convergent in view of part (a). Taking this +into account, it is not difficult to see that there exist constants ω > 0 and µ ∈ (0, 1) +such that +∥xk+1 − xk∥ ≤ ωµk +holds for all sufficiently large k ∈ N. Hence, for given integers ℓ > k > 0 large enough, +we therefore obtain +∥xℓ+1 − xk∥ ≤ +ℓ +� +j=k +∥xj+1 − xj∥ ≤ ω +ℓ +� +j=k +µj ≤ ωµk +∞ +� +j=0 +µj = +ω +1 − µµk. +Taking the limit ℓ → ∞ yields +∥xk − x∗∥ ≤ +ω +1 − µµk +for all large enough k ∈ N. This completes the proof of the (local) R-linear conver- +gence of {xk} to its limit x∗. +We note that similar rate-of-convergence results can be obtained for the more +general case where the desingularization function is given by χ(t) = ctκ for some +κ ∈ (0, 1]. The easiest way to see that is to modify the previous proof and to apply, +for example, [1, Lemma 1]. +5 +Conclusions +In this paper, we have shown that convergence of the whole sequence generated by +proximal gradient methods applied to the composite optimization problem (P) can +be achieved whenever the gradient of the smooth function f is locally Lipschitz con- +tinuous while the objective function ψ possesses the KL property at all points of its +domain. For our analysis, we neither needed a priori boundedness of iterates and +stepsizes nor any additional convexity assumptions. Our findings also gave rise to the +statement of associated rate-or-convergence results. +Several generalizations of the proximal gradient method involving, e.g., inertial +terms or Bregman distances, see [5, 14–16] and the references therein, have been +investigated in the presence of global Lipschitzness of the gradient associated with +the smooth term, as well as the KL property. 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Wets. +Variational Analysis. +Springer, 2009. +doi:10.1007/978-3-642-02431-3. +22 + diff --git a/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/load_file.txt b/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6000ee2eb0781a9499fb117116520592f92d5a96 --- /dev/null +++ b/i9E4T4oBgHgl3EQfSwzD/content/tmp_files/load_file.txt @@ -0,0 +1,862 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf,len=861 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='05002v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='OC] 12 Jan 2023 Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka–Łojasiewicz Property without Global Lipschitz Assumptions Xiaoxi Jia∗ Christian Kanzow† Patrick Mehlitz‡ January 13, 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We consider a composite optimization problem where the sum of a con- tinuously differentiable and a merely lower semicontinuous function has to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The proximal gradient algorithm is the classical method for solving such a problem numer- ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The corresponding global convergence and local rate-of-convergence theory typically assumes, besides some technical conditions, that the smooth function has a globally Lip- schitz continuous gradient and that the objective function satisfies the Kurdyka–Łojasiewicz property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Though this global Lipschitz assumption is satisfied in several applications where the objective function is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', quadratic, this requirement is very restrictive in the non- quadratic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Some recent contributions therefore try to overcome this global Lipschitz condition by replacing it with a local one, but, to the best of our knowledge, they still require some extra condition in order to obtain the desired global and rate-of-convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The aim of this paper is to show that the local Lipschitz assumption together with the Kurdyka–Łojasiewicz property is sufficient to recover these convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Non-Lipschitz Optimization, Nonsmooth Optimization, Proximal Gradient Method, Kurdyka–Łojasiewicz Property, Rate-of-Convergence AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 49J52, 90C30 1 Introduction In this paper, we are concerned with problems from composite optimization where the sum of a continuously differentiable function f and a merely lower semicontinuous function φ has to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Problems of this type appear quite frequently in ∗University of Würzburg, Institute of Mathematics, 97074 Würzburg, Germany, xiaoxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='jia@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='de †University of Würzburg, Institute of Mathematics, 97074 Würzburg, Germany, kanzow@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='de, ORCID: 0000-0003-2897-2509 ‡Brandenburgische Technische Universität Cottbus-Senftenberg, Institute of Mathematics, 03046 Cottbus, Germany, mehlitz@b-tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='de, ORCID: 0000-0002-9355-850X 1 many practically relevant areas like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', machine learning, data compression, matrix completion, and image processing, see [10,18,19,24,30,33], where, typically, f models a tracking-type term while φ is used to promote sparse structures in the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For an algorithmic treatment of such problems, it seems a nearby idea to exploit the composite form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', differentiability of f on the one hand and additional struc- tural properties of the function φ on the other hand (typically, the nonsmoothness encapsulated within φ is of specific type in all aforementioned applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' More precisely, the so-called proximal mapping of the function φ has to be available, which is typically the case in the aforementioned practically relevant scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The idea behind the definition of proximal mappings is to interrelate the search for minimizers (or at least stationary points) with a fixed-point problem, and to apply a fixed-point iteration to the proximal mapping in order to tackle the minimization of the under- lying function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Combining the available oracles for f and φ in order to construct an algorithm to minimize f + φ led to the development of so-called proximal gradient methods which date back to [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' It is worth to note that proximal gradient algo- rithms can be interpreted as so-called forward-backward splitting methods which are far older, see [17,37] for their origins and [6] for a modern view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Popular instances of proximal gradient methods are the iterative shrinkage/threshold algorithm (ISTA) and its accelerated version (FISTA = fast ISTA), see [8], where φ has to be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The monograph [7] presents a nice overview of existing results addressing proximal gradient methods where the nonsmooth part enjoys convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' It has been pointed out in the seminal works [4,13] that the convergence theory for proximal gradient methods can be extended to situations where the nonsmooth part φ is merely lower semicontinuous and not necessarily convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In both afore- mentioned papers, the analysis, which covers both (global) convergence and rate- of-convergence results, requires a so-called descent lemma as well as the celebrated Kurdyka–Łojasiewicz property, originating from [29,31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The majority of available convergence results regarding proximal gradient methods seems to indicate that the price we have to pay for allowing φ to be nonsmooth is that the gradient ∇f of the smooth part has to be globally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This requirement, which holds naturally when f is a (convex) quadratic function (as indicated above, this happens to be the case in many standard applications from image processing and data sci- ence), turns out to be rather restrictive in the non-quadratic situation which also is of practical interest, see Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us review some contributions where the authors try to get rid of this global Lip- schitz assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' First, we would like to mention [5] where composite optimization problems with convex functions f and φ are considered without postulating global Lipschitzness of ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' It is shown that local Lipschitz continuity of ∇f is enough to obtain rate-of-convergence results for the iterates generated by a Bregman-type prox- imal gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' However, the authors of [5] require the additional assumption that there is a constant L > 0 such that Lh − f is convex, where h is a convex func- tion which defines the Bregman distance (let us mention that h equals the squared Euclidean norm in our setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This convexity-type condition is satisfied in a couple of practically relevant situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The approach of [5] was generalized to the noncon- vex setting in [14] using, once again, a local Lipschitz assumption on ∇f, as well as 2 the slightly stronger assumption (in order to deal with the nonconvexity) that there exist a constant L > 0 and a convex function h such that both Lh − f and Lh + f are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us emphasize that this constant L plays a central role in the de- sign of the corresponding proximal-type methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' More precisely, it is used explicitly for the determination of the stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In the recent paper [21], global convergence results are proven under a local Lipschitz assumption on ∇f (without postulating any of the convexity-type conditions from above), but the authors assume (a priori) boundedness of iterates and stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The present paper is based on [28] where the authors show global convergence results for proximal gradient methods in the sense that every accumulation point is shown to be a suitable stationary point of the composite optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The analysis in [28] is based on the local Lipschitz continuity of ∇f, and does not require the iterates to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' An extension of this work, using a nonmonotone line search, is given in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In contrast to most existing papers on proximal gradient methods, however, convergence of the entire sequence is not addressed in [22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, no associated rate-of-convergence results could be given ([22] presents some standard worst-case rate-of-convergence results addressing the difference of two sub- sequent iterates along convergent subsequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The aim of this paper is to fill this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' More precisely, we show that the entire sequence generated by the proximal gradient method converges to a limit with a suitable rate, provided that this point satisfies the aforementioned Kurdyka–Łojasiewicz property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The underlying conver- gence theory is still based on a merely local Lipschitz assumption on ∇f, neither its global Lipschitzness nor the (a priori) boundedness of the iterates and stepsizes is presumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To this end, we stress that our analysis is not based on any kind of descent lemma, which is in contrast to the contributions [5,14] mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The paper is organized as follows: In Section 2, we formally introduce the model problem of interest and provide some necessary notation as well as background ma- terial from generalized differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The proximal gradient method together with the global convergence properties known from [28] are stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The con- vergence and rate-of-convergence analysis is then given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We close with some final remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 2 Problem Setting and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 Problem Setting Throughout the paper, we investigate the numerical treatment of the composite op- timization problem min x ψ(x) := f(x) + φ(x), x ∈ X, (P) where f : X → R is continuously differentiable, φ: X → R := R ∪ {∞} is lower semi- continuous (possibly infinite-valued and nondifferentiable), and X denotes a Euclidean space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', a real and finite-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since we do not want to deal with trivial situations, we assume that there exist points in X where the value of φ is 3 finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us underline that X is chosen to be Euclidean because this allows to cover applications from matrix analysis like low-rank optimization or matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In order to minimize the function ψ: X → R in (P), we will exploit its composite structure which allows for gradient steps with respect to the continuously differen- tiable function f on the one hand and so-called proximal steps with respect to φ on the other hand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', we rely on a splitting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Throughout the last decades, experiments on numerous practically relevant optimization problems have shown that splitting methods are superior to the direct applications of standard methods from nonsmooth optimization to the function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 Basic Notation Throughout the paper, the Euclidean space X will be equipped with the inner product ⟨·, ·⟩: X × X → R and the associated norm ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Given a set A ⊂ X and an element x ∈ X, we use A+x := x+A := {x}+A := {x+a | a ∈ A} for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, dist(x, A) := inf{∥y − x∥ | y ∈ A} denotes the distance of the point x to the set A with dist(x, ∅) := ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For given ε > 0, Bε(x) := {y ∈ X | ∥y − x∥ ≤ ε} denotes the closed ε-ball around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The continuous linear operator f ′(x): X → R denotes the derivative of the con- tinuously differentiable function f : X → R at x ∈ X, and we will make use of ∇f(x) := f ′(x)∗1 where f ′(x)∗: R → X is the adjoint of f ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This way, ∇f is a mapping from X to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We further say that a sequence {xk} ⊂ X converges Q-linearly to x∗ ∈ X if there is a constant c ∈ (0, 1) such that the inequality ∥xk+1 − x∗∥ ≤ c∥xk − x∗∥ holds for all sufficiently large k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, {xk} is said to converge R-linearly to x∗ if we have lim sup k→∞ ∥xk − x∗∥1/k < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that this R-linear convergence holds if there exist constants ω > 0 and µ ∈ (0, 1) such that ∥xk − x∗∥ ≤ ωµk holds for all sufficiently large k ∈ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', if the expression ∥xk − x∗∥ is dominated by a Q-linearly convergent null sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3 Generalized Differentiation The following concepts are standard in variational analysis, and we refer the interested reader to the monographs [34,39] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us fix a merely lower semicontinuous function ϑ: X → R and pick x ∈ dom ϑ where dom ϑ := {x ∈ X | ϑ(x) < ∞} denotes the domain of ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the set �∂ϑ(x) := � η ∈ X ���� lim inf y→x, y̸=x ϑ(y) − ϑ(x) − ⟨η, y − x⟩ ∥y − x∥ ≥ 0 � 4 is called the regular (or Fréchet) subdifferential of ϑ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, the set ∂ϑ(x) := � η ∈ X ����� ∃{xk}, {ηk} ⊂ X: xk → x, ϑ(xk) → ϑ(x), ηk → η, ηk ∈ �∂ϑ(xk) ∀k ∈ N � is well known as the limiting (or Mordukhovich) subdifferential of ϑ at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Clearly, we always have �∂ϑ(x) ⊂ ∂ϑ(x) by construction of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Whenever ϑ is a convex function, equality holds, and both subdifferentials coincide with the subdifferential of convex analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', �∂ϑ(x) = ∂ϑ(x) = {η ∈ X | ∀y ∈ dom ϑ: ϑ(y) ≥ ϑ(x) + ⟨η, y − x⟩} is valid in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' By definition of the regular subdifferential, it is clear that whenever x∗ ∈ dom ϑ is a local minimizer of ϑ, then 0 ∈ �∂ϑ(x∗) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The latter fact is known as Fermat’s rule, see [34, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='30(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Thus, the inclusion 0 ∈ ∂ϑ(x∗) is a necessary optimality condition for x∗ being a local minimizer of ϑ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that, for ϑ being convex, this necessary optimality condition is also sufficient for (global) minimality of x∗ for ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us now apply this to the special case where ϑ := ψ is the sum of the contin- uously differentiable function f and a merely lower semicontinuous function φ, as it happens to be the case when investigating (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Whenever x ∈ dom φ is fixed, the sum rule ∂(f + φ)(x) = ∇f(x) + ∂φ(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) holds due to the assumed continuous differentiability of f, see [34, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='30(ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Application of Fermat’s rule therefore shows that the optimality condition 0 ∈ ∇f(x∗) + ∂φ(x∗) holds at any local minimizer x∗ ∈ dom φ of the composite optimization problem (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Any point x∗ ∈ dom φ satisfying this necessary optimality condition will be called an M-stationary point of (P) due to the appearance of the limiting (or Mordukhovich) subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We next introduce the famous Kurdyka–Łojasiewicz property that was already mentioned in Section 1 and which plays a central role in our subsequent convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The version of this property stated below is a generalization of the classical Kurdyka–Łojasiewicz inequality for nonsmooth functions as introduced in [3,11,12] and afterwards used in the local convergence analysis of several nonsmooth optimiza- tion methods, see [2,4,13,15,16,35,36] for a couple of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let g : X → R be lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We say that g has the KL property, where KL abbreviates Kurdyka–Łojasiewicz, at x∗ ∈ {x ∈ X | ∂g(x) ̸= ∅} if there exist a constant η > 0, a neighborhood U ⊂ X of x∗, and a continuous concave function χ: [0, η] → [0, ∞) which is continuously differentiable on (0, η) and satisfies χ(0) = 0 as well as χ′(t) > 0 for all t ∈ (0, η) such that the so-called KL inequality χ′� g(x) − g(x∗) � dist � 0, ∂g(x) � ≥ 1 holds for all x ∈ U ∩ � x ∈ X | g(x∗) < g(x) < g(x∗) + η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The function χ from above is referred to as the desingularization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 5 We note that there exist classes of functions where the KL property holds with the corresponding desingularization function given by χ(t) := ctκ for κ ∈ (0, 1] and some constant c > 0, where the parameter κ is called the KL exponent, see [12,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 3 A Proximal Gradient Method and its Global Con- vergence Properties This section begins with a formal description of a proximal gradient method for the composite optimization problem (P), and then summarizes the associated global convergence properties established in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that our proximal gradient method uses a line search which is important to get global convergence properties without a global Lipschitz assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We start with a precise statement of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 (Proximal Gradient Method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Require: τ > 1, 0 < γmin ≤ γmax < ∞, δ ∈ (0, 1), x0 ∈ dom φ 1: Set k := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 2: while A suitable termination criterion is violated at iteration k do 3: Choose γ0 k ∈ [γmin, γmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 4: For i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', compute a solution xk,i of min x f(xk) + ⟨∇f(xk), x − xk⟩ + γk,i 2 ∥x − xk∥2 + φ(x), x ∈ X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) with γk,i := τ iγ0 k, until the acceptance criterion ψ(xk,i) ≤ ψ(xk) − δγk,i 2 ∥xk,i − xk∥2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 5: Denote by ik := i the terminal value, and set γk := γk,ik and xk+1 := xk,ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 6: Set k ← k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 7: end while 8: return xk Our convergence analysis requires some technical assumptions as well as a local Lipschitz condition on the gradient of the continuously differentiable function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (a) The function ψ is bounded from below on dom φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (b) The function φ is bounded from below by an affine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (c) The function ∇f : X → X is locally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Keeping in mind that our goal is to minimize the function ψ in (P), Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (a) is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (b) is employed to guarantee existence of solutions for the appearing subproblems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To be precise, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (b) implies that the objective function of the subproblem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) is, for fixed k, i ∈ N, 6 coercive, and therefore always attains a global minimizer xk,i (which does not need to be unique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Finally, the local Lipschitz condition for ∇f from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (c) will play a crucial role especially in Section 4 where we consider situations where a sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 converges as a whole and give associated rate- of-convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In the following, we recall the central global convergence properties of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 whose proofs can be found in [28, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that, throughout our analysis of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, we implicitly assume that this method generates an infinite sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For a discussion of a practical termination criterion, we refer to [28, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' First, we recall that the stepsize rule in Step 4 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 is always finite if the current iterate is not already stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, the overall method is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Consider a fixed iteration k ∈ N of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, assume that xk is not an M-stationary point of (P), and suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the inner loop in Step 4 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 is finite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', we have γk = γk,ik for some finite index ik ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The following result summarizes some of the properties of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 that will later be used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (a) and (b) hold, and let {xk} be a sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the following statements hold: (a) ∥xk+1 − xk∥ → 0 as k → ∞, (b) for any convergent subsequence {xk}K, γk∥xk+1 − xk∥ →K 0 holds as k →K ∞, (c) if, additionally, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (c) is valid, then for any convergent subse- quence {xk}K, {γk}K is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Finally, we restate the main global convergence result for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, see again [28, Section 3] for the corresponding details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then each accumulation point of a sequence {xk} generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 is an M-stationary point of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that [28, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1] shows that a result like Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5 also holds with- out any Lipschitz condition regarding ∇f, but it then requires a slightly stronger condition for the nonsmooth function φ, namely the continuity of φ on its domain (this condition holds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', if φ is the indicator function of a constraint set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Our analysis in Section 4, however, requires the local Lipschitz condition for the gradient ∇f, so we decided to treat it as a standing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We close this section by mentioning two classes of examples where the standard global Lipschitz assumption on the gradient of f is typically violated, whereas a local Lipschitz condition is often satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (Augmented Lagrangian Methods) Consider the constrained optimization problem min x f(x) + φ(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' c(x) ∈ C, 7 where f : X → R and φ: X → R are as in (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In addition, we have some constraints defined by a continuously differentiable function c: X → Y, where Y is another Eu- clidean space, and a nonempty, closed, and convex set C ⊂ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Given a current iterate xk ∈ X and a corresponding Lagrange multiplier estimate λk ∈ Y, augmented Lagrangian techniques then compute the next iterate xk+1 by solving (approximately) the subproblem min x f(x) + φ(x) + ρk 2 dist2 � c(x) + λk ρk , C � , x ∈ X for some penalty parameter ρk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since the squared distance function y �→ dist2(y, C) is continuously differentiable by convexity of C, see [6, Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='30], this subproblem has exactly the structure of the composite optimization problem (P) and can therefore, in principle, be solved by a proximal gradient method, see [20,23,26,27] for suitable realizations of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Assuming that the gradient of the smooth part of this objective function (with respect to the variable x) is globally Lipschitz continuous, however, is pretty strong is this setting and, basically, requires the constraint function c to be linear and the set C to be polyhedral, whereas local Lipschitzness of this gradient holds under mild conditions on the smoothness of f and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The following example makes use of conjugate functions, see [6, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since, within this paper, they only occur in this particular application, we refrain from stating their precise definitions and properties, and refer the interested reader to the excellent monographs [6,7,39] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (Dual Proximal Gradient Methods) Consider the (primal) optimization problem min x g(x) + h(Ax), x ∈ X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3) where both functions g : X → R and h: Y → R are lower semicontinuous and convex while possessing nonempty domains, and A: X → Y is a linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Above, Y is another Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that none of the functions g or h is assumed to be (continuously) differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The (Fenchel) dual problem of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3) is given by min y g∗(A∗y) + h∗(−y), y ∈ Y (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4) with the two conjugate functions g∗: X → R and h∗ : Y → R being lower semicontinu- ous and convex, and A∗ : Y → X being the adjoint of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Under suitable assumptions, the pair (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4) enjoys strong duality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', the optimal objective function values of these problems coincide, see [38], which motivates to solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4) instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3) in some applications where the conjugate functions are explicitly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Assuming, in addition, that g is uniformly convex, it is known that g∗ is real- valued everywhere and continuously differentiable with a globally Lipschitz continuous 8 gradient, see [39, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Consequently, as promoted in [9], a standard proximal gradient algorithm can be applied to the dual problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' On the other hand, if g is only strictly convex, then the domain of g∗ is, in general, no longer the entire space, but g∗ can still be shown to be continuously differentiable on the interior of its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Its gradient, however, is no longer guaranteed to be globally Lipschitz continuous on the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 4 Convergence Analysis in the Presence of the KL Property The aim of this section is to show convergence of the entire sequence {xk} generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 provided that there exists an accumulation point x∗ which, in addition, satisfies the KL property, and to present associated rate-of-convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The proofs of these results are based on a local Lipschitz assumption on ∇f only, without the a priori assumption that the whole sequence {xk} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Based on some recent contributions in the area of proximal gradient and related first-order methods, it seems reasonable to expect such a result to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For example, [13,35] consider a whole class of first-order methods and investigate their (essentially local) convergence showing, in particular, that the entire sequence {xk} generated by their methods stays within a certain neighborhood of a solution provided that the KL property holds at this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Their approach is not directly applicable to our situation since, on the one hand, we do not use the a priori assumption that our iterates are bounded, and, on the other hand, because the adaption of the methods considered in [13, 35] to the proximal gradient setting would result in an algorithm with a constant stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' However, having an accumulation point of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 satisfying the KL property, we know from the local Lipschitz assumption on ∇f that a respective global Lipschitz condition holds in a suitable neighborhood of this point, which then can be used to verify that the stepsizes computed by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This – more or less heuristic – idea fortifies us to believe that one can also get convergence and rate- of-convergence results under the KL property in the presence of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The following analysis is a careful mathematical realization of this somewhat vague idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We begin with a result which shows that, locally around an accumulation point of the sequence {xk}, the associated stepsizes γk remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This observation and its proof are related to [28, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that this statement is essentially different from the boundedness of stepsizes along convergent subsequences of iterates which is inherent in the presence of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 hold, let {xk} be any sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, and let x∗ be an accumulation point of this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then, for any ρ > 0, there is a constant ¯γρ > 0 (usually depending on ρ) such that γk ≤ ¯γρ holds for all k ∈ N such that xk ∈ Bρ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' First, recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3 that the stepsize γk is well-defined for each k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let ρ > 0 be fixed, and recall that the assumed local Lipschitz continuity of ∇f implies that this gradient mapping is (globally) Lipschitz continuous on the compact set B2ρ(x∗) (note that we took 2ρ as the radius of this ball here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us denote the corresponding Lipschitz constant by L2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since x∗ is an accumulation point of the sequence {xk}, there are infinitely many iterates of this sequence belonging to Bρ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Now, assume, by contradiction, that there is a subsequence {γk}K with xk ∈ Bρ(x∗) for all k ∈ K such that {γk}K is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Without loss of generality, we may assume that γk →K ∞, that the subsequence of iterates {xk}K converges to some point ¯x (not necessarily equal to x∗), and that, for each k ∈ K, the acceptance criterion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) is violated in the first iteration of the inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then, for the trial stepsize ˆγk := γk/τ = τ ik−1γ0 k, we also have ˆγk →K ∞, whereas the corresponding trial vector ˆxk := xk,ik−1 does not satisfy the acceptance criterion from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', we have ψ(ˆxk) > ψ(xk) − δ ˆγk 2 ∥ˆxk − xk∥2 ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) On the other hand, since ˆxk solves the corresponding subproblem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) with ˆγk in place of γk,i, we have ⟨∇f(xk), ˆxk − xk⟩ + ˆγk 2 ∥ˆxk − xk∥2 + φ(ˆxk) − φ(xk) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) We claim that this, in particular, implies ˆxk →K ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In fact, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2), the Cauchy- Schwarz inequality, and the fact that {ψ(xk)} is monotonically decreasing by con- struction of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, we obtain ˆγk 2 ∥ˆxk − xk∥2 ≤ ∥∇f(xk)∥∥ˆxk − xk∥ + φ(xk) − φ(ˆxk) = ∥∇f(xk)∥∥ˆxk − xk∥ + ψ(xk) − f(xk) − φ(ˆxk) ≤ ∥∇f(xk)∥∥ˆxk − xk∥ + ψ(x0) − f(xk) − φ(ˆxk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since f is continuously differentiable and −φ is bounded from above by an affine function in view of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 (b), the above estimate implies ∥ˆxk − xk∥ →K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In fact, if {∥ˆxk − xk∥}K would be unbounded, then the left-hand side would grow more rapidly than the right-hand side, and if {∥ˆxk − xk∥}K would be bounded, but staying away, at least on a subsequence, from zero by a positive number, the right- hand side would be bounded, whereas the left-hand side would be unbounded on the corresponding subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Consequently, we have ∥ˆxk − xk∥ →K 0, and since xk →K ¯x, this implies ˆxk →K ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In particular, since ¯x ∈ Bρ(x∗), this implies that, for all sufficiently large k ∈ K, we have both xk ∈ B2ρ(x∗) and ˆxk ∈ B2ρ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let us fix some k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Using the mean-value theorem yields the existence of a point ξk on the line segment connecting xk with ˆxk such that ψ(ˆxk) − ψ(xk) = f(ˆxk) + φ(ˆxk) − f(xk) − φ(xk) = ⟨∇f(ξk), ˆxk − xk⟩ + φ(ˆxk) − φ(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 10 Substituting the resulting expression for φ(ˆxk) − φ(xk) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) yields ⟨∇f(xk) − ∇f(ξk), ˆxk − xk⟩ + ˆγk 2 ∥ˆxk − xk∥2 + ψ(ˆxk) − ψ(xk) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3) Exploiting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1), we therefore obtain ˆγk 2 ∥ˆxk − xk∥2 ≤ −⟨∇f(xk) − ∇f(ξk), ˆxk − xk⟩ + ψ(xk) − ψ(ˆxk) ≤ ∥∇f(xk) − ∇f(ξk)∥∥ˆxk − xk∥ + δ ˆγk 2 ∥ˆxk − xk∥2 which can be rewritten as (1 − δ)ˆγk 2 ∥ˆxk − xk∥ ≤ ∥∇f(xk) − ∇f(ξk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since ξk in an element from the line connecting xk and ˆxk, it follows that ξk ∈ B2ρ(x∗) for all k ∈ K sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, the Lipschitz continuity of ∇f on this ball yields (1 − δ)ˆγk 2 ∥ˆxk − xk∥ ≤ L2ρ∥xk − ξk∥ ≤ L2ρ∥xk − ˆxk∥ for all sufficiently large k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since ˆxk ̸= xk in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1), this implies that {ˆγk}K is bounded which, in turn, yields the boundedness of the subsequence {γk}K, contradicting our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We next show that the entire sequence {ψ(xk)} converges to ψ(x∗), where x∗ is an arbitrary accumulation point of a sequence {xk} generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Note that this result is not completely obvious since ψ is only lower semicontinuous but not continuous in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Indeed, this property results from the construction of the iterates xk+1 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 be satisfied, and let x∗ be an accumulation point of a sequence {xk} generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the entire sequence {ψ(xk)} converges to ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let {xk}K be a subsequence converging to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' By means of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4 (a), we also have xk+1 →K x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since ψ is lower semicontinuous, we then obtain ψ(x∗) ≤ lim inf k→K∞ ψ(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4) On the other hand, by construction, the entire sequence {ψ(xk)} is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since it is also bounded from below by ψ(x∗) as a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4), it follows that the whole sequence {ψ(xk)} converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' It remains to show that its limit is equal to (the lower bound) ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To this end, we first note that xk+1 solves the subproblem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) with stepsize γk in place of γk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, we have ⟨∇f(xk), xk+1 − xk⟩ + γk 2 ∥xk+1 − xk∥2 + φ(xk+1) ≤ ⟨∇f(xk), x∗ − xk⟩ + γk 2 ∥x∗ − xk∥2 + φ(x∗) 11 for each k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Taking the upper limit as k →K ∞, and using the continuity of ∇f as well as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4, we obtain lim sup k→K∞ φ(xk+1) ≤ φ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4) and using the continuity of f yields ψ(xk+1) →K ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since {ψ(xk)} converges, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' All results stated so far are independent of the KL property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The remaining part of our analysis, however, is heavily based on the assumption that our objective function ψ satisfies the KL property at a given accumulation point x∗ of a sequence {xk} generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In particular, let η > 0 be the corresponding constant from the definition of the associated desingularization function χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, we will assume that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4, we can find a sufficiently large index ˆk ∈ N such that sup k≥ˆk ∥xk+1 − xk∥ ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5) We then define ρ := η + 1 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6) as well as the compact set Cρ := Bρ(x∗) ∩ Lψ(x0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7) where Lψ(x0) := {x ∈ X | ψ(x) ≤ ψ(x0)} is the sublevel set of ψ with respect to x0, the starting point exploited in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' By monotonicity of {ψ(xk)}, we have {xk} ⊂ Lψ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Finally, throughout the section, let Lρ > 0 be a (global) Lipschitz constant of ∇f on Cρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Finally, in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, we have γk ≤ ¯γρ ∀xk ∈ Cρ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='8) with some suitable upper bound ¯γρ > 0 (depending on our choice of ρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Using this notation, we can formulate the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 hold, and let {xk} be any sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Suppose that {xk}K is a subsequence converging to some limit point x∗, and that ψ has the KL property at x∗ with desingularization function χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then there is a sufficiently large constant k0 ∈ K such that the corresponding constant α := ∥xk0 − x∗∥ + � 8 � ψ(xk0) − ψ(x∗) � δγmin + 2 � ¯γρ + Lρ � δγmin χ � ψ(xk0) − ψ(x∗) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9) satisfies α < 1 2, where ρ > 0 and ¯γρ > 0 are the constants defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='8), respectively, while Lρ > 0 is a Lipschitz constant of ∇f on Cρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7), and δ > 0 as well as γmin > 0 are the parameters from Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The statement follows from the fact that each summand on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9) can be made arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This is clear for the first one since the subsequence {xk}K converges to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This is also true for the second summand as a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Finally, the third one can be made arbitrarily small since we have ψ(xk) → ψ(x∗) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2, taking into account that the desingularization function χ is continuous at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, the statement follows by taking an index k0 ∈ K sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We next state another technical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 hold, and let {xk} be any sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Suppose that {xk}K is a subsequence converging to some limit point x∗, and that ψ has the KL property at x∗ with desingularization function χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then dist � 0, ∂ψ(xk+1) � ≤ � ¯γρ + Lρ � ∥xk+1 − xk∥ holds for all sufficiently large k ∈ N such that xk ∈ Bα(x∗), where α < 1 2 denotes the constant from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9), ¯γρ > 0 is the constant from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='8), and Lρ > 0 is the Lipschitz constant of ∇f on Cρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For any k ∈ N, since xk+1 is a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1), we obtain 0 ∈ ∇f(xk) + γk(xk+1 − xk) + ∂φ(xk+1) from the corresponding M-stationary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This implies γk(xk − xk+1) + ∇f(xk+1) − ∇f(xk) ∈ ∇f(xk+1) + ∂φ(xk+1) = ∂ψ(xk+1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='10) for all k ∈ N, where we used the sum rule (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1) for the limiting subdifferential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Now, take an arbitrary index k ∈ N sufficiently large such that xk ∈ Bα(x∗) and k ≥ ˆk, where ˆk is the index from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3, we have α ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Therefore, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1 shows that γk ≤ ¯γρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='11) Moreover, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6), and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3, we get ∥xk+1 − x∗∥ ≤ ∥xk+1 − xk∥ + ∥xk − x∗∥ ≤ η + α ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, xk, xk+1 ∈ Cρ holds with the compact set Cρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Therefore, we have ��∇f(xk+1) − ∇f(xk) �� ≤ Lρ∥xk+1 − xk∥ by definition of Lρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='11), we thus obtain dist � 0, ∂ψ(xk+1) � ≤ ��γk(xk − xk+1) + ∇f(xk+1) − ∇f(xk) �� ≤ γk∥xk+1 − xk∥ + Lρ∥xk+1 − xk∥ ≤ (¯γρ + Lρ � ∥xk+1 − xk∥ for all k ∈ N satisfying k ≥ ˆk and xk ∈ Bα(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 13 The following result shows that the entire sequence {xk}, generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1, already converges to one of its accumulation points x∗ provided that the objective function ψ satisfies the KL property at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The proof combines our previous results with a technique used in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 hold, and let {xk} be any sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Suppose that {xk}K is a subsequence converging to some limit point x∗, and that ψ has the KL property at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the entire sequence {xk} converges to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2, we know that the whole sequence {ψ(xk)} is monoton- ically decreasing and converging to ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This implies that ψ(xk) ≥ ψ(x∗) holds for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Now, suppose we have ψ(xk) = ψ(x∗) for some index k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then, by monotonic- ity, we also get ψ(xk+1) = ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Consequently, we obtain from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) that 0 ≤ δγmin 2 ∥xk+1 − xk∥2 ≤ ψ(xk) − ψ(xk+1) = 0 and, thus, xk+1 = xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since, by assumption, the subsequence {xk}K converges to x∗, this implies that xk = x∗ for all k ∈ N sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In particular, we have convergence of the entire (eventually constant) sequence {xk} to x∗ in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For the remainder of this proof, we can therefore assume that ψ(xk) > ψ(x∗) holds for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We then let α ∈ (0, 1/2) be the constant from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9), and k0 ∈ K be the corresponding iteration index which is used in the definition of α, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We then have 0 < ψ(xk) − ψ(x∗) ≤ ψ(xk0) − ψ(x∗) for all k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Without loss of generality, we may also assume that k0 ≥ ˆk (the latter being the index defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5)) and that k0 is sufficiently large to satisfy ψ(xk0) < ψ(x∗) + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='12) Let χ: [0, η] → [0, ∞) be the desingularization function which comes along with the validity of the KL property at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Due to χ(0) = 0 and χ′(t) > 0 for all t ∈ (0, η), we obtain χ � ψ(xk) − ψ(x∗) � ≥ 0 ∀k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='13) We now claim that the following two statements hold for all k ≥ k0: (a) xk ∈ Bα(x∗), (b) ∥xk0 − x∗∥ + �k i=k0 ∥xi+1 − xi∥ ≤ α, which is equivalent to k � i=k0 ∥xi+1−xi∥ ≤ � 8 � ψ(xk0) − ψ(x∗) � δγmin + 2 � ¯γρ + Lρ � δγmin χ � ψ(xk0)−ψ(x∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='14) We verify these two statements jointly by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For k = k0, statement (a) holds simply by the definition of α in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Furthermore, the acceptance criterion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) together with the monotonicity of {ψ(xk)} implies ∥xk0+1 − xk0∥ ≤ � 2 � ψ(xk0) − ψ(xk0+1) � δγmin ≤ � 2 � ψ(xk0) − ψ(x∗) � δγmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='15) 14 In particular, this shows that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='14) holds for k = k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Suppose that both statements hold for some k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Using the triangle inequality, the induction hypothesis, and the definition of α, we obtain ∥xk+1 − x∗∥ ≤ k � i=k0 ∥xi+1 − xi∥ + ∥xk0 − x∗∥ ≤ α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', statement (a) holds for k + 1 in place of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The verification of the induction step for (b) is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To this end, first note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='12) implies ψ(x∗) < ψ(xi) < ψ(x∗) + η ∀i ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='16) Since ψ has the KL property at x∗, we have χ′� ψ(xi) − ψ(x∗) � dist � 0, ∂ψ(xi) � ≥ 1 ∀i ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='17) Since xi ∈ Bα(x∗) for all i ∈ {k0, k0 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' , k} by our induction hypothesis, we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4 and obtain (after a simple index shift) dist � 0, ∂ψ(xi) � ≤ � ¯γρ + Lρ � ∥xi − xi−1∥ ∀i ∈ {k0 + 1, k0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' , k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='17), we therefore obtain χ′� ψ(xi) − ψ(x∗) � ≥ 1 � ¯γρ + Lρ � ∥xi − xi−1∥ ∀i ∈ {k0 + 1, k0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' , k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='18) To simplify some of the subsequent formulas, we follow [13] and introduce the short- hand notation ∆i,j := χ � ψ(xi) − ψ(x∗) � − χ � ψ(xj) − ψ(x∗) � for i, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The assumed concavity of χ then implies ∆i,i+1 ≥ χ′� ψ(xi) − ψ(x∗) �� ψ(xi) − ψ(xi+1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='19) Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='18), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='19), and the acceptance criterion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2), we therefore get ∆i,i+1 ≥ χ′� ψ(xi) − ψ(x∗) �� ψ(xi) − ψ(xi+1) � ≥ ψ(xi) − ψ(xi+1) (¯γρ + Lρ)∥xi − xi−1∥ ≥ δγmin 2(¯γρ + Lρ) ∥xi+1 − xi∥2 ∥xi − xi−1∥ = β∥xi+1 − xi∥2 ∥xi − xi−1∥ for all i ∈ {k0+1, k0+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' , k+1}, where we used the constant β := δγmin 2(¯γρ+Lρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Noting that a + b ≥ 2 √ ab holds for all real numbers a, b ≥ 0, we therefore obtain 1 β∆i,i+1 + ∥xi − xi−1∥ ≥ 2 � 1 β∆i,i+1∥xi − xi−1∥ ≥ 2∥xi+1 − xi∥ 15 for all i ∈ {k0 + 1, k0 + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' , k + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Summation yields 2 k+1 � i=k0+1 ∥xi+1 − xi∥ ≤ k+1 � i=k0+1 ∥xi − xi−1∥ + 1 β k+1 � i=k0+1 ∆i,i+1 = k � i=k0+1 ∥xi+1 − xi∥ + ∥xk0+1 − xk0∥ + 1 β ∆k0+1,k+2 ≤ k+1 � i=k0+1 ∥xi+1 − xi∥ + ∥xk0+1 − xk0∥ + 1 β∆k0+1,k+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Subtracting the first summand from the right-hand side, exploiting the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='15), and using the nonnegativity as well as monotonicity of the desingularization function χ, we obtain k+1 � i=k0+1 ∥xi+1 − xi∥ ≤ � 2 � ψ(xk0) − ψ(x∗) � δγmin + 1 β χ � ψ(xk0) − ψ(x∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Adding the term ∥xk0+1 − xk0∥ to both sides and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='15) once again, we get k+1 � i=k0 ∥xi+1 − xi∥ ≤ � 8 � ψ(xk0) − ψ(x∗) � δγmin + 1 β χ � ψ(xk0) − ψ(x∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, statement (b) holds for k + 1 in place of k, and this completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In particular, it follows from (a) that xk ∈ Bα(x∗) for all k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Taking k → ∞ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='14) therefore shows that {xk} is a Cauchy sequence and, thus, convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since we already know that x∗ is an accumulation point, it follows that the entire sequence {xk} converges to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We finally state our rate-of-convergence result for one particular class of desin- gularization functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The result holds for a more general class of such functions, and we comment on this after the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To keep the notation simple and since this result, having in mind the previous ones, is more or less a standard observation, we decided to state this rate-of-convergence result in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2 hold, and let {xk} be any sequence generated by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Suppose that {xk}K is a subsequence converging to some limit point x∗, and that ψ has the KL property at x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Then the entire sequence {xk} converges to x∗, and if the corresponding desingularization function has the form χ(t) = ct1/2 for some c > 0, the following statements hold: (a) the sequence {ψ(xk)} converges Q-linearly to ψ(x∗), (b) the sequence {xk} converges R-linearly to x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5, we only need to verify the quantitative statements (a) and (b) of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 16 As noted at the beginning of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='5, we may assume, without loss of generality, that ψ(xk) > ψ(x∗) holds for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2, we then have xk ∈ Bα(x∗) ∩ � x ∈ dom φ | ψ(x∗) < ψ(x) < ψ(x∗) + η � for all k ∈ N sufficiently large, where α > 0 is the constant from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='9) and η > 0 denotes the constant from the definition of the desingularization function χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Since ψ satisfies the KL property at x∗ with χ(t) = ct1/2, we have 1 ≤ χ′� ψ(xk+1) − ψ(x∗) � dist � 0, ∂ψ(xk+1) � = c 2 � ψ(xk+1) − ψ(x∗) �−1/2 dist � 0, ∂ψ(xk+1) � for all sufficiently large k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Taking into account Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='4, this yields 1 ≤ c(¯γρ + Lρ) 2 � ψ(xk+1) − ψ(x∗) �−1/2∥xk+1 − xk∥ for all k ∈ N sufficiently large, where ¯γρ > 0 is the constant from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='8) and Lρ > 0 is the global Lipschitz constant of ∇f on Cρ from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Rearranging this expression yields ∥xk+1 − xk∥ ≥ 2 c(¯γρ + Lρ) � ψ(xk+1) − ψ(x∗) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='20) On the other hand, by the acceptance criterion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) and γk ≥ γmin, we have ψ(xk+1) − ψ(xk) ≤ −δγmin 2 ∥xk+1 − xk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='21) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='20) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='21), we obtain � ψ(xk+1) − ψ(x∗) � − � ψ(xk) − ψ(x∗) � = ψ(xk+1) − ψ(xk) ≤ −δγmin 2 ∥xk+1 − xk∥2 ≤ − 2δγmin c2(¯γρ + Lρ)2 � ψ(xk+1) − ψ(x∗) � = −σ � ψ(xk+1) − ψ(x∗) � for all k ∈ N sufficiently large, where we used the constant σ := 2δγmin c2(¯γρ+Lρ)2 for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Rearranging these terms yields ψ(xk+1) − ψ(x∗) ≤ 1 1 + σ � ψ(xk) − ψ(x∗) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='22) for all k ∈ N large enough, which shows that the sequence {ψ(xk)} converges Q- linearly to ψ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' To verify statement (b), observe that the descent test (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='2) and the monotonicity of the sequence {ψ(xk)} yield δγmin 2 ∥xk+1 − xk∥2 ≤ ψ(xk) − ψ(xk+1) ≤ ψ(xk) − ψ(x∗) =: ψk, 17 and that the sequence {ψk} is Q-linearly convergent in view of part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Taking this into account, it is not difficult to see that there exist constants ω > 0 and µ ∈ (0, 1) such that ∥xk+1 − xk∥ ≤ ωµk holds for all sufficiently large k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Hence, for given integers ℓ > k > 0 large enough, we therefore obtain ∥xℓ+1 − xk∥ ≤ ℓ � j=k ∥xj+1 − xj∥ ≤ ω ℓ � j=k µj ≤ ωµk ∞ � j=0 µj = ω 1 − µµk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Taking the limit ℓ → ∞ yields ∥xk − x∗∥ ≤ ω 1 − µµk for all large enough k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' This completes the proof of the (local) R-linear conver- gence of {xk} to its limit x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' We note that similar rate-of-convergence results can be obtained for the more general case where the desingularization function is given by χ(t) = ctκ for some κ ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' The easiest way to see that is to modify the previous proof and to apply, for example, [1, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' 5 Conclusions In this paper, we have shown that convergence of the whole sequence generated by proximal gradient methods applied to the composite optimization problem (P) can be achieved whenever the gradient of the smooth function f is locally Lipschitz con- tinuous while the objective function ψ possesses the KL property at all points of its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' For our analysis, we neither needed a priori boundedness of iterates and stepsizes nor any additional convexity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Our findings also gave rise to the statement of associated rate-or-convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Several generalizations of the proximal gradient method involving, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=', inertial terms or Bregman distances, see [5, 14–16] and the references therein, have been investigated in the presence of global Lipschitzness of the gradient associated with the smooth term, as well as the KL property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' Keeping our findings in mind, it might be promising to check whether our technique of proof can be applied in these settings to weaken the appearing Lipschitz assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E4T4oBgHgl3EQfSwzD/content/2301.05002v1.pdf'} +page_content=' References [1] F.' metadata={'source': 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0000000000000000000000000000000000000000..1130f9a5c33541db1709c7794eea0a19d4aad9af --- /dev/null +++ b/itAzT4oBgHgl3EQfpP1U/content/tmp_files/2301.01609v1.pdf.txt @@ -0,0 +1,1789 @@ +Published at Deep RL Workshop, NeurIPS 2022 +EMERGENT COLLECTIVE INTELLIGENCE FROM +MASSIVE-AGENT COOPERATION AND COMPETITION +Hanmo Chen1,*,‡, Stone Tao2,∗, Jiaxin Chen3,†, Weihan Shen3, +Xihui Li1,‡, Sikai Cheng4,‡, Xiaolong Zhu3, Xiu Li1 +1Tsinghua University, Shenzhen International Graduate School, 2University of California, San Diego +3Parametrix.ai, 4Georgia Institute of Technology +{chm20,xh-li21}@mails.tsinghua.edu.cn, stao@ucsd.edu +{jiaxinchen,weihanshen,xiaolongzhu}@chaocanshu.ai +scheng326@gatech.edu.cn, li.xiu@sz.tsinghua.edu.cn +ABSTRACT +Inspired by organisms evolving through cooperation and competition between dif- +ferent populations on Earth, we study the emergence of artificial collective intel- +ligence through massive-agent reinforcement learning. To this end, We propose +a new massive-agent reinforcement learning environment, Lux, where dynamic +and massive agents in two teams scramble for limited resources and fight off the +darkness. In Lux, we build our agents through the standard reinforcement learn- +ing algorithm in curriculum learning phases and leverage centralized control via a +pixel-to-pixel policy network. As agents co-evolve through self-play, we observe +several stages of intelligence, from the acquisition of atomic skills to the develop- +ment of group strategies. Since these learned group strategies arise from individ- +ual decisions without an explicit coordination mechanism, we claim that artificial +collective intelligence emerges from massive-agent cooperation and competition. +We further analyze the emergence of various learned strategies through metrics +and ablation studies, aiming to provide insights for reinforcement learning imple- +mentations in massive-agent environments. +1 +INTRODUCTION +Complex group and social behaviors widely exist in humans and animals on Earth. In a vast ecosys- +tem, the simultaneous cooperation and competition between populations and the changing environ- +ment serve as a natural driving force for the co-evolution of massive numbers of organisms (Wolpert +& Tumer, 1999; Dawkins & Krebs, 1979). This large-scale co-evolution between populations has +enabled group strategies for tasks individuals cannot accomplish (Ha & Tang, 2022). Inspired by this +self-organizing mechanism in nature, i.e., collective intelligence emerges from massive-agent coop- +eration and competition, we propose to simulate the emergence of collective intelligence through +training reinforcement learning agents in a massive-agent environment. We hope this can become +a stepping stone toward massive-agent reinforcement learning research and an inspiring method for +complex massive-agent problems. +Recent progress in multi-agent reinforcement learning (MARL) demonstrates its potential to com- +plete complex tasks through multi-agent cooperation, such as playing StarCraft2 (Vinyals et al., +2019) and DOTA2 (Berner et al., 2019). However, the number of agents is still limited to dozens in +those scenarios, far away from natural populations. To support large-scale multi-agent cooperation +and competition, we reintroduce the massive-agent setting into multi-agent reinforcement learning. +To this end, we propose Lux, a cooperative and competitive environment where hundreds of agents +in two populations scramble for limited resources and fight off the darkness. We believe Lux is a +suitable testbench for experimenting with collective intelligence because it provides an open envi- +ronment for hundreds of agents to cooperate, compete and evolve. +∗Equal contribution. +†Corresponding author. +‡Work done as research intern at Parametrix.ai. +1 +arXiv:2301.01609v1 [cs.AI] 4 Jan 2023 + +Published at Deep RL Workshop, NeurIPS 2022 +From the algorithmic perspective, the massive-agent setting poses great difficulties to reinforcement +learning algorithms since the credit assignment problem becomes increasingly challenging. Some +research (Lowe et al., 2017) focuses on the credit assignment problem between multi-agents, how- +ever, it lacks the scalability to massive-agent scenarios. To overcome that, we present a centralized +control solution for Lux using a pixel-to-pixel modeling architecture (Han et al., 2019) coupled with +Proximal Policy Optimization (PPO) (Schulman et al., 2017) algorithm. Using that solution, we +avoid the problem of credit assignment, with up to a 90% win rate versus the state-of-the-art policy +(Isaiah et al., 2021) proposed by the Toad Brigade team (TB) which won first place in the Lux AI +competition on Kaggle1. +Through self-play and curriculum learning phases, we observe several stages of the massive-agent +co-evolution, from atomic skills such as moving and building to group strategies such as efficient +territory occupation and long-term resource management. Note that group strategies arise from indi- +vidual decisions without any explicit coordination mechanism or hierarchy, demonstrating how col- +lective intelligence arises with co-evolution. Through quantitative analyses, further evidence shows +that collective intelligence can emerge from massive-agent cooperation and competition, leading to +behaviors beyond our expectations. For example, agents learn to stand in a diagonal row and move +as a whole to segment off parts of the map as shown in Figure 1. Without any prior knowledge, +this efficient strategy emerges from spontaneous exploration. Furthermore, we perform a detailed +ablation study to illustrate some implementation techniques which may be helpful in massive-agent +reinforcement learning. +(a) Blue is our policy and Yellow is TB. +(b) Yellow is our policy and Blue is TB. +Figure 1: Two episodes between our policy and TB where our Workers stand in a diagonal row. Our +agents discover it as an efficient way to expand the territory and limit the enemy’s movement. +Our main contributions are 1) we reintroduce massive-agent reinforcement learning as a scenario +for studying collective intelligence and propose a new environment, Lux, as a starting point. 2) +we provide evidence that collective intelligence emerges from co-evolution through massive agents’ +cooperation and competition in Lux. 3) we discuss the implementation details of our solution, which +may provide valuable insights into massive-agent reinforcement learning. +2 +RELATED WORK +Multi-Agent Environments. Many environments such as Multi-agent Particle Environment (MPE) +(Lowe et al., 2017) and Google Research Football (Kurach et al., 2020) are proposed to study multi- +agent cooperation and competition. For multi-agent cooperation, StarCraft Multi-Agent Challenge +(SMAC) (Samvelyan et al., 2019) provides a common testbench. However, SMAC focuses on de- +centralized micromanagement scenarios with only approximately 30 agents in play. In massive- +agent environments, Neural MMO (Suarez et al., 2021) is an open-ended Massively Multi-player +Online (MMO) game environment with up to 1024 agents. MAgent (Zheng et al., 2018) is a grid +world environment that supports up to a million agents. We propose Lux, a massive-agent rein- +forcement learning environment, which can support thousands of agents simultaneously acting at +one step. Unlike previous massive-agent environments, Lux incorporates Real-Time-Strategy (RTS) +game dynamics that are similar to Battlecode (2022) and MiniRTS (Tian et al., 2017). Moreover, +Lux scales up the number of agents with frequent spawns and deaths, which opens up the potential +for complex strategies in such a large-scale and highly dynamic scenario. +1https://www.kaggle.com/c/lux-ai-2021/ +2 + +Published at Deep RL Workshop, NeurIPS 2022 +Credit Assignment in MARL. Credit assignment between agents (Chang et al., 2003) is a crucial +challenge in multi-agent cooperation. Several value-based multi-agent algorithms (Sunehag et al., +2017; Rashid et al., 2018; Iqbal & Sha, 2018)) decompose global value into individual values using +a linear model or neural network, which can be viewed as an implicit way of credit assignment. +Another way of doing this is computing an agent-specific advantage function. For example, Foerster +et al. (2017) uses counterfactual regret to measure contributions to the team. In complex games, +Berner et al. (2019) and Ye et al. (2020) use hand-crafted team-based rewards for each agent as an +explicit method of credit assignment. Compared to the implicit value decomposition method, this +explicit reward-shaping method requires prior domain knowledge and lacks generalization ability. +However, both of them are limited to small population scenarios and are hard to scale to massive and +dynamic agents. Han et al. (2019) handles this problem using grid-wise centralized policy instead of +decentralized policy. It uses a convolutional neural network to map from pixel-wise observations to +actions over each pixel, which avoids the credit assignment problem while achieving efficient multi- +agent collaboration. Following this, we adapt this pixel-to-pixel architecture to the Lux environment +with the PPO algorithm (Schulman et al., 2017) and curriculum learning phases. +Collective Intelligence and Emergence Behaviors. +Collective Intelligence, including self- +organization and emergent behaviors (Wolpert & Tumer, 1999; Woolley et al., 2010), has a long +history connected with biological and economic studies. Research on emergent behavior usually +emphasizes that group strategies emerge from multi-agent co-evolution in a designed environment +rather than hand-crafted collaboration mechanisms. Baker et al. (2019) uses reinforcement learn- +ing agents and autocurricula (Leibo et al., 2019) in the Hide-and-seek environment, leading to the +emergence of tool use. Yang et al. (2018) proposes using million-agent reinforcement learning to +study how the agents’ grouping behaviors will change with the environmental resources. Zheng +et al. (2021) uses a two-level, deep RL framework to train agents and a social planner in an eco- +nomic environment, where optimal taxation policy emerges as the result of co-adaptation. Johanson +et al. (2022) studies the emergence of bartering behavior in a microeconomics-based environment +with producers and consumers. Dynamics in those environments usually induce agents’ behaviors +within human comprehension, thus limiting the possible emergent strategies. Since RTS games pro- +vide a perfect Petri dish for collective intelligence, our study absorbs RTS game dynamics into the +environment where simple rules may induce complex group strategies. +3 +LUX +Like the Earth, a suitable environment for collective intelligence to evolve must support massive +agents’ competition and cooperation. For that purpose, we propose an open-sourced environment +Lux, where hundreds of agents in two teams compete for resources and build cities as illustrated in +Figure 2. +Figure 2: A snapshot of Lux with hundreds of agents in two teams. Workers can collect resources +and build CityTiles. At night, CityTiles and Workers need fuel to stay alive and will be consumed by +darkness if fuel runs out. The team that owns more cities wins in the end. +3 + +Resources +Uranium +Coal +Tree +Agents +Workers +CityTilesPublished at Deep RL Workshop, NeurIPS 2022 +Setups. The map is a 2D square grid of size 12 to 32, scattered with different resources. An +episode consists of 360 turns, split into 9 Day/Night cycles of 30 days and 10 nights. There are +two basic units named Worker and CityTile. Each team starts with one Worker and one CityTile. +Workers can collect resources and build CityTiles and CityTiles can also build new Workers. Workers +collect adjacent resources automatically and convert resources to fuel when standing upon a friendly +CityTile. At night, CityTiles and Workers consume fuel to stay alive and will be consumed by +darkness if the fuel runs out. At the end of the game, the team with more CityTiles wins. More +details about the environment are in Appendix A.1. +Observation and Action Space. For observation, each team has perfect information about the +game state, including the global map, its own, and the opponent’s information. For actions, each +team needs to make decisions for every Worker and CityTile. A Worker can move in 4 cardinal +directions and build a CityTile when it has enough resources. Workers however cannot move onto +a tile with an enemy CityTile or Worker. A CityTile can build a Worker or research to increase the +team’s research points. Sufficient research points unlock the ability for the team’s Workers to mine +high-level resources that convert to more fuel like coal and uranium. +MARL in Lux. For MARL research, Lux raises a challenging situation for multi-agent modeling +and the credit assignment problem. Distinguished from other environments, the number of agents +in Lux is massive and dynamic. For a 32 × 32 map, the number of agents in a team can rise to +1000. Moreover, Workers and CityTiles are built and lost all the time, bringing difficulty for multi- +agent modeling of dynamic agents. A carefully-designed credit assignment scheme may be useful +in small-scale problems; however, with massive and dynamic agents, it becomes impractical due to +the combinatorial complexity. Furthermore, the win-or-lose sparse reward throws another challenge +on the hard exploration. +RTS in Lux. At first sight, Lux seems like a pocket-sized RTS game like StarCraft2. Agents in Lux +need to balance economic decisions and individual control, which requires high-level coordination +between hundreds of agents. However, the major difference between Lux and RTS games is the +way of controlling units. In RTS games, the low-level unit actions are executed by fixed rules, +which allows human players or AI to focus on macro-strategies and economic decisions. In Lux, +atomic actions such as moving and building are all controlled by the learned policy, resulting in an +action space of approximately 10180, magnitudes beyond StarCraft2 (Vinyals et al., 2019). Thus, a +successful policy needs to learn atomic skills and group strategies together, which is significant in +the emergence of collective intelligence. +4 +METHODOLOGY +Overall, our policy is trained using the standard algorithm PPO (Schulman et al., 2017) with Gen- +eralized Advantage Estimation (GAE) (Schulman et al., 2016). For massive-agent coordination, we +use a pixel-to-pixel architecture as the centralized policy (Han et al., 2019; Isaiah et al., 2021), tak- +ing both observations and actions as images and using the ResNet (He et al., 2016) structure as the +backbone. To address the sparse reward problem, we design three phases with different rewards as a +progressive curriculum. For clarity, we refer to the “agent” as each unit on the map and refer to the +“policy” as the centralized policy network that controls every agent. +4.1 +PIXEL-TO-PIXEL ARCHITECTURE +We model the massive-agent control problem using a centralized policy with a pixel-to-pixel archi- +tecture (Han et al., 2019). The policy network takes images as input observations and outputs actions +over each pixel in the form of an action map. More implementation details are in Appendix A.2. +Policy Network Architecture. The architecture of our policy network is pictured in Figure 3. The +input image (C ×H ×W) consists of C channels containing information about itself, the opponent, +and the global state. We use a ResNet-style convolutional network as the backbone. For actions, we +use a convolutional layer with kernel size 1 and output channels as the action dimensions. Moreover, +we use a flattened layer and a fully-connected layer for the value estimation as the critic. A valid +action mask is used to eliminate unnecessary exploration. +4 + +Published at Deep RL Workshop, NeurIPS 2022 +Figure 3: Policy network architecture. C is the input channels and H, W denote the map height +and width. E is the feature map channel through the backbone. Output channels AWorker/CityTile are +the corresponding action dimensions. +Why Centralized Policy. In Lux, our policy needs to control hundreds of agents each time step. +While the decentralized policy in MARL is computationally efficient and easy to scale, it needs a +carefully-designed credit assignment mechanism. In Lux, however, as agents are massive and dy- +namic, the credit assignment problem becomes increasingly challenging. To avoid that, we adopt a +centralized policy controlling every agent over the map. This pixel-to-pixel architecture with a con- +volutional network leverages the advantage of centralized and decentralized methods. Convolutional +layers work as a parameter-sharing mechanism across agents, similar to shared policy networks in +decentralized methods. This parameter-sharing mechanism improves learning efficiency via data +reuse. Furthermore, the deep stacked structure provides a large receptive field for global information +extraction and multi-agent communication, which naturally avoids the trouble of credit assignment +(Han et al., 2019). +4.2 +CURRICULUM TRAINING PHASES +The objective of agents in Lux is to own more CityTiles than the opponent, but the final result only +provides a sparse reward (1 for win, −1 for lose), resulting in the hard exploration problem (Badia +et al., 2020). Reward shaping is a common method to handle this problem in reinforcement learning. +However, hand-crafted rewards can easily direct agents into specific behavioral patterns with limited +strategies. Hence, we design three phases with different rewards as a progressive curriculum. First, +we use a dense reward to guide the policy towards basic skills. Then we gradually reduce the learning +signals and utilize the sparse reward to encourage the policy to explore more diversified strategies. +Phase 1: Dense Rewards for Basic Skills. At first, we use hand-crafted dense rewards to encourage +basic skills. Specifically, four kinds of behaviors are given rewards, namely, the increase of Workers +and CityTiles, Research Points and fuel. More details are in Appendix A.2. +Phase 2: Sparse Reward with Scaled Signals. In Phase 2, a reward is given only when an episode +ends. However, our policy still needs guidance through long-term reasoning. We modify the reward +with a slight signal about the win condition, i.e., ± +� +|Nself − Nop|, where Nself and Nop denote +the number of our own and the enemy’s CityTiles, encouraging to own more CityTiles for the win. +Phase 3: Win-or-Lose Sparse Reward. The win-or-lose sparse reward (1 for win and −1 for lose) +is applied in the final phase. After human-designed guidance in the first two phases, the win-or-lose +sparse reward encourages our policy to explore more advanced strategies. +5 +EMERGENT COLLECTIVE INTELLIGENCE +Through massive-agent cooperation and competition, we have observed three stages of our agents’ +evolution. Training from scratch, agents quickly acquire atomic skills such as collecting resources +and building cities. After around 5 million episodes, an elementary level of coordination appears on +the regional scale with dozens of agents. As training proceeds, the coordination expands from re- +5 + +Action Dimension +Input +Representation +Output +E×H×W +C×HxW +EXI +Conv +Resources +Residual +Valid'Mask +Blocks *8 +(k=1) +A +Softmax +[0000 +Worker. +10100 +0000 +Workers +Worker +0001 +Actions +CityTiles +Softmax +0000 +10100 +CityTile +0000 +CityTile +Lo011] +Global-Time +FC + Expand +ActionsPublished at Deep RL Workshop, NeurIPS 2022 +gional to global scope, which includes long-term economic decisions and precise control of hundreds +of agents. Those global strategies naturally arise from individual decisions due to massive-agent in- +teraction and co-evolution without any explicit coordination mechanism, signifying the emergence +of collective intelligence. +5.1 +ATOMIC SKILLS +The first step of our agents is to get a grasp of atomic skills. Guided by dense rewards, Workers +learn to move toward resources to collect fuel, and build and fuel the CityTiles, as shown in Figure +4a. However, at this stage agents are more likely to work alone and unable to make group decisions. +For example, Workers tend to build more CityTiles than they can support, leading to a sudden loss +of large cities as illustrated in Figure 4b as they run out of fuel. +(a) Workers collect resources and build CityTiles. +(b) CityTiles run out of fuel and collapse. +Figure 4: Illustration of atomic skills in a self-play episode. Agents acquire atomic skills such as +collecting resources and building CityTiles. However, due to a lack of group coordination, CityTiles +often burn out fuel and collapse. +5.2 +REGIONAL COORDINATION +As training proceeds, regional coordination appears, which involves dozens of agents in a local area. +For example, agents learn to carefully choose locations before building a CityTile and develop self- +organizing patterns for occupying resources efficiently. We describe a few examples of regional +strategies: +Construction Planning. As CityTiles built next to each other can share fuel and reduce cost at night, +agents gradually learn that the locations of CityTiles are important in city survival and fuel saving. +We find that agents discover several patterns of construction planning as visualized in Figure 5: 1) +build CityTiles near the resources for quicker access to fuel sources. 2) build CityTiles in a long row +to form cities that act like the Great Wall to prevent enemies’ aggression. 3) build CityTiles in blocks +to reduce fuel costs at night. +(a) CityTiles built near resources. +(b) CityTiles built in a row. +(c) CityTiles built in blocks. +Figure 5: Three emergent patterns of construction planning. a) build near resources for quick +access to fuel. b) build in a row as the Great Wall for defense. c) build in blocks to save fuel. +We use the city survival ratio (the final number of CityTiles divided by the most number of CityTiles +in one episode) to measure how these building patterns work quantitatively. As shown in Figure 6a, +the regional-scale construction planning effectively helps CityTiles fight off the darkness. +Territory Division. We have also observed a self-organizing structure where several Workers stand +in a diagonal row shown in Figure 1. Those Workers simultaneously move forward and backward +6 + +3Published at Deep RL Workshop, NeurIPS 2022 +(a) City survival ratio: final number divided by +the most number of CityTiles in one episode. +(b) Five-diagonal: how many times five or more +Workers stand in a diagonal row in one episode. +Figure 6: Quantitative analysis of region coordination using city survival ratio and five- +diagonal. Both metrics are evaluated using self-play. a) The city survival ratio increases as training +continues, indicating that the regional construction planning effectively helps CityTiles live long and +prosper. b) The frequency of the five-diagonal shape increases during the course of training, which +demonstrates the gradual acquisition of this strategy. +as a whole to keep the formation, and when any of them die, a new Worker nearby will fill in. +In this shape, they can effectively guard and expand the team’s territory and limit the enemy’s +movement. We measure a statistic called Five-Diagonal (how many times five or more Workers +stand in a diagonal row in one game) to investigate how often this strategy is utilized. Results in +Figure 6b illustrate that the frequency our agents use this strategy generally increases with training +in the long term, indicating it is an acquired strategy rather than a circumstance. +5.3 +GLOBAL STRATEGIES +As in micro-management scenarios of SMAC (Samvelyan et al., 2019), regional coordination of +dozens of agents is often found in multi-agent cooperation. However, our agents go far beyond that, +achieving much larger-scale coordination between hundreds of agents. We provide interpretation +and analysis of several global strategies as follows: +Sustainable Development. A key component in Lux is the balance of city development and re- +source consumption. In the early stages, the rapidly growing cities often face severe fuel shortages. +Gradually, our policy learns to develop cities at a sustainable speed in tune with fuel production +depending on the resource distribution and the opponent’s behavior. Another phenomenon we have +observed is the retention of trees. As trees are the only renewable resource in Lux, forest protection +is significant in securing long-term fuel supplies. Our agents intentionally preserve trees from exces- +sive deforestation and build CityTiles near the woods in defense of the enemy’s aggression. Another +metric, total wood collect (the total collected woods divided by originally spawned woods) is used to +measure how this forest protection strategy influences our fuel supplies. Results in Figure 7a show +how these protection strategies significantly improve the utilization efficiency of wood, resulting in +our agent collecting more than 500% of the original wood on the map at times. +(a) Total wood collect: total wood collected di- +vided by originally spawn woods. +(b) Total fuel: total fuel storage in one episode +using our latest model in a fixed map. +Figure 7: Quantitative analysis of global strategies using total wood collect and total fuel. The +metrics are evaluated using self-play. a) The utilization efficiency of wood increases as our policy +grows the sense of forest protection. b) In one episode, our fuel storage accumulates until turn 240. +After that, it tries to build more CityTiles for the win. +All In For The Win. Another surprising strategy is that when the episode is about to end, our policy +will rapidly harvest all the protected trees and try to build as many CityTiles as possible for the win +7 + +Phase 1 + Phase 2 +Phase 3 +1.0 +Ratio +0.8 +0.6 + Survival +0.4 +City +0.2 +0.0 +0 +2 +3 +4 +5 +6 +7 +8 +9 +1 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +# Episodes (× 106)Phase 1 +Phase 2 +Phase 3 +7 +6 +5 +Diagonal +4 +D +Five +1 +0 +0 +2 +3 +4 +5 +6 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +# Episodes (× 106)Phase 1 + Phase 2 +Phase 3 +5 + Collect +4 +Wood +3 +Total +0 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +# Episodes (× 106)60,000 +50,000 +40,000 +Fuel +30,000 +20,000 +10,000 +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +240 +260 +280 +300 +320 +340 360 +# TurnsPublished at Deep RL Workshop, NeurIPS 2022 +as shown in Figure 7b. Furthermore, we observe that sometimes cities retain very little fuel at the +end of an episode, evidence that almost all the resources have been fully utilized, as any resources +left at the end would be a waste. Efficiently using all remaining resources before the end is very +challenging because it needs the overall calculation of total fuel consumption by all Workers and +CityTiles, in addition to precise control of every agent. We think this strategy perfectly demonstrates +the emergence of collective intelligence through the combination of long-term economic decisions +and massive-agent mobilization. +6 +EXPERIMENTS +In this section, we perform ablation studies to reflect on our policy implementation and general +reinforcement learning algorithms under massive-agent settings: 1) we investigate the necessity of +curriculum learning phases by training with different procedures. Results demonstrate that our cur- +riculum design can help tackle the hard exploration problem caused by sparse rewards in the early +stages of training and encourage the emergence of complex strategies beyond human design. 2) we +further demonstrate the generalization ability of our model across different map sizes. When eval- +uating on maps of size 32, the policy trained on size 12 still retains some basic strategies. After a +fine-tuning phase of only 1.8 million episodes, the transferred policy achieves a 90% win rate against +TB on maps of size 32. The results indicate our model can learn generalizable representations suit- +able for the environment through learned spatial structures via convolutional layers. 3) we compare +our centralized policy against a standard decentralized solution with carefully-designed team-based +rewards. The centralized policy achieves a 98% win rate. See implementation details in Appendix +A.2. The decentralized policy implementation and experiment are in Appendix A.3. +6.1 +DESIGN OF CURRICULUM LEARNING PHASES +We perform experiments to investigate the necessity of our curriculum learning phases. Five differ- +ent procedures are applied: a) Only Phase 1; b) Phase 1 and 2 without Phase 3; c) Phase 1 and 3, +without Phase 2; d) Phase 1, 2, and 3 (the original procedure); e) Only Phase 3. +Figure 8: The win rate curves from different training phases. All win rates are evaluated against +TB on maps of size 12. Training with direct sparse rewards results in a 0% win rate, while training +only with Phase 1 dense rewards converges at around 50%. Compared to Phase 1+2, Phase 1+3 +improves slower and results in a lower win rate of around 70%. Phase 1+2 achieves an 85% win- +rate, and Phase 1+2+3 further boosts it to above 90%. +Phase 1 utilizes dense rewards which are fundamental for helping the centralized policy acquire +atomic skills for individual agents. As shown in Figure 8, our policy can hardly learn any basic +skills when directly training with sparse rewards, resulting in a win rate of around zero due to the +hard exploration problem. +Phase 2 utilizes a scaled sparse reward which plays two roles in the whole learning procedure, +accelerating learning and improving performance. First, continuous learning with dense rewards +converges at a 50% win rate, but the win rate rapidly rises to 70% with Phase 2. On the other hand, +without Phase 2, switching from Phase 1 to Phase 3 is more challenging with a lower performance +even after training for a longer period. This shows that the scaled sparse reward can work as a proper +8 + + Phase 1+2 +Phase 1+2+3 +Phase 1+3 +Phase 3 Only +Phase +1 +100% +%06 +80% +Phase 3 from 2 +Phase 2 +70% +100% +rate +60% +50% +90% +Win +40% +30% +%08 +20% +10% +20.2 +21.4 +19.0 19.4 +19.8 +20.6 +21.0 +非 Episodes (× 10°) +0% +5 +8 +10 +11 +12 +13 +14 +15 +16 +17 +1819 +0 +1 +2 +3 +4 +7 +9 +6 +非 Episodes +(× 106)Published at Deep RL Workshop, NeurIPS 2022 +transition between dense rewards and a win-or-loss sparse reward (applied in Phase 3) as it explicitly +tells the policy that owning more cities is the key to winning. +Phase 3 utilizes a sparse win-loss reward which further boosts the final performance to above 90%. +As the Phase 2 training converges to a win rate of 85% without Phase 3, the win-loss sparse reward +pushes our policy to go further and explore, resulting in an overall 90% win rate. +6.2 +GENERALIZATION OF REPRESENTATIONS FOR REINFORCEMENT LEARNING +We provide clear evidence that the learned representations from the convolutional neural network +and reinforcement learning algorithms can be generalized to different map sizes. First, we directly +transfer the policy net trained on maps of size 12 to size 32. As shown in Figure 9, basic skills +are retained on larger maps such as Workers collecting and fueling cities, even showcasing some +structured city construction planning to surround and protect wood resources. Secondly, after an ad- +ditional fine-tuning phase of around 1 million episodes, the policy quickly adapts to larger maps and +achieves an overall 90% win rate against TB, while training from scratch uses 1.6 million episodes +for a 20% win rate as shown in Figure 10. +Figure 9: Illustration of the general- +ization ability in a self-play episode. +When the policy trained on maps of size +12 is directly transferred to 32, some +strategies are retained such as Workers +building and fueling cities with plans. +Figure 10: The win-rate curves on 32 × 32 maps of +training from scratch and transfer. Both are evalu- +ated against TB. After a fine-tuning phase of 1 million +steps, the transferred policy achieves a 90% win rate, +while training from scratch uses 1.6 million episodes +for a 20% win rate. +The results demonstrate the generalization ability of our model, which provides insights into speed- +ing up policy training on large maps: we can pre-train our policy on small maps and then transfer +it to large maps with a fine-tuning phase. This procedure significantly reduces the training time +because smaller maps are faster for environment simulation and network update. For example, the +Lux environment simulation on CPU is 2.5× slower on maps of size 32 than size 12, and the policy +network update on GPU is 5× slower on maps of size 32. As training on maps of size 12 is both +time-saving and computationally efficient, our “Pre-train and Fine-tune” scheme achieves a higher +win rate with fewer training hours. +7 +DISCUSSION AND FUTURE WORK +We have demonstrated that collective intelligence can emerge from massive-agent cooperation and +competition. As proof of concept, we propose Lux, an environment hosting hundreds of agents and +incorporating RTS game dynamics. Through standard reinforcement learning algorithms and pixel- +to-pixel centralized modeling, we observe several stages of agents’ strategy evolution. Our agents +exhibit ambitious group strategies based on accurate individual control of massive agents without +explicit coordination mechanisms, signifying the emergence of collective intelligence. +We hope our work with Lux will be a stepping stone toward artificial collective intelligence. In +Lux, we observe the number of agents can reach up to 2000 in a single timestep, but this still pales +in comparison to the millions or even billions of organisms cooperating and competing in nature. +The Lux environment can be easily extended to host more agents as the experiments in Appendix +A.4, but simulation and inference become extremely slow reaching the million-agent level. Going +forward, the environment design and engineering as well as the training algorithm need a lot of +9 + +02 +.2Phase +Learning from Scratch - Learning from Transfer Model +100% +90% +80% +70% +rate +60% +50% +u!N +40% +30% +20% +10% +0% +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +# Episodes (× 106)Published at Deep RL Workshop, NeurIPS 2022 +modifications to adapt to such a scale. We also acknowledge that the RTS game dynamics in Lux +may not directly coincide with real-world problems. 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URL https://arxiv.org/abs/2108.02755. +A +APPENDIX +A.1 +DETAILED RULES OF LUX +For ease of understanding, the environment rules including unit types and action spaces are simpli- +fied in the main text. In this part, we provide a detailed description of the environment design and +rules. The version of Lux we use is compatible with the version on Kaggle Lux AI S1 competition2. +The rules can also be found at https://www.lux-ai.org/specs-2021 and the following +text is a reformatted and slightly modified version of the original rules. +Background. The night is dark and full of terrors. Two teams must fight off the darkness, collect +resources, and advance through the ages. Daytime finds a desperate rush to gather and build the +resources to carry you through the impending night. Plan and expand carefully – any city that fails +to produce enough light will be consumed by darkness. +Environment. In the Lux AI Challenge Season 1, two competing teams control a team of Units and +CityTiles that collect resources to fuel their Cities, with the main objective to own as many CityTiles +as possible at the end of the turn-based game. Both teams have complete information about the entire +game state and use that information to optimize resource collection, compete for scarce resources +against the opponent, and build cities to gain points. Each competitor must program their policy in +their language of choice. Each turn, your agent gets 3 seconds to submit their actions, excess time +is not saved across turns. In each game, you are given a pool of 60 seconds that is tapped into each +time you go over a turn’s 3-second limit. Upon using up all 60 seconds and going over the 3-second +limit, your agent freezes and can no longer submit additional actions. +The Map. The world of Lux is represented as a 2D grid. Coordinates increase east (right) and south +(down). The map is always a square and can be 12, 16, 24, or 32 tiles long. The (0, 0) coordinate is +at the top left. +Figure 11: The specification of the map in Lux. +The map has various features including Resources (Wood, Coal, Uranium), Units (Workers, Carts), +CityTiles, and Roads. In order to prevent maps from favoring one player over another, it is guaranteed +that maps are always symmetric by vertical or horizontal reflection. Each player will start with a +single CityTile and a single Worker on that CityTile. +Resources. There are 3 kinds of resources: Wood, Coal, and Uranium (in order of increasing fuel +efficiency). These resources are collected by Workers, then dropped off once a Worker moves on +2https://www.kaggle.com/c/lux-ai-2021/ +12 + +(0,0) +(0,y) +(x,0) +(x,y)Published at Deep RL Workshop, NeurIPS 2022 +Table 1: The specifications of resource collection and convert. +Resource Type +Research Points Pre-requisite +Fuel Value per Unit +Units Collected per Turn +Wood +0 +1 +20 +Coal +50 +10 +5 +Uranium +200 +40 +2 +top of a CityTile to then be converted into fuel for the city. Some resources require research points +before they are possible to collect. Wood in particular can regrow. Each turn, every wood tile’s +wood amount increases by 2.5% of its current wood amount rounded up. Wood tiles that have been +depleted will not regrow. Only wood tiles with less than 500 wood will regrow. +Collection Mechanics. At the end of each turn, Workers automatically receive resources from +all adjacent (North, East, South, West, or Center) resource tiles they can collect resources from +according to the current symmetric formula: +Iterating over uranium, coal, then wood resources: +• Each unit makes resource collection requests to collect an even number of resources from +each adjacent tile of the current iterated resource such that the collected amount takes the +unit’s cargo above capacity. E.g. Worker with 60 wood adjacent to 3 wood tiles asks for 14 +from each, receives 40 wood, and wastes 2. +• All tiles of the current iterated resource then try to fulfill requests. If they can’t, they make +sure all unfulfilled requests get an equal amount, and the leftover is wasted. E.g. if 4 +Workers are mining a tile of 25 wood, but one of them is only asking for 5 while the others +are asking for 20 wood each, then first all Workers get 5 wood each, leaving 5 wood left +over for 3 more Workers with space left. This can be evenly distributed by giving 1 wood +each to the last 3 Workers, leaving 2 wood left that is then wasted. +Workers cannot mine while on CityTiles. Instead, if there is at least one Worker on a CityTile, that +CityTile will automatically collect adjacent resources at the same rate as a Worker each turn and +directly convert it all to fuel. The collection mechanic for a CityTile is the same as a Worker and you +can treat a CityTile as an individual Worker collecting resources. +Actions. Units and CityTiles can perform actions each turn given certain conditions. In general, all +actions are simultaneously applied and are validated against the state of the game at the start of a +turn. The next few sections describe the Units and CityTiles in detail. +CityTiles. A CityTile is a building that takes up one tile of space. Adjacent CityTiles collectively +form a City. Each CityTile can perform a single action provided the CityTile has a Cooldown < 1. +Actions: +• Build Worker - Build Worker unit on top of this CityTile (cannot build a Worker if the +current number of owned Workers + carts equals the number of owned CityTiles). +• Build Cart - Build Carts unit on top of this CityTile (cannot build a cart if the current number +of owned Workers + carts equals the number of owned CityTiles). +• Research - Increase your team’s Research Points by 1. +Units. There are two unit types, Workers, and Carts. Every unit can perform a single action once +they have a Cooldown < 1. All units can choose the move action and move in any of the 5 direc- +tions, North, East, South, West, or Center. Moreover, all units can carry raw resources gained from +automatic mining or resource transfer. Workers are capped at 100 units of resources and Carts are +capped at 2000 units of resources. Whenever a unit moves on top of a friendly CityTile, the City that +CityTile forms converts all carried resources into fuel. +There can be at most one unit on tiles without a CityTile. Moreover, units cannot move on top of the +opposing team’s CityTiles. However, units can stack on top of each other on a friendly CityTile. If +two units attempt to move to the same tile that is not a CityTile, this is considered a collision, and +the move action is canceled. +13 + +Published at Deep RL Workshop, NeurIPS 2022 +Table 2: The specifications of cooldown. +Unit Type +Base Cooldown +CityTile +10 +Worker +2 +Cart +3 +Workers. Actions: +• Move - Move the unit in one of 5 directions, North, East, South, West, or Center. +• Pillage - Reduce the Road level of the tile the unit is on by 0.5. +• Transfer - Send any amount of a single resource-type from a unit’s cargo to another (start- +of-turn) adjacent Unit, up to the latter’s cargo capacity. Excess is returned to the original +unit. +• Build CityTile - Build a CityTile right under this Worker, provided the Worker has 100 total +resources of any type in their cargo (full cargo), and the tile is empty. If the building is +successful, all carried resources are consumed, and a new CityTile is built with 0 starting +resources. +Carts. Actions: +• Move - Move the unit in one of 5 directions, North, East, South, West, Center. +• Transfer - Send any amount of a single resource-type from a unit’s cargo to another (start- +of-turn) adjacent Unit, up to the latter’s cargo capacity. Excess is returned to the original +unit. +Cooldown. CityTiles, Workers, and Carts all have a cooldown mechanic after each action. Units +and CityTiles can only act when they have Cooldown < 1. At the end of each turn, after Road has +been built and pillaged, each unit’s Cooldown decreases by 1 and decreases by the level of the Road +the unit is on at the end of the turn. CityTiles are not affected by road levels, and cooldown always +decreases by 1. The minimum Cooldown is 0. After an action is performed, the unit’s Cooldown +will increase by a Base Cooldown, as specified in Table 2. +Roads. As Carts travel across the map, they start to create roads that allow all Units to move faster. +At the end of each turn, Cart will upgrade the road level of the tile it ends on by 0.75. The higher +the road level, the faster Units can move and perform actions. All tiles start with a road level of 0 +and are capped at 6. Moreover, CityTiles automatically have a max road level of 6. Workers can +also destroy roads via the pillage action which reduces road levels by 0.5 each time. If a City is +consumed by darkness, the road level of all tiles in the City’s CityTiles will go back to 0. +Day/Night Cycle. +The Day/Night cycle consists of a 40-turn cycle, the first 30 turns being day +turns, the last 10 being night turns. There are 360 turns in a match, forming 9 cycles. During +the night, Units and Cities need to produce light to survive. Each turn of the night, each Unit and +CityTile will consume an amount of fuel, see Table 3 for rates. Units in particular will use their +carried resources to produce light whereas CityTiles will use their fuel to produce light. Workers +and Carts will only need to consume resources if they are not on a CityTile. When outside the City, +Workers and Carts must consume whole units of resources to satisfy their night needs, e.g. if a +Worker carries 1 wood and 5 uranium on them, they will consume a full wood for 1 fuel, then a full +unit of uranium to fulfill the last 3 fuel requirements, wasting 37 fuel. Units will always consume +the least efficient resources first. +Lastly, at night, Units gain 2× more Base Cooldown. Should any Unit during the night run out of +fuel, they will be removed from the game and disappear into the night forever. Should a City run +out of fuel, however, the entire City with all of the CityTiles it owns will fall into darkness and be +removed from the game. +Game Resolution order. To help avoid confusion over smaller details of how each turn is resolved, +we provide the game resolution order here and how actions are applied. Actions in the game are +first all validated against the current game state to see if they are valid. Then the actions, along with +game events, are resolved in the following order and simultaneously within each step: +14 + +Published at Deep RL Workshop, NeurIPS 2022 +Table 3: The specifications of file burn, nadj denotes the number of adjacent friendly CityTiles. +Unit Type +Fuel Burn in City +Fuel Burn Outside City +CityTile +23 − 5 × nadj +N/A +Worker +0 +10 +Cart +0 +4 +1. CityTile actions along with increased cooldown. +2. Unit actions along with increased cooldown. +3. Roads are created. +4. Resource collection. +5. Resource drops on CityTiles. +6. If night time, make Units consume resources and CityTiles consume fuel. +7. Regrow wood tiles that are not depleted to 0. +8. Cooldowns are handled/computed for each unit and CityTile. +The only exception to the validation criteria is that units may move smoothly between spaces, mean- +ing if two units are adjacent, they can swap places in one turn. Otherwise, actions such as one unit +building a CityTile, then another unit moving on top of the new CityTile, are not allowed as the +current state does not have this newly built city and units cannot move on top of other units outside +of CityTiles. +Win Conditions. After 360 turns the winner is whichever team has the most CityTiles on the map. +If that is a tie, then whichever team has the most units owned on the board wins. If still a tie, the +game is marked as a tie. A game may end early if a team no longer has any more Units or CityTiles. +Then the other team wins. +A.2 +ADDITIONAL IMPLEMENTATION DETAILS +Detailed information on our policy implementation is illustrated in this section, including feature +engineering, network design, and reinforcement learning algorithm implementation. +PPO implementation. Standard PPO loss (Schulman et al., 2017) is used as the policy loss to +optimize the policy net and to estimate the advantage, we apply GAE (Schulman et al., 2016) with +the trajectory length as 32. For Phase 1 training with dense rewards, we set the GAE parameter +λ = 0.95, the discount factor γ = 0.99. For Phase 2 and 3 training with sparse rewards, we set +λ = 1, γ = 1. We apply PPO2 with a clip operation and set the clipping ratio ϵ = 0.1. Mean +squared loss is used to optimize the critic’s value head and the weight of the value loss is set as 0.5. +The entropy loss is also computed to encourage exploration with a coefficient of 0.1. We use Adam +(Kingma & Ba, 2014) as the optimizer with the learning rate 1e − 4. For computational resources, +we use an NVIDIA-V100 GPU and 600 CPU cores. +Input. Input features fed into our policy network consists of two parts: 1) the vector containing +global information such as timestep and total fuel. Features in the global information vector and +their corresponding data specifications are listed in Table 4. 2) the image input containing the map +information and the locations of our own and enemy’s Workers and CityTiles are listed in Table 5. +Dense Reward Design. The detailed dense reward design is listed in Table 6. Four types of rewards +are given for specific behaviors of CityTiles and Workers. The total reward is the sum of four sub- +rewards. +Data Preprocessing. The whole data preprocessing procedure is illustrated in Figure 12. The global +information vector is split into two parts, the one-hot features of dimension 51 and the other features +of dimension 18. The one-hot features are fed into a linear layer with an output dimension of 9 +for embedding. For unifying input into image shapes, the embedding vector of length 9 and the +other global information vector of length 18 are expanded as the image sizes in separate channels, +where every pixel in each channel is of the same value. After that, these expanded image features +(9×H ×W and 18×H ×W, H and W are the map height and width) along with the original image +15 + +Published at Deep RL Workshop, NeurIPS 2022 +Table 4: Input features Part 1: global information vector. +Feature Description +Type +Range +Normalization Coefficient +Current Cycle +One-Hot +[0,8] +N/A +Current Turn In This Cycle +One-Hot +[0,39] +N/A +If At Night +One-Hot +[0,1] +N/A +Own CityTile Numbers +Int +[0,1024] +100 +Enemy CityTile Numbers +Int +[0,1024] +100 +Own Unit Numbers +Int +[0,1024] +100 +Enemy Unit Numbers +Int +[0,1024] +100 +Own Research Points +Int +[0,200] +200 +Enemy Research Points +Int +[0,200] +200 +Own Total Fuel +Int +[0,106] +2300 +Enemy Total Fuel +Int +[0,106] +2300 +Own Average Fuel Per CityTile +Float +[0,104] +230 +Enemy Average Fuel Per CityTile +Float +[0,104] +230 +Own Total Fuel Cost +Int +[0,105] +230 +Enemy Total Fuel Cost +Int +[0,105] +230 +Own Average Fuel Cost Per CityTile +Float +[0,23] +23 +Enemy Average Fuel Cost Per CityTile +Float +[0,23] +23 +If Own Team Can Collect Coal +Bool +[0,1] +N/A +If Enemy Team Can Collect Coal +Bool +[0,1] +N/A +If Own Team Can Collect Uranium +Bool +[0,1] +N/A +If Enemy Team Can Collect Uranium +Bool +[0,1] +N/A +features ( 37×H ×W) are passed through separate convolutional neural networks with a kernel size +of 1. Then three parts of input images are concatenated together in the channel dimension. Through +another convolutional layer with a kernel size of 1, input channels as 64, and output channels as 128, +the tensors as the ResNet backbone input is in shape 128 × H × W. +Figure 12: The Data Preprocessing Procedure. Global information is split into two parts, one-hot +and others. One-hot vector is first embedded through a fully-connected layer and then expands as the +map size. After passing the 1 × 1 Conv2D, one-hot and other features are concatenated in channels, +going through another 1 × 1 Conv2D. The global features and map features are concatenated with +another 1 × 1 Conv2D with input channels as 64 and output channels as 128. +16 + +Global information vector +Map-feature +(69) +(37 * H * W) +人 +One-hot (51) +Others (18) +FC +Embedding (9) +Expand +Expand +1×1 Conv2D (9 * H * W) +1×1 Conv2D (18 * H * W) +Concatenate (27 * H * W) +1×1 Conv2D (27 *H *W) +Concatenate (64 * H * W) +1×1 Conv2D (128 * H * W) +OutputPublished at Deep RL Workshop, NeurIPS 2022 +Table 5: Input features Part 2: image features. +Feature Description +Type +Normalization Coefficient +If No Worker +Bool +N/A +If Own Worker +Bool +N/A +If Enemy Worker +Bool +N/A +If No Cart +Bool +N/A +If Own Cart +Bool +N/A +If Enemy Cart +Bool +N/A +If No CityTile +Bool +N/A +If Own CityTile +Bool +N/A +If Enemy CityTile +Bool +N/A +Road Level +Float +6 +Worker Cooldown +Float +10 +If Worker Can Act +Bool +N/A +Cart Cooldown +Float +10 +If Cart Can Act +Bool +N/A +CityTile Cooldown +Float +10 +If CityTile Can Act +Bool +N/A +If It is Resource +Bool +N/A +Wood Amount +Int +100 +If Wood Can Regrow +Bool +N/A +Coal Amount +Int +100 +Uranium Amount +Int +100 +Worker Wood Carry Amount +Int +100 +Worker Coal Carry Amount +Int +100 +Worker Uranium Carry Amount +Int +100 +If Worker Reaches Carry Limit +Bool +N/A +Cart Wood Carry Amount +Int +100 +Cart Coal Carry Amount +Int +100 +Cart Uranium Carry Amount +Int +100 +CityTile Fuel Cost +Int +100 +CityTile Average Fuel +Float +230 +If CityTile Can Survive Tonight +Bool +N/A +Fuel CityTile Needed to Survive +Int +230 +If Worker is at CityTile +Bool +N/A +If Cart is at CityTile +Bool +N/A +X Relative Distance to Center +Float +N/A +Y Relative Distance to Center +Float +N/A +Table 6: Design details of Phase 1: dense rewards. +Units +Behaviors +Weights +CityTiles +Research Points Increases +0.01 +Units Built +0.5 +Workers +Fuel Increases +0.0001 +CityTiles Built +1 +ResNet Backbone. The ResNet backbone consists of 8 Residual blocks. Each Residual block +comprises two convolutional layers and a Squeeze-and-Excitation(SE) layer. The detailed structure +of the Residual Block is shown in Figure 13. +Output. After the ResNet backbone, we use multiple heads for the output actions and value. The +learned representation from the ResNet backbone is first passed through a Spectral Normalization +layer. For the action head, we use three separate heads for the Workers, Carts and CityTiles. Each +head is a convolutional layer with kernel size as 1 and output channels as the corresponding action +17 + +Published at Deep RL Workshop, NeurIPS 2022 +Figure 13: Residual Block Design and Squeeze-and-Excitation Layer. Each Residual block con- +sists two convolutional layers (kernel size = 5, padding = 2, stride = 1) and LeakyReLU as the +activation function. Squeeze-and-Excitation(SE) layer consists of a 2D Average Pooling, and two +fully-connected layers. +dimensions (19 for Worker, 17 for Cart and 4 for CityTile). For the critic’s head, we use an Average +Pooling to transform the representation of size 128 × H × W to a vector of length 128. Then we +use a fully-connective layer to get a single value for the critic’s estimation. +Figure 14: Output actions and value. First through a spectral normalization layer, three action +heads, and a value head are appended. 1 × 1 Conv2D is used for output actions of Worker, Cart, and +CityTile. AvgPool2D and a fully-connected layer are used for value estimation. +Valid Action Mask. Valid action mask is a common technique in reinforcement learning to elim- +inate unnecessary explorations and accelerate the learning process. We calculate the valid action +mask based on the following rules: +• Workers: All actions are invalid when cooldown > 1. When cooldown < 1, moving to +enemy CityTiles or tiles with other Workers on it is invalid; moving to friendly CityTile is +always valid; and building a CityTile is valid only when its resource achieves 100. Doing +nothing is always valid. +• CityTiles: All actions are invalid when cooldown > 1. When cooldown < 1, building a +Worker is valid when the number of workers are less than the number of CityTiles; research +is valid when the team’s research point < 200. Doing nothing is always valid. +A.3 +DECENTRALIZED POLICY IMPLEMENTATION AND ABLATION STUDIES +In this section, detailed information on our decentralized policy implementation is described, in- +cluding input features, network design, and rule-based agents. +18 + +SE Block +Residual Block +Input (128 * H * W) +Input +(128 *H * W) +5×5 Conv2D +AvgPool2D +(128) +FC +(8) +LReLU +ReLU +(8) +5×5Conv2D +FC +(128) +LReLU +Sigmoid +(128) +SE Block +Expand +(128* H * W) +X +LReLU +ReLU +Output (128* H * W) +Output +(128* H * W)Representation +(128 * H * W) +Spectral Norm +(128 * H * W) +1×1 Conv2D +1×1 Conv2D +1×1 Conv2D +AvgPool2D +(19* H *W) +(17 * H *W) +(4* H* W) +(128) +Cart Actions +CityTile Actions +Value +Worker ActionsPublished at Deep RL Workshop, NeurIPS 2022 +Table 7: Decentralized policy input: global information. +Feature Description +Type +Range +Number of Agents Observed +Int +320 +Global CityTile Number +One-hot +[0,320] +Global Unit Number +One-hot +[0,320] +Current Cycle +One-hot +[0,9] +Current Turn in this cycle +One-hot +[0,39] +If At Night +Bool +N/A +Own Research Point +Int +[0,200] +If Own Team Can Collect Coal +Bool +N/A +If Own Team Can Collect Uranium +Bool +N/A +Table 8: Decentralized policy input: self information. +Feature Description +Type +Range +Location X +Int +[0,31] +Location Y +Int +[0,31] +Location X +One-Hot +[0,31] +Location Y +One-Hot +[0,31] +Type +Bool +N/A +If At City +Bool +N/A +Alive +Bool +[N/A +Cooldown +One-Hot +[0,9] +If At Night +Bool +N/A +Wood Carry Amount +Int +[0,100] +Coal Carry Amount +Int +[0,100] +Uranium Carry Amount +Int +[0,100] +Wood Carry Amount +One-Hot +[0,100] +Coal Carry Amount +One-Hot +[0,100] +Uranium Carry Amount +One-Hot +[0,100] +Fuel +Int +[0,4000] +Fuel +One-Hot +[0,4000] +Input. The input of our decentralized policy can be divided into four parts: global information +(listed in Table 7), self information (listed in Table 8), other agents (team and enemy) information +(listed in Table 9) and map information (listed in Table 10). +Reward Shaping. Each agent receives three types of rewards: 1) Its own reward to encourage cer- +tain behaviors such as survival, building cities, collecting resources, and fueling cities. 2) CityTile +reward. Though the CityTile is a rule-based reward, this reward is used to guide the Workers’ behav- +ior to support the CityTiles. 3) Team reward. It consists of a final win reward and average reward of +the team to encourage cooperation among agents. +Network Design. The input is split into seven parts, i.e., global features, self features, friend Worker +features, friend CityTile features, enemy Worker features, enemy CityTile features, and image fea- +tures. For the former six vector features, we use six different two-layer fully-connected networks for +feature extraction. And for those features involving multiple units, we perform max pooling along +units. For the image features, we use three convolutional layers and flatten the learned representa- +tions. Then those representations are concatenated together and passed through two fully-connected +layers for the actions and values. +Rule-based Agents. For simplicity, we only use Worker as reinforcement learning agents and +CityTiles as rule-based agents. The decision rules of CityTiles are simple and intuitive: 1) Build +a Worker if a CityTile can. 2) If it cannot build a Worker and the team’s research point < 200, +research. 3) Do nothing otherwise. +19 + +Published at Deep RL Workshop, NeurIPS 2022 +Table 9: Decentralized policy input: other agents’ information. +Other Agents +Feature Description +Type +Range +Own Worker × 160 +Is Friend +Bool +N/A +Location X +Int +[0,31] +Location Y +Int +[0,31] +Distance +Int +[0,62] +If At City +Bool +N/A +Cooldown +One-Hot +[0,3] +Wood Carry Amount +Int +[0,100] +Coal Carry Amount +Int +[0,100] +Uranium Carry Amount +Int +[0,100] +Own CityTile × 160 +Is Friend +Bool +N/A +Location X +Int +[0,31] +Location Y +Int +[0,31] +Distance +Int +[0,62] +Cooldown +One-Hot +[0,9] +Average Fuel Per CityTile +Float +[0,2300] +Fuel Cost Per Night +Int +[0,23] +If Can Survive Tonight +Bool +N/A +Fuel Needed to Survive Tonight +Int +[0,230] +Enemy Worker × 160 +Is Friend +Bool +N/A +Location X +Int +[0,31] +Location Y +Int +[0,31] +Distance +Int +[0,62] +If At City +Bool +N/A +Cooldown +One-Hot +[0,3] +Wood Carry Amount +Int +[0,100] +Coal Carry Amount +Int +[0,100] +Uranium Carry Amount +Int +[0,100] +Enemy CityTile × 160 +Is Friend +Bool +N/A +Location X +Int +[0,31] +Location Y +Int +[0,31] +Distance +Int +[0,62] +Cooldown +One-Hot +[0,9] +Average Fuel Per CityTile +Float +[0,2300] +Fuel Cost Per Night +Int +[0,23] +If Can Survive Tonight +Bool +N/A +Fuel Needed to Survive Tonight +Int +[0,230] +Compared with Decentralized Control. To illustrate our pixel-to-pixel centralized control solu- +tion, we perform a comparative experiment with the decentralized control solution. Competing with +the decentralized control solution, the centralized control solution achieves 98% win rate comput- +ing by 100 runs. As shown in Figure 16, the decentralized policy can acquire basic skills such +as collecting and building CityTiles. Encouraged by the team-based reward, decentralized agents +even acquired a basic level of regional cooperation. However, since the cooperation is induced by +pre-engineered rewards, it can only be applied to special scenarios and cannot be extended to other +complex situations. For example, in a local map with woods, the decentralized agents are at an +advantage initially, but due to their cooperation lacking adaptivity, our agents gradually build cities +surrounding them and limit their development to gain the advantage. As a result, at Turn 120, the +centralized policy has taken control of every resource on the map. Moreover, more group strategies +emerged from the evolution of the centralized policy, for instance, being aggressive in sending some +Workers to occupy and protect the key resources from its opponent. +20 + +Published at Deep RL Workshop, NeurIPS 2022 +Figure 15: Decentralized policy net architecture. For the global and self features, we use two +fully-connected layers for extraction. For unit features and CityTile features, we use two linear +layers and then apply MaxPooling along the units. Three convolutional layers with a flattened and +linear layer are used for map features. Then the outputs are concatenated together, using two fully- +connected layers for the output actions and value. +Figure 16: One episode between the decentralized and centralized policy. Yellow is the decen- +tralized policy, and Blue is the centralized policy. In a local battle, Blue is at an advantage at first, +but with better coordination, Yellow turns things around within just 30 turns. +21 + +Unit feature +Global feature +Self feature +Citytile feature +Map feature +(17 *15*15) +(698) +(405) +(9 * 160 Units) +(9 * 160 Cities) +Padding = 1 +FC + ReLU +FC + ReLU +FC + ReLU +4×4 Conv2D +FC + ReLU +(128) +(128) +(128 * 160 Units) +(128 * 160 Cities) +(32 *14 *14) +ReLU +Stride = 2 +4×4 Conv2D +FC + ReLU +FC + ReLU +FC + ReLU +FC + ReLU +(64) +(64) +(64 * 160 Units) +(64 * 160 Cities) +(32 *7*7) +ReLU +3×3 Conv2D +Stride = 2 +MaxPool +MaxPool +(64) +(16*4*4) +(64) +ReLU +Flatten + FC +(128) +FC + ReLU +(512) +FC + ReLU +(128) +FC +FC +(7) + (1) +Value +ActionsTurn 100 +Turn 115 +Decentralized +Centralized +Turn 130 +AdvantagePublished at Deep RL Workshop, NeurIPS 2022 +Table 10: Decentralized policy input: map information. +Map Information +Feature Description +Type +Range +Resource Map +Is Wood Here +Bool +N/A +Wood Reserves +Int +[0,1000] +Is Coal Here +Bool +N/A +Coal Reserves +Int +[0,1000] +Is Uranium Here +Bool +N/A +Uranium Reserves +Int +[0,1000] +Worker Map +Is Friend Worker +Bool +N/A +Is Enemy Worker +Bool +N/A +Worker Cooldown +Int +[0,3] +Worker Wood Carry Amount +Int +[0,100] +Worker Coal Carry Amount +Int +[0,100] +Worker Uranium Carry Amount +Int +[0,100] +City Map +Is Friend CityTile +Bool +N/A +Is Enemy CityTile +Bool +N/A +CityTile Cooldown +Int +[0,9] +Average Fuel Per CityTile +Float +[0,2300] +Fuel Cost Per Night +Int +[0,23] +If Can Survive Tonight +Bool +N/A +Fuel Needed to Survive Tonight +Int +[0,230] +Road Level +Bool +N/A +Table 11: Decentralized policy reward design. +Reward Type +Feature Description +Weights +Worker Reward +Worker Death Penalty +−1 +Worker Survive One Night Turn +0.05 +Worker Survive Ten Night Turns +0.5 +Worker Build a CityTile +1 per CityTile +Built City Fuel Saving +0.05 × (23−Fuel Cost) +Worker Fuel Increase +0.005 per fuel +Worker Fuel Donation +0.01 per fuel +CityTile Reward +CityTile Death penalty +−1 +CityTile Survive One Night Turn +0.05 +CityTile Survive Ten Night Turns +0.5 +CityTile Research Point Increase +0.02 per point +Research Point reaches 50 +1 +Research Point reaches 200 +4 +CityTile Build a Worker +1 +Team Reward +Final Win Reward +100 +Team Average Reward +0.1× average of friend reward +A.4 +MORE GENERALIZATION STUDIES +In section 6.2, we demonstrate the generalization of our model by transferring the policy trained +on maps of size 12 to size 32. More studies are conducted to further investigate the generalization +ability of our proposed model on larger maps. Results show that even transferred to larger maps, our +model still retains a surprising ability of massive-agent coordination. +We use the model trained on 32 × 32 maps as the base model and evaluate it on different map +sizes without fine-tuning. On maps of sizes 48 and 64, our policy shows the fantastic mastery of +massive-agent coordination as shown in Figure 17. +22 + +Published at Deep RL Workshop, NeurIPS 2022 +Figure 17: Policy transfer on maps of size 64. There are 1069 CityTiles and 1059 units for the +orange team, and 779 CityTiles and 768 units for the blue team. In larger maps, our policy demon- +strates the generalization ability of coordination between thousands of agents. +Figure 18: Policy transfer on maps of size 128. The large map and the cooldown mechanism +limit the ability to build large cities fulfilling the map. However, our policy still exhibits skills and +strategies they acquire on 32 × 32 maps. This large-scale setting eventually causes the web viewer +unresponsive. +23 + + Page Unresponsive +WARNINGS (2) +Relav ersions +You can wait for it to become responsive or exit the page. +Lux Al Challenge Viewer +Tile Properties +Wait +Exit page +(83, 67) +General +otal City Tiles +80759 +.0/6.0 +City Growth +Turn 27 +Debug ModePublished at Deep RL Workshop, NeurIPS 2022 +Furthermore, we make a bold attempt on the 128 × 128 maps. However, due to the cooldown +mechanism, agents can hardly travel across the map within 360 turns, which makes it impossible to +build large cities fulfilling the map like they do in 32 × 32 maps. Moreover, in a larger map, the +environment simulation is much slower, which takes about 30 minutes for one episode. It indicates +that although Lux has the scalability for millions of agents, the game core and the dynamics need a +lot of modification to adapt to larger scales. Nevertheless, we find our agents still exhibit skills and +strategies they acquire on 32 × 32 maps as in Figure 18, which demonstrates the potential of our +method at a million-agent scale. +24 + diff --git a/itAzT4oBgHgl3EQfpP1U/content/tmp_files/load_file.txt b/itAzT4oBgHgl3EQfpP1U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ff2831b719c91f424f75ac3144f0693c7622985 --- /dev/null +++ b/itAzT4oBgHgl3EQfpP1U/content/tmp_files/load_file.txt @@ -0,0 +1,1093 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf,len=1092 +page_content='Published at Deep RL Workshop, NeurIPS 2022 EMERGENT COLLECTIVE INTELLIGENCE FROM MASSIVE-AGENT COOPERATION AND COMPETITION Hanmo Chen1,*,‡, Stone Tao2,∗, Jiaxin Chen3,†, Weihan Shen3, Xihui Li1,‡, Sikai Cheng4,‡, Xiaolong Zhu3, Xiu Li1 1Tsinghua University, Shenzhen International Graduate School, 2University of California, San Diego 3Parametrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='ai, 4Georgia Institute of Technology {chm20,xh-li21}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='cn, stao@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='edu {jiaxinchen,weihanshen,xiaolongzhu}@chaocanshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='ai scheng326@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='cn, li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='xiu@sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='cn ABSTRACT Inspired by organisms evolving through cooperation and competition between dif- ferent populations on Earth, we study the emergence of artificial collective intel- ligence through massive-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To this end, We propose a new massive-agent reinforcement learning environment, Lux, where dynamic and massive agents in two teams scramble for limited resources and fight off the darkness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Lux, we build our agents through the standard reinforcement learn- ing algorithm in curriculum learning phases and leverage centralized control via a pixel-to-pixel policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As agents co-evolve through self-play, we observe several stages of intelligence, from the acquisition of atomic skills to the develop- ment of group strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Since these learned group strategies arise from individ- ual decisions without an explicit coordination mechanism, we claim that artificial collective intelligence emerges from massive-agent cooperation and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We further analyze the emergence of various learned strategies through metrics and ablation studies, aiming to provide insights for reinforcement learning imple- mentations in massive-agent environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 1 INTRODUCTION Complex group and social behaviors widely exist in humans and animals on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In a vast ecosys- tem, the simultaneous cooperation and competition between populations and the changing environ- ment serve as a natural driving force for the co-evolution of massive numbers of organisms (Wolpert & Tumer, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Dawkins & Krebs, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This large-scale co-evolution between populations has enabled group strategies for tasks individuals cannot accomplish (Ha & Tang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Inspired by this self-organizing mechanism in nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', collective intelligence emerges from massive-agent coop- eration and competition, we propose to simulate the emergence of collective intelligence through training reinforcement learning agents in a massive-agent environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We hope this can become a stepping stone toward massive-agent reinforcement learning research and an inspiring method for complex massive-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Recent progress in multi-agent reinforcement learning (MARL) demonstrates its potential to com- plete complex tasks through multi-agent cooperation, such as playing StarCraft2 (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019) and DOTA2 (Berner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, the number of agents is still limited to dozens in those scenarios, far away from natural populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To support large-scale multi-agent cooperation and competition, we reintroduce the massive-agent setting into multi-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To this end, we propose Lux, a cooperative and competitive environment where hundreds of agents in two populations scramble for limited resources and fight off the darkness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We believe Lux is a suitable testbench for experimenting with collective intelligence because it provides an open envi- ronment for hundreds of agents to cooperate, compete and evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' ∗Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' ‡Work done as research intern at Parametrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='01609v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='AI] 4 Jan 2023 Published at Deep RL Workshop, NeurIPS 2022 From the algorithmic perspective, the massive-agent setting poses great difficulties to reinforcement learning algorithms since the credit assignment problem becomes increasingly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Some research (Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) focuses on the credit assignment problem between multi-agents, how- ever, it lacks the scalability to massive-agent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To overcome that, we present a centralized control solution for Lux using a pixel-to-pixel modeling architecture (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019) coupled with Proximal Policy Optimization (PPO) (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Using that solution, we avoid the problem of credit assignment, with up to a 90% win rate versus the state-of-the-art policy (Isaiah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2021) proposed by the Toad Brigade team (TB) which won first place in the Lux AI competition on Kaggle1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Through self-play and curriculum learning phases, we observe several stages of the massive-agent co-evolution, from atomic skills such as moving and building to group strategies such as efficient territory occupation and long-term resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Note that group strategies arise from indi- vidual decisions without any explicit coordination mechanism or hierarchy, demonstrating how col- lective intelligence arises with co-evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Through quantitative analyses, further evidence shows that collective intelligence can emerge from massive-agent cooperation and competition, leading to behaviors beyond our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, agents learn to stand in a diagonal row and move as a whole to segment off parts of the map as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Without any prior knowledge, this efficient strategy emerges from spontaneous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Furthermore, we perform a detailed ablation study to illustrate some implementation techniques which may be helpful in massive-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (a) Blue is our policy and Yellow is TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (b) Yellow is our policy and Blue is TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 1: Two episodes between our policy and TB where our Workers stand in a diagonal row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Our agents discover it as an efficient way to expand the territory and limit the enemy’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Our main contributions are 1) we reintroduce massive-agent reinforcement learning as a scenario for studying collective intelligence and propose a new environment, Lux, as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) we provide evidence that collective intelligence emerges from co-evolution through massive agents’ cooperation and competition in Lux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3) we discuss the implementation details of our solution, which may provide valuable insights into massive-agent reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2 RELATED WORK Multi-Agent Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Many environments such as Multi-agent Particle Environment (MPE) (Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) and Google Research Football (Kurach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2020) are proposed to study multi- agent cooperation and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For multi-agent cooperation, StarCraft Multi-Agent Challenge (SMAC) (Samvelyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019) provides a common testbench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, SMAC focuses on de- centralized micromanagement scenarios with only approximately 30 agents in play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In massive- agent environments, Neural MMO (Suarez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2021) is an open-ended Massively Multi-player Online (MMO) game environment with up to 1024 agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' MAgent (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2018) is a grid world environment that supports up to a million agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We propose Lux, a massive-agent rein- forcement learning environment, which can support thousands of agents simultaneously acting at one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Unlike previous massive-agent environments, Lux incorporates Real-Time-Strategy (RTS) game dynamics that are similar to Battlecode (2022) and MiniRTS (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, Lux scales up the number of agents with frequent spawns and deaths, which opens up the potential for complex strategies in such a large-scale and highly dynamic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='com/c/lux-ai-2021/ 2 Published at Deep RL Workshop, NeurIPS 2022 Credit Assignment in MARL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Credit assignment between agents (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2003) is a crucial challenge in multi-agent cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Several value-based multi-agent algorithms (Sunehag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Rashid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Iqbal & Sha, 2018)) decompose global value into individual values using a linear model or neural network, which can be viewed as an implicit way of credit assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Another way of doing this is computing an agent-specific advantage function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, Foerster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2017) uses counterfactual regret to measure contributions to the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In complex games, Berner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2019) and Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2020) use hand-crafted team-based rewards for each agent as an explicit method of credit assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Compared to the implicit value decomposition method, this explicit reward-shaping method requires prior domain knowledge and lacks generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, both of them are limited to small population scenarios and are hard to scale to massive and dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2019) handles this problem using grid-wise centralized policy instead of decentralized policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' It uses a convolutional neural network to map from pixel-wise observations to actions over each pixel, which avoids the credit assignment problem while achieving efficient multi- agent collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Following this, we adapt this pixel-to-pixel architecture to the Lux environment with the PPO algorithm (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) and curriculum learning phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Collective Intelligence and Emergence Behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Collective Intelligence, including self- organization and emergent behaviors (Wolpert & Tumer, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Woolley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2010), has a long history connected with biological and economic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Research on emergent behavior usually emphasizes that group strategies emerge from multi-agent co-evolution in a designed environment rather than hand-crafted collaboration mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2019) uses reinforcement learn- ing agents and autocurricula (Leibo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019) in the Hide-and-seek environment, leading to the emergence of tool use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2018) proposes using million-agent reinforcement learning to study how the agents’ grouping behaviors will change with the environmental resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2021) uses a two-level, deep RL framework to train agents and a social planner in an eco- nomic environment, where optimal taxation policy emerges as the result of co-adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Johanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2022) studies the emergence of bartering behavior in a microeconomics-based environment with producers and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Dynamics in those environments usually induce agents’ behaviors within human comprehension, thus limiting the possible emergent strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Since RTS games pro- vide a perfect Petri dish for collective intelligence, our study absorbs RTS game dynamics into the environment where simple rules may induce complex group strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3 LUX Like the Earth, a suitable environment for collective intelligence to evolve must support massive agents’ competition and cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For that purpose, we propose an open-sourced environment Lux, where hundreds of agents in two teams compete for resources and build cities as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 2: A snapshot of Lux with hundreds of agents in two teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers can collect resources and build CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At night, CityTiles and Workers need fuel to stay alive and will be consumed by darkness if fuel runs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The team that owns more cities wins in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3 Resources Uranium Coal Tree Agents Workers CityTilesPublished at Deep RL Workshop, NeurIPS 2022 Setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The map is a 2D square grid of size 12 to 32, scattered with different resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' An episode consists of 360 turns, split into 9 Day/Night cycles of 30 days and 10 nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There are two basic units named Worker and CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each team starts with one Worker and one CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers can collect resources and build CityTiles and CityTiles can also build new Workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers collect adjacent resources automatically and convert resources to fuel when standing upon a friendly CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At night, CityTiles and Workers consume fuel to stay alive and will be consumed by darkness if the fuel runs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At the end of the game, the team with more CityTiles wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' More details about the environment are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Observation and Action Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For observation, each team has perfect information about the game state, including the global map, its own, and the opponent’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For actions, each team needs to make decisions for every Worker and CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A Worker can move in 4 cardinal directions and build a CityTile when it has enough resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers however cannot move onto a tile with an enemy CityTile or Worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A CityTile can build a Worker or research to increase the team’s research points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Sufficient research points unlock the ability for the team’s Workers to mine high-level resources that convert to more fuel like coal and uranium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' MARL in Lux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For MARL research, Lux raises a challenging situation for multi-agent modeling and the credit assignment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Distinguished from other environments, the number of agents in Lux is massive and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For a 32 × 32 map, the number of agents in a team can rise to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, Workers and CityTiles are built and lost all the time, bringing difficulty for multi- agent modeling of dynamic agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A carefully-designed credit assignment scheme may be useful in small-scale problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' however, with massive and dynamic agents, it becomes impractical due to the combinatorial complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Furthermore, the win-or-lose sparse reward throws another challenge on the hard exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' RTS in Lux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At first sight, Lux seems like a pocket-sized RTS game like StarCraft2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Agents in Lux need to balance economic decisions and individual control, which requires high-level coordination between hundreds of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, the major difference between Lux and RTS games is the way of controlling units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In RTS games, the low-level unit actions are executed by fixed rules, which allows human players or AI to focus on macro-strategies and economic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Lux, atomic actions such as moving and building are all controlled by the learned policy, resulting in an action space of approximately 10180, magnitudes beyond StarCraft2 (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Thus, a successful policy needs to learn atomic skills and group strategies together, which is significant in the emergence of collective intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 4 METHODOLOGY Overall, our policy is trained using the standard algorithm PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) with Gen- eralized Advantage Estimation (GAE) (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For massive-agent coordination, we use a pixel-to-pixel architecture as the centralized policy (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Isaiah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2021), tak- ing both observations and actions as images and using the ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2016) structure as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To address the sparse reward problem, we design three phases with different rewards as a progressive curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For clarity, we refer to the “agent” as each unit on the map and refer to the “policy” as the centralized policy network that controls every agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 PIXEL-TO-PIXEL ARCHITECTURE We model the massive-agent control problem using a centralized policy with a pixel-to-pixel archi- tecture (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The policy network takes images as input observations and outputs actions over each pixel in the form of an action map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' More implementation details are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Policy Network Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The architecture of our policy network is pictured in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The input image (C ×H ×W) consists of C channels containing information about itself, the opponent, and the global state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We use a ResNet-style convolutional network as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For actions, we use a convolutional layer with kernel size 1 and output channels as the action dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, we use a flattened layer and a fully-connected layer for the value estimation as the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A valid action mask is used to eliminate unnecessary exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 4 Published at Deep RL Workshop, NeurIPS 2022 Figure 3: Policy network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' C is the input channels and H, W denote the map height and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' E is the feature map channel through the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Output channels AWorker/CityTile are the corresponding action dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Why Centralized Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Lux, our policy needs to control hundreds of agents each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' While the decentralized policy in MARL is computationally efficient and easy to scale, it needs a carefully-designed credit assignment mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Lux, however, as agents are massive and dy- namic, the credit assignment problem becomes increasingly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To avoid that, we adopt a centralized policy controlling every agent over the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This pixel-to-pixel architecture with a con- volutional network leverages the advantage of centralized and decentralized methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Convolutional layers work as a parameter-sharing mechanism across agents, similar to shared policy networks in decentralized methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This parameter-sharing mechanism improves learning efficiency via data reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Furthermore, the deep stacked structure provides a large receptive field for global information extraction and multi-agent communication, which naturally avoids the trouble of credit assignment (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 CURRICULUM TRAINING PHASES The objective of agents in Lux is to own more CityTiles than the opponent, but the final result only provides a sparse reward (1 for win, −1 for lose), resulting in the hard exploration problem (Badia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Reward shaping is a common method to handle this problem in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, hand-crafted rewards can easily direct agents into specific behavioral patterns with limited strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Hence, we design three phases with different rewards as a progressive curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' First, we use a dense reward to guide the policy towards basic skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then we gradually reduce the learning signals and utilize the sparse reward to encourage the policy to explore more diversified strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 1: Dense Rewards for Basic Skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At first, we use hand-crafted dense rewards to encourage basic skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Specifically, four kinds of behaviors are given rewards, namely, the increase of Workers and CityTiles, Research Points and fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' More details are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 2: Sparse Reward with Scaled Signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Phase 2, a reward is given only when an episode ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, our policy still needs guidance through long-term reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We modify the reward with a slight signal about the win condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', ± � |Nself − Nop|, where Nself and Nop denote the number of our own and the enemy’s CityTiles, encouraging to own more CityTiles for the win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 3: Win-or-Lose Sparse Reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The win-or-lose sparse reward (1 for win and −1 for lose) is applied in the final phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After human-designed guidance in the first two phases, the win-or-lose sparse reward encourages our policy to explore more advanced strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 5 EMERGENT COLLECTIVE INTELLIGENCE Through massive-agent cooperation and competition, we have observed three stages of our agents’ evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Training from scratch, agents quickly acquire atomic skills such as collecting resources and building cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After around 5 million episodes, an elementary level of coordination appears on the regional scale with dozens of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=" As training proceeds, the coordination expands from re- 5 Action Dimension Input Representation Output E×H×W C×HxW EXI Conv Resources Residual Valid'Mask Blocks *8 (k=1) A Softmax [0000 Worker." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 10100 0000 Workers Worker 0001 Actions CityTiles Softmax 0000 10100 CityTile 0000 CityTile Lo011] Global-Time FC + Expand ActionsPublished at Deep RL Workshop, NeurIPS 2022 gional to global scope, which includes long-term economic decisions and precise control of hundreds of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Those global strategies naturally arise from individual decisions due to massive-agent in- teraction and co-evolution without any explicit coordination mechanism, signifying the emergence of collective intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 ATOMIC SKILLS The first step of our agents is to get a grasp of atomic skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Guided by dense rewards, Workers learn to move toward resources to collect fuel, and build and fuel the CityTiles, as shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, at this stage agents are more likely to work alone and unable to make group decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, Workers tend to build more CityTiles than they can support, leading to a sudden loss of large cities as illustrated in Figure 4b as they run out of fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (a) Workers collect resources and build CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (b) CityTiles run out of fuel and collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 4: Illustration of atomic skills in a self-play episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Agents acquire atomic skills such as collecting resources and building CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, due to a lack of group coordination, CityTiles often burn out fuel and collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 REGIONAL COORDINATION As training proceeds, regional coordination appears, which involves dozens of agents in a local area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, agents learn to carefully choose locations before building a CityTile and develop self- organizing patterns for occupying resources efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We describe a few examples of regional strategies: Construction Planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As CityTiles built next to each other can share fuel and reduce cost at night, agents gradually learn that the locations of CityTiles are important in city survival and fuel saving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We find that agents discover several patterns of construction planning as visualized in Figure 5: 1) build CityTiles near the resources for quicker access to fuel sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) build CityTiles in a long row to form cities that act like the Great Wall to prevent enemies’ aggression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3) build CityTiles in blocks to reduce fuel costs at night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (a) CityTiles built near resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (b) CityTiles built in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (c) CityTiles built in blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 5: Three emergent patterns of construction planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' a) build near resources for quick access to fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' b) build in a row as the Great Wall for defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' c) build in blocks to save fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We use the city survival ratio (the final number of CityTiles divided by the most number of CityTiles in one episode) to measure how these building patterns work quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As shown in Figure 6a, the regional-scale construction planning effectively helps CityTiles fight off the darkness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Territory Division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We have also observed a self-organizing structure where several Workers stand in a diagonal row shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Those Workers simultaneously move forward and backward 6 3Published at Deep RL Workshop, NeurIPS 2022 (a) City survival ratio: final number divided by the most number of CityTiles in one episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (b) Five-diagonal: how many times five or more Workers stand in a diagonal row in one episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 6: Quantitative analysis of region coordination using city survival ratio and five- diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Both metrics are evaluated using self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' a) The city survival ratio increases as training continues, indicating that the regional construction planning effectively helps CityTiles live long and prosper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' b) The frequency of the five-diagonal shape increases during the course of training, which demonstrates the gradual acquisition of this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' as a whole to keep the formation, and when any of them die, a new Worker nearby will fill in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In this shape, they can effectively guard and expand the team’s territory and limit the enemy’s movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We measure a statistic called Five-Diagonal (how many times five or more Workers stand in a diagonal row in one game) to investigate how often this strategy is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Results in Figure 6b illustrate that the frequency our agents use this strategy generally increases with training in the long term, indicating it is an acquired strategy rather than a circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3 GLOBAL STRATEGIES As in micro-management scenarios of SMAC (Samvelyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2019), regional coordination of dozens of agents is often found in multi-agent cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, our agents go far beyond that, achieving much larger-scale coordination between hundreds of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We provide interpretation and analysis of several global strategies as follows: Sustainable Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A key component in Lux is the balance of city development and re- source consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In the early stages, the rapidly growing cities often face severe fuel shortages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Gradually, our policy learns to develop cities at a sustainable speed in tune with fuel production depending on the resource distribution and the opponent’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Another phenomenon we have observed is the retention of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As trees are the only renewable resource in Lux, forest protection is significant in securing long-term fuel supplies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Our agents intentionally preserve trees from exces- sive deforestation and build CityTiles near the woods in defense of the enemy’s aggression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Another metric, total wood collect (the total collected woods divided by originally spawned woods) is used to measure how this forest protection strategy influences our fuel supplies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Results in Figure 7a show how these protection strategies significantly improve the utilization efficiency of wood, resulting in our agent collecting more than 500% of the original wood on the map at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (a) Total wood collect: total wood collected di- vided by originally spawn woods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (b) Total fuel: total fuel storage in one episode using our latest model in a fixed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 7: Quantitative analysis of global strategies using total wood collect and total fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The metrics are evaluated using self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' a) The utilization efficiency of wood increases as our policy grows the sense of forest protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' b) In one episode, our fuel storage accumulates until turn 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After that, it tries to build more CityTiles for the win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' All In For The Win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Another surprising strategy is that when the episode is about to end, our policy will rapidly harvest all the protected trees and try to build as many CityTiles as possible for the win 7 Phase 1 Phase 2 Phase 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 Survival 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 City 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 0 2 3 4 5 6 7 8 9 1 10 11 12 13 14 15 16 17 18 19 # Episodes (× 106)Phase 1 Phase 2 Phase 3 7 6 5 Diagonal 4 D Five 1 0 0 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 # Episodes (× 106)Phase 1 Phase 2 Phase 3 5 Collect 4 Wood 3 Total 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 # Episodes (× 106)60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 Fuel 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='000 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 # TurnsPublished at Deep RL Workshop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' NeurIPS 2022 as shown in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Furthermore, we observe that sometimes cities retain very little fuel at the end of an episode, evidence that almost all the resources have been fully utilized, as any resources left at the end would be a waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Efficiently using all remaining resources before the end is very challenging because it needs the overall calculation of total fuel consumption by all Workers and CityTiles, in addition to precise control of every agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We think this strategy perfectly demonstrates the emergence of collective intelligence through the combination of long-term economic decisions and massive-agent mobilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 6 EXPERIMENTS In this section, we perform ablation studies to reflect on our policy implementation and general reinforcement learning algorithms under massive-agent settings: 1) we investigate the necessity of curriculum learning phases by training with different procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Results demonstrate that our cur- riculum design can help tackle the hard exploration problem caused by sparse rewards in the early stages of training and encourage the emergence of complex strategies beyond human design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) we further demonstrate the generalization ability of our model across different map sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' When eval- uating on maps of size 32, the policy trained on size 12 still retains some basic strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After a fine-tuning phase of only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='8 million episodes, the transferred policy achieves a 90% win rate against TB on maps of size 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The results indicate our model can learn generalizable representations suit- able for the environment through learned spatial structures via convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3) we compare our centralized policy against a standard decentralized solution with carefully-designed team-based rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The centralized policy achieves a 98% win rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' See implementation details in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The decentralized policy implementation and experiment are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 DESIGN OF CURRICULUM LEARNING PHASES We perform experiments to investigate the necessity of our curriculum learning phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Five differ- ent procedures are applied: a) Only Phase 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' b) Phase 1 and 2 without Phase 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' c) Phase 1 and 3, without Phase 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' d) Phase 1, 2, and 3 (the original procedure);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' e) Only Phase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 8: The win rate curves from different training phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' All win rates are evaluated against TB on maps of size 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Training with direct sparse rewards results in a 0% win rate, while training only with Phase 1 dense rewards converges at around 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Compared to Phase 1+2, Phase 1+3 improves slower and results in a lower win rate of around 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 1+2 achieves an 85% win- rate, and Phase 1+2+3 further boosts it to above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 1 utilizes dense rewards which are fundamental for helping the centralized policy acquire atomic skills for individual agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As shown in Figure 8, our policy can hardly learn any basic skills when directly training with sparse rewards, resulting in a win rate of around zero due to the hard exploration problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 2 utilizes a scaled sparse reward which plays two roles in the whole learning procedure, accelerating learning and improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' First, continuous learning with dense rewards converges at a 50% win rate, but the win rate rapidly rises to 70% with Phase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' On the other hand, without Phase 2, switching from Phase 1 to Phase 3 is more challenging with a lower performance even after training for a longer period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This shows that the scaled sparse reward can work as a proper 8 Phase 1+2 Phase 1+2+3 Phase 1+3 Phase 3 Only Phase 1 100% %06 80% Phase 3 from 2 Phase 2 70% 100% rate 60% 50% 90% Win 40% 30% %08 20% 10% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 非 Episodes (× 10°) 0% 5 8 10 11 12 13 14 15 16 17 1819 0 1 2 3 4 7 9 6 非 Episodes (× 106)Published at Deep RL Workshop, NeurIPS 2022 transition between dense rewards and a win-or-loss sparse reward (applied in Phase 3) as it explicitly tells the policy that owning more cities is the key to winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Phase 3 utilizes a sparse win-loss reward which further boosts the final performance to above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As the Phase 2 training converges to a win rate of 85% without Phase 3, the win-loss sparse reward pushes our policy to go further and explore, resulting in an overall 90% win rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 GENERALIZATION OF REPRESENTATIONS FOR REINFORCEMENT LEARNING We provide clear evidence that the learned representations from the convolutional neural network and reinforcement learning algorithms can be generalized to different map sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' First, we directly transfer the policy net trained on maps of size 12 to size 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As shown in Figure 9, basic skills are retained on larger maps such as Workers collecting and fueling cities, even showcasing some structured city construction planning to surround and protect wood resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Secondly, after an ad- ditional fine-tuning phase of around 1 million episodes, the policy quickly adapts to larger maps and achieves an overall 90% win rate against TB, while training from scratch uses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 million episodes for a 20% win rate as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 9: Illustration of the general- ization ability in a self-play episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' When the policy trained on maps of size 12 is directly transferred to 32, some strategies are retained such as Workers building and fueling cities with plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 10: The win-rate curves on 32 × 32 maps of training from scratch and transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Both are evalu- ated against TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After a fine-tuning phase of 1 million steps, the transferred policy achieves a 90% win rate, while training from scratch uses 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 million episodes for a 20% win rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The results demonstrate the generalization ability of our model, which provides insights into speed- ing up policy training on large maps: we can pre-train our policy on small maps and then transfer it to large maps with a fine-tuning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This procedure significantly reduces the training time because smaller maps are faster for environment simulation and network update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, the Lux environment simulation on CPU is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5× slower on maps of size 32 than size 12, and the policy network update on GPU is 5× slower on maps of size 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As training on maps of size 12 is both time-saving and computationally efficient, our “Pre-train and Fine-tune” scheme achieves a higher win rate with fewer training hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 7 DISCUSSION AND FUTURE WORK We have demonstrated that collective intelligence can emerge from massive-agent cooperation and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As proof of concept, we propose Lux, an environment hosting hundreds of agents and incorporating RTS game dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Through standard reinforcement learning algorithms and pixel- to-pixel centralized modeling, we observe several stages of agents’ strategy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Our agents exhibit ambitious group strategies based on accurate individual control of massive agents without explicit coordination mechanisms, signifying the emergence of collective intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We hope our work with Lux will be a stepping stone toward artificial collective intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Lux, we observe the number of agents can reach up to 2000 in a single timestep, but this still pales in comparison to the millions or even billions of organisms cooperating and competing in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The Lux environment can be easily extended to host more agents as the experiments in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4, but simulation and inference become extremely slow reaching the million-agent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Going forward, the environment design and engineering as well as the training algorithm need a lot of 9 02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2Phase Learning from Scratch - Learning from Transfer Model 100% 90% 80% 70% rate 60% 50% u!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N 40% 30% 20% 10% 0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='7 # Episodes (× 106)Published at Deep RL Workshop, NeurIPS 2022 modifications to adapt to such a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We also acknowledge that the RTS game dynamics in Lux may not directly coincide with real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, with Lux as a blueprint, economic rules and dynamics like Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' (2021) can be incorporated, which may provide some reference for economic decisions and policies in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' REFERENCES Adri`a Puigdom`enech Badia, Pablo Sprechmann, Alex Vitvitskyi, Zhaohan Daniel Guo, Bilal Piot, Steven Kapturowski, Olivier Tieleman, Mart´ın Arjovsky, Alexander Pritzel, Andrew Bolt, and Charles Blundell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Never give up: Learning directed exploration strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Stephan Zheng, Alexander Trott, Sunil Srinivasa, David C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Parkes, and Richard Socher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The AI economist: Optimal economic policy design via two-level deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CoRR, abs/2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='02755, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='org/abs/2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='02755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1 DETAILED RULES OF LUX For ease of understanding, the environment rules including unit types and action spaces are simpli- fied in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In this part, we provide a detailed description of the environment design and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The version of Lux we use is compatible with the version on Kaggle Lux AI S1 competition2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The rules can also be found at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='lux-ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='org/specs-2021 and the following text is a reformatted and slightly modified version of the original rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The night is dark and full of terrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Two teams must fight off the darkness, collect resources, and advance through the ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Daytime finds a desperate rush to gather and build the resources to carry you through the impending night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Plan and expand carefully – any city that fails to produce enough light will be consumed by darkness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In the Lux AI Challenge Season 1, two competing teams control a team of Units and CityTiles that collect resources to fuel their Cities, with the main objective to own as many CityTiles as possible at the end of the turn-based game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Both teams have complete information about the entire game state and use that information to optimize resource collection, compete for scarce resources against the opponent, and build cities to gain points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each competitor must program their policy in their language of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each turn, your agent gets 3 seconds to submit their actions, excess time is not saved across turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In each game, you are given a pool of 60 seconds that is tapped into each time you go over a turn’s 3-second limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Upon using up all 60 seconds and going over the 3-second limit, your agent freezes and can no longer submit additional actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The world of Lux is represented as a 2D grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Coordinates increase east (right) and south (down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The map is always a square and can be 12, 16, 24, or 32 tiles long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The (0, 0) coordinate is at the top left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 11: The specification of the map in Lux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The map has various features including Resources (Wood, Coal, Uranium), Units (Workers, Carts), CityTiles, and Roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In order to prevent maps from favoring one player over another, it is guaranteed that maps are always symmetric by vertical or horizontal reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each player will start with a single CityTile and a single Worker on that CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There are 3 kinds of resources: Wood, Coal, and Uranium (in order of increasing fuel efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' These resources are collected by Workers, then dropped off once a Worker moves on 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='com/c/lux-ai-2021/ 12 (0,0) (0,y) (x,0) (x,y)Published at Deep RL Workshop, NeurIPS 2022 Table 1: The specifications of resource collection and convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Resource Type Research Points Pre-requisite Fuel Value per Unit Units Collected per Turn Wood 0 1 20 Coal 50 10 5 Uranium 200 40 2 top of a CityTile to then be converted into fuel for the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Some resources require research points before they are possible to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Wood in particular can regrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each turn, every wood tile’s wood amount increases by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5% of its current wood amount rounded up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Wood tiles that have been depleted will not regrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Only wood tiles with less than 500 wood will regrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Collection Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At the end of each turn, Workers automatically receive resources from all adjacent (North, East, South, West, or Center) resource tiles they can collect resources from according to the current symmetric formula: Iterating over uranium, coal, then wood resources: Each unit makes resource collection requests to collect an even number of resources from each adjacent tile of the current iterated resource such that the collected amount takes the unit’s cargo above capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Worker with 60 wood adjacent to 3 wood tiles asks for 14 from each, receives 40 wood, and wastes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' All tiles of the current iterated resource then try to fulfill requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If they can’t, they make sure all unfulfilled requests get an equal amount, and the leftover is wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' if 4 Workers are mining a tile of 25 wood, but one of them is only asking for 5 while the others are asking for 20 wood each, then first all Workers get 5 wood each, leaving 5 wood left over for 3 more Workers with space left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This can be evenly distributed by giving 1 wood each to the last 3 Workers, leaving 2 wood left that is then wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers cannot mine while on CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Instead, if there is at least one Worker on a CityTile, that CityTile will automatically collect adjacent resources at the same rate as a Worker each turn and directly convert it all to fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The collection mechanic for a CityTile is the same as a Worker and you can treat a CityTile as an individual Worker collecting resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units and CityTiles can perform actions each turn given certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In general, all actions are simultaneously applied and are validated against the state of the game at the start of a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The next few sections describe the Units and CityTiles in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A CityTile is a building that takes up one tile of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Adjacent CityTiles collectively form a City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each CityTile can perform a single action provided the CityTile has a Cooldown < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Actions: Build Worker - Build Worker unit on top of this CityTile (cannot build a Worker if the current number of owned Workers + carts equals the number of owned CityTiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Build Cart - Build Carts unit on top of this CityTile (cannot build a cart if the current number of owned Workers + carts equals the number of owned CityTiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Research - Increase your team’s Research Points by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There are two unit types, Workers, and Carts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Every unit can perform a single action once they have a Cooldown < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' All units can choose the move action and move in any of the 5 direc- tions, North, East, South, West, or Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, all units can carry raw resources gained from automatic mining or resource transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers are capped at 100 units of resources and Carts are capped at 2000 units of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Whenever a unit moves on top of a friendly CityTile, the City that CityTile forms converts all carried resources into fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There can be at most one unit on tiles without a CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, units cannot move on top of the opposing team’s CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, units can stack on top of each other on a friendly CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If two units attempt to move to the same tile that is not a CityTile, this is considered a collision, and the move action is canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 13 Published at Deep RL Workshop, NeurIPS 2022 Table 2: The specifications of cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Unit Type Base Cooldown CityTile 10 Worker 2 Cart 3 Workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Actions: Move - Move the unit in one of 5 directions, North, East, South, West, or Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Pillage - Reduce the Road level of the tile the unit is on by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Transfer - Send any amount of a single resource-type from a unit’s cargo to another (start- of-turn) adjacent Unit, up to the latter’s cargo capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Excess is returned to the original unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Build CityTile - Build a CityTile right under this Worker, provided the Worker has 100 total resources of any type in their cargo (full cargo), and the tile is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If the building is successful, all carried resources are consumed, and a new CityTile is built with 0 starting resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Carts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Actions: Move - Move the unit in one of 5 directions, North, East, South, West, Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Transfer - Send any amount of a single resource-type from a unit’s cargo to another (start- of-turn) adjacent Unit, up to the latter’s cargo capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Excess is returned to the original unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CityTiles, Workers, and Carts all have a cooldown mechanic after each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units and CityTiles can only act when they have Cooldown < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At the end of each turn, after Road has been built and pillaged, each unit’s Cooldown decreases by 1 and decreases by the level of the Road the unit is on at the end of the turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CityTiles are not affected by road levels, and cooldown always decreases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The minimum Cooldown is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After an action is performed, the unit’s Cooldown will increase by a Base Cooldown, as specified in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Roads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As Carts travel across the map, they start to create roads that allow all Units to move faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' At the end of each turn, Cart will upgrade the road level of the tile it ends on by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The higher the road level, the faster Units can move and perform actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' All tiles start with a road level of 0 and are capped at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, CityTiles automatically have a max road level of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers can also destroy roads via the pillage action which reduces road levels by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If a City is consumed by darkness, the road level of all tiles in the City’s CityTiles will go back to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Day/Night Cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The Day/Night cycle consists of a 40-turn cycle, the first 30 turns being day turns, the last 10 being night turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There are 360 turns in a match, forming 9 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' During the night, Units and Cities need to produce light to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each turn of the night, each Unit and CityTile will consume an amount of fuel, see Table 3 for rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units in particular will use their carried resources to produce light whereas CityTiles will use their fuel to produce light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Workers and Carts will only need to consume resources if they are not on a CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' When outside the City, Workers and Carts must consume whole units of resources to satisfy their night needs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' if a Worker carries 1 wood and 5 uranium on them, they will consume a full wood for 1 fuel, then a full unit of uranium to fulfill the last 3 fuel requirements, wasting 37 fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units will always consume the least efficient resources first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Lastly, at night, Units gain 2× more Base Cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Should any Unit during the night run out of fuel, they will be removed from the game and disappear into the night forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Should a City run out of fuel, however, the entire City with all of the CityTiles it owns will fall into darkness and be removed from the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Game Resolution order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To help avoid confusion over smaller details of how each turn is resolved, we provide the game resolution order here and how actions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Actions in the game are first all validated against the current game state to see if they are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then the actions, along with game events, are resolved in the following order and simultaneously within each step: 14 Published at Deep RL Workshop, NeurIPS 2022 Table 3: The specifications of file burn, nadj denotes the number of adjacent friendly CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Unit Type Fuel Burn in City Fuel Burn Outside City CityTile 23 − 5 × nadj N/A Worker 0 10 Cart 0 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CityTile actions along with increased cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Unit actions along with increased cooldown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Roads are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Resource collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Resource drops on CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If night time, make Units consume resources and CityTiles consume fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Regrow wood tiles that are not depleted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Cooldowns are handled/computed for each unit and CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The only exception to the validation criteria is that units may move smoothly between spaces, mean- ing if two units are adjacent, they can swap places in one turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Otherwise, actions such as one unit building a CityTile, then another unit moving on top of the new CityTile, are not allowed as the current state does not have this newly built city and units cannot move on top of other units outside of CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Win Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After 360 turns the winner is whichever team has the most CityTiles on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If that is a tie, then whichever team has the most units owned on the board wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' If still a tie, the game is marked as a tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A game may end early if a team no longer has any more Units or CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then the other team wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2 ADDITIONAL IMPLEMENTATION DETAILS Detailed information on our policy implementation is illustrated in this section, including feature engineering, network design, and reinforcement learning algorithm implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' PPO implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Standard PPO loss (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2017) is used as the policy loss to optimize the policy net and to estimate the advantage, we apply GAE (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', 2016) with the trajectory length as 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For Phase 1 training with dense rewards, we set the GAE parameter λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='95, the discount factor γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For Phase 2 and 3 training with sparse rewards, we set λ = 1, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We apply PPO2 with a clip operation and set the clipping ratio ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Mean squared loss is used to optimize the critic’s value head and the weight of the value loss is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The entropy loss is also computed to encourage exploration with a coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We use Adam (Kingma & Ba, 2014) as the optimizer with the learning rate 1e − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For computational resources, we use an NVIDIA-V100 GPU and 600 CPU cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Input features fed into our policy network consists of two parts: 1) the vector containing global information such as timestep and total fuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Features in the global information vector and their corresponding data specifications are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) the image input containing the map information and the locations of our own and enemy’s Workers and CityTiles are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Dense Reward Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The detailed dense reward design is listed in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Four types of rewards are given for specific behaviors of CityTiles and Workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The total reward is the sum of four sub- rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Data Preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The whole data preprocessing procedure is illustrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The global information vector is split into two parts, the one-hot features of dimension 51 and the other features of dimension 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The one-hot features are fed into a linear layer with an output dimension of 9 for embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For unifying input into image shapes, the embedding vector of length 9 and the other global information vector of length 18 are expanded as the image sizes in separate channels, where every pixel in each channel is of the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After that, these expanded image features (9×H ×W and 18×H ×W, H and W are the map height and width) along with the original image 15 Published at Deep RL Workshop, NeurIPS 2022 Table 4: Input features Part 1: global information vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Feature Description Type Range Normalization Coefficient Current Cycle One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='8] N/A Current Turn In This Cycle One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='39] N/A If At Night One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1] N/A Own CityTile Numbers Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1024] 100 Enemy CityTile Numbers Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1024] 100 Own Unit Numbers Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1024] 100 Enemy Unit Numbers Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1024] 100 Own Research Points Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='200] 200 Enemy Research Points Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='200] 200 Own Total Fuel Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='106] 2300 Enemy Total Fuel Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='106] 2300 Own Average Fuel Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='104] 230 Enemy Average Fuel Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='104] 230 Own Total Fuel Cost Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='105] 230 Enemy Total Fuel Cost Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='105] 230 Own Average Fuel Cost Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='23] 23 Enemy Average Fuel Cost Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='23] 23 If Own Team Can Collect Coal Bool [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1] N/A If Enemy Team Can Collect Coal Bool [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1] N/A If Own Team Can Collect Uranium Bool [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1] N/A If Enemy Team Can Collect Uranium Bool [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1] N/A features ( 37×H ×W) are passed through separate convolutional neural networks with a kernel size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then three parts of input images are concatenated together in the channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Through another convolutional layer with a kernel size of 1, input channels as 64, and output channels as 128, the tensors as the ResNet backbone input is in shape 128 × H × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 12: The Data Preprocessing Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Global information is split into two parts, one-hot and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' One-hot vector is first embedded through a fully-connected layer and then expands as the map size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After passing the 1 × 1 Conv2D, one-hot and other features are concatenated in channels, going through another 1 × 1 Conv2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The global features and map features are concatenated with another 1 × 1 Conv2D with input channels as 64 and output channels as 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 16 Global information vector Map-feature (69) (37 * H * W) 人 One-hot (51) Others (18) FC Embedding (9) Expand Expand 1×1 Conv2D (9 * H * W) 1×1 Conv2D (18 * H * W) Concatenate (27 * H * W) 1×1 Conv2D (27 *H *W) Concatenate (64 * H * W) 1×1 Conv2D (128 * H * W) OutputPublished at Deep RL Workshop, NeurIPS 2022 Table 5: Input features Part 2: image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Feature Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Normalization Coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If No Worker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Own Worker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Enemy Worker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If No Cart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Own Cart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Enemy Cart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If No CityTile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Own CityTile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Enemy CityTile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Road Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Worker Cooldown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Worker Can Act ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Cart Cooldown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Cart Can Act ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='CityTile Cooldown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If CityTile Can Act ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If It is Resource ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Wood Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Wood Can Regrow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Coal Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Uranium Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Worker Wood Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Worker Coal Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Worker Uranium Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Worker Reaches Carry Limit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Cart Wood Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Cart Coal Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Cart Uranium Carry Amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='CityTile Fuel Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='CityTile Average Fuel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If CityTile Can Survive Tonight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Fuel CityTile Needed to Survive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Int ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Worker is at CityTile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='If Cart is at CityTile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Bool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='X Relative Distance to Center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Y Relative Distance to Center ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Float ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Table 6: Design details of Phase 1: dense rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Units Behaviors Weights CityTiles Research Points Increases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='01 Units Built 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 Workers Fuel Increases 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0001 CityTiles Built 1 ResNet Backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The ResNet backbone consists of 8 Residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each Residual block comprises two convolutional layers and a Squeeze-and-Excitation(SE) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The detailed structure of the Residual Block is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' After the ResNet backbone, we use multiple heads for the output actions and value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The learned representation from the ResNet backbone is first passed through a Spectral Normalization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For the action head, we use three separate heads for the Workers, Carts and CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each head is a convolutional layer with kernel size as 1 and output channels as the corresponding action 17 Published at Deep RL Workshop, NeurIPS 2022 Figure 13: Residual Block Design and Squeeze-and-Excitation Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each Residual block con- sists two convolutional layers (kernel size = 5, padding = 2, stride = 1) and LeakyReLU as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Squeeze-and-Excitation(SE) layer consists of a 2D Average Pooling, and two fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' dimensions (19 for Worker, 17 for Cart and 4 for CityTile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For the critic’s head, we use an Average Pooling to transform the representation of size 128 × H × W to a vector of length 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then we use a fully-connective layer to get a single value for the critic’s estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 14: Output actions and value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' First through a spectral normalization layer, three action heads, and a value head are appended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 1 × 1 Conv2D is used for output actions of Worker, Cart, and CityTile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' AvgPool2D and a fully-connected layer are used for value estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Valid Action Mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Valid action mask is a common technique in reinforcement learning to elim- inate unnecessary explorations and accelerate the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We calculate the valid action mask based on the following rules: Workers: All actions are invalid when cooldown > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' When cooldown < 1, moving to enemy CityTiles or tiles with other Workers on it is invalid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' moving to friendly CityTile is always valid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' and building a CityTile is valid only when its resource achieves 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Doing nothing is always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' CityTiles: All actions are invalid when cooldown > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' When cooldown < 1, building a Worker is valid when the number of workers are less than the number of CityTiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' research is valid when the team’s research point < 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Doing nothing is always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3 DECENTRALIZED POLICY IMPLEMENTATION AND ABLATION STUDIES In this section, detailed information on our decentralized policy implementation is described, in- cluding input features, network design, and rule-based agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 18 SE Block Residual Block Input (128 * H * W) Input (128 *H * W) 5×5 Conv2D AvgPool2D (128) FC (8) LReLU ReLU (8) 5×5Conv2D FC (128) LReLU Sigmoid (128) SE Block Expand (128* H * W) X LReLU ReLU Output (128* H * W) Output (128* H * W)Representation (128 * H * W) Spectral Norm (128 * H * W) 1×1 Conv2D 1×1 Conv2D 1×1 Conv2D AvgPool2D (19* H *W) (17 * H *W) (4* H* W) (128) Cart Actions CityTile Actions Value Worker ActionsPublished at Deep RL Workshop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' NeurIPS 2022 Table 7: Decentralized policy input: global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Feature Description Type Range Number of Agents Observed Int 320 Global CityTile Number One-hot [0,320] Global Unit Number One-hot [0,320] Current Cycle One-hot [0,9] Current Turn in this cycle One-hot [0,39] If At Night Bool N/A Own Research Point Int [0,200] If Own Team Can Collect Coal Bool N/A If Own Team Can Collect Uranium Bool N/A Table 8: Decentralized policy input: self information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Feature Description Type Range Location X Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location X One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Type Bool N/A If At City Bool N/A Alive Bool [N/A Cooldown One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='9] If At Night Bool N/A Wood Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Coal Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Uranium Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Wood Carry Amount One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Coal Carry Amount One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Uranium Carry Amount One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Fuel Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4000] Fuel One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4000] Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The input of our decentralized policy can be divided into four parts: global information (listed in Table 7), self information (listed in Table 8), other agents (team and enemy) information (listed in Table 9) and map information (listed in Table 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Reward Shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Each agent receives three types of rewards: 1) Its own reward to encourage cer- tain behaviors such as survival, building cities, collecting resources, and fueling cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) CityTile reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Though the CityTile is a rule-based reward, this reward is used to guide the Workers’ behav- ior to support the CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3) Team reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' It consists of a final win reward and average reward of the team to encourage cooperation among agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Network Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The input is split into seven parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=', global features, self features, friend Worker features, friend CityTile features, enemy Worker features, enemy CityTile features, and image fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For the former six vector features, we use six different two-layer fully-connected networks for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' And for those features involving multiple units, we perform max pooling along units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For the image features, we use three convolutional layers and flatten the learned representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then those representations are concatenated together and passed through two fully-connected layers for the actions and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Rule-based Agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For simplicity, we only use Worker as reinforcement learning agents and CityTiles as rule-based agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The decision rules of CityTiles are simple and intuitive: 1) Build a Worker if a CityTile can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 2) If it cannot build a Worker and the team’s research point < 200, research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 3) Do nothing otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 19 Published at Deep RL Workshop, NeurIPS 2022 Table 9: Decentralized policy input: other agents’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Other Agents Feature Description Type Range Own Worker × 160 Is Friend Bool N/A Location X Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Distance Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='62] If At City Bool N/A Cooldown One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3] Wood Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Coal Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Uranium Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Own CityTile × 160 Is Friend Bool N/A Location X Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Distance Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='62] Cooldown One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='9] Average Fuel Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2300] Fuel Cost Per Night Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='23] If Can Survive Tonight Bool N/A Fuel Needed to Survive Tonight Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='230] Enemy Worker × 160 Is Friend Bool N/A Location X Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Distance Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='62] If At City Bool N/A Cooldown One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3] Wood Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Coal Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Uranium Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Enemy CityTile × 160 Is Friend Bool N/A Location X Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Location Y Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='31] Distance Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='62] Cooldown One-Hot [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='9] Average Fuel Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2300] Fuel Cost Per Night Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='23] If Can Survive Tonight Bool N/A Fuel Needed to Survive Tonight Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='230] Compared with Decentralized Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' To illustrate our pixel-to-pixel centralized control solu- tion, we perform a comparative experiment with the decentralized control solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Competing with the decentralized control solution, the centralized control solution achieves 98% win rate comput- ing by 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As shown in Figure 16, the decentralized policy can acquire basic skills such as collecting and building CityTiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Encouraged by the team-based reward, decentralized agents even acquired a basic level of regional cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, since the cooperation is induced by pre-engineered rewards, it can only be applied to special scenarios and cannot be extended to other complex situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For example, in a local map with woods, the decentralized agents are at an advantage initially, but due to their cooperation lacking adaptivity, our agents gradually build cities surrounding them and limit their development to gain the advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' As a result, at Turn 120, the centralized policy has taken control of every resource on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, more group strategies emerged from the evolution of the centralized policy, for instance, being aggressive in sending some Workers to occupy and protect the key resources from its opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 20 Published at Deep RL Workshop, NeurIPS 2022 Figure 15: Decentralized policy net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For the global and self features, we use two fully-connected layers for extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' For unit features and CityTile features, we use two linear layers and then apply MaxPooling along the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Three convolutional layers with a flattened and linear layer are used for map features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Then the outputs are concatenated together, using two fully- connected layers for the output actions and value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 16: One episode between the decentralized and centralized policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Yellow is the decen- tralized policy, and Blue is the centralized policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In a local battle, Blue is at an advantage at first, but with better coordination, Yellow turns things around within just 30 turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Unit feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Global feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Self feature ' metadata={'source': 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Cities) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Padding = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4×4 Conv2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='MaxPool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(64) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(16*4*4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(64) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Flatten + FC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(512) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC + ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(128) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='FC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='ActionsTurn 100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Turn 115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Decentralized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Centralized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='Turn 130 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='AdvantagePublished at Deep RL Workshop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' NeurIPS 2022 Table 10: Decentralized policy input: map information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Map Information Feature Description Type Range Resource Map Is Wood Here Bool N/A Wood Reserves Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1000] Is Coal Here Bool N/A Coal Reserves Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1000] Is Uranium Here Bool N/A Uranium Reserves Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1000] Worker Map Is Friend Worker Bool N/A Is Enemy Worker Bool N/A Worker Cooldown Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='3] Worker Wood Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Worker Coal Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] Worker Uranium Carry Amount Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='100] City Map Is Friend CityTile Bool N/A Is Enemy CityTile Bool N/A CityTile Cooldown Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='9] Average Fuel Per CityTile Float [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2300] Fuel Cost Per Night Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='23] If Can Survive Tonight Bool N/A Fuel Needed to Survive Tonight Int [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='230] Road Level Bool N/A Table 11: Decentralized policy reward design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Reward Type Feature Description Weights Worker Reward Worker Death Penalty −1 Worker Survive One Night Turn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='05 Worker Survive Ten Night Turns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 Worker Build a CityTile 1 per CityTile Built City Fuel Saving 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='05 × (23−Fuel Cost) Worker Fuel Increase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='005 per fuel Worker Fuel Donation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='01 per fuel CityTile Reward CityTile Death penalty −1 CityTile Survive One Night Turn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='05 CityTile Survive Ten Night Turns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='5 CityTile Research Point Increase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='02 per point Research Point reaches 50 1 Research Point reaches 200 4 CityTile Build a Worker 1 Team Reward Final Win Reward 100 Team Average Reward 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='1× average of friend reward A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='4 MORE GENERALIZATION STUDIES In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='2, we demonstrate the generalization of our model by transferring the policy trained on maps of size 12 to size 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' More studies are conducted to further investigate the generalization ability of our proposed model on larger maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Results show that even transferred to larger maps, our model still retains a surprising ability of massive-agent coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' We use the model trained on 32 × 32 maps as the base model and evaluate it on different map sizes without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' On maps of sizes 48 and 64, our policy shows the fantastic mastery of massive-agent coordination as shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 22 Published at Deep RL Workshop, NeurIPS 2022 Figure 17: Policy transfer on maps of size 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' There are 1069 CityTiles and 1059 units for the orange team, and 779 CityTiles and 768 units for the blue team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' In larger maps, our policy demon- strates the generalization ability of coordination between thousands of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Figure 18: Policy transfer on maps of size 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' The large map and the cooldown mechanism limit the ability to build large cities fulfilling the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, our policy still exhibits skills and strategies they acquire on 32 × 32 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' This large-scale setting eventually causes the web viewer unresponsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' 23 Page Unresponsive WARNINGS (2) Relav ersions You can wait for it to become responsive or exit the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Lux Al Challenge Viewer Tile Properties Wait Exit page (83, 67) General otal City Tiles 80759 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content='0 City Growth Turn 27 Debug ModePublished at Deep RL Workshop, NeurIPS 2022 Furthermore, we make a bold attempt on the 128 × 128 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' However, due to the cooldown mechanism, agents can hardly travel across the map within 360 turns, which makes it impossible to build large cities fulfilling the map like they do in 32 × 32 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Moreover, in a larger map, the environment simulation is much slower, which takes about 30 minutes for one episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' It indicates that although Lux has the scalability for millions of agents, the game core and the dynamics need a lot of modification to adapt to larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfpP1U/content/2301.01609v1.pdf'} +page_content=' Nevertheless, we find our agents still exhibit skills and strategies they acquire on 32 × 32 maps as in Figure 18, which demonstrates the potential of our method at a million-agent scale.' metadata={'source': 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+1,1450 @@ +Robust Bayesian Target Value Optimization +Johannes G. Hoffer∗, Sascha Ranftl†, Bernhard C. Geiger‡ +January 12, 2023 +Abstract +We consider the problem of finding an input to a stochastic black +box function such that the scalar output of the black box function is as +close as possible to a target value in the sense of the expected squared +error. While the optimization of stochastic black boxes is classic in (ro- +bust) Bayesian optimization, the current approaches based on Gaussian +processes predominantly focus either on i) maximization/minimization +rather than target value optimization or ii) on the expectation, but not +the variance of the output, ignoring output variations due to stochasticity +in uncontrollable environmental variables. In this work, we fill this gap +and derive acquisition functions for common criteria such as the expected +improvement, the probability of improvement, and the lower confidence +bound, assuming that aleatoric effects are Gaussian with known variance. +Our experiments illustrate that this setting is compatible with certain +extensions of Gaussian processes, and show that the thus derived acquisi- +tion functions can outperform classical Bayesian optimization even if the +latter assumptions are violated. An industrial use case in billet forging is +presented. +1 +Introduction +Inverse problems, where one aims to find parameters of a system either explain- +ing or guaranteeing certain behavior, are ubiquitous in science and industry. +Consider, for example, process control in manufacturing, where a part or mate- +rial undergoes a specific manufacturing process characterized by tuneable design +variables. These design variables should be optimized such that the output of +the manufacturing process is as close as possible to a defined target. However, +manufacturing processes are often influenced by uncertainties of different kinds, +such as material imperfections, variation within process tolerances, seasonal ef- +fects, or limited accuracy for controlling process variables. These uncertainties, +∗voestalpine Böhler Aerospace GmbH & Co KG, Mariazellerstraße 25, Kapfenberg, Austria +†Institute of Theoretical Physics-Computational Physics, Graz University of Technology, +8010 Graz, Austria +‡Know-Center GmbH, Inffeldgasse 13, Graz, Austria +1 +arXiv:2301.04344v1 [cs.LG] 11 Jan 2023 + +which are often summarized as environmental effects, need to be taken into +account when solving inverse problems. +Standard Gaussian processes (GPs) are capable of solving inverse problems +under uncertainties, and may comprise distinct kinds of uncertainties [Ranftl +and von der Linden, 2021, Ranftl et al., 2020], e.g., aleatoric and epistemic +uncertainties. +Indeed, several acquisition functions have been proposed for +“noisy” [Gramacy and Lee, 2011, Huang et al., 2006, Letham et al., 2019, Picheny +et al., 2010], and “robust” [Kirschner et al., 2020, Bogunovic et al., 2018] +Bayesian optimization (BO), cf. Section 3. However, the majority of the previ- +ous works on GP-based BO does not distinguish between epistemic and aleatoric +uncertainties, which arise from finiteness of training data and stochasticity in +the relationship between input and output, respectively. Those works that do +either focus on maximization/minimization settings rather than on target value +optimization, or their understanding of robustness against aleatoric effects is +limited to optimizing the expected output of the black box function, ignoring +its variance due to aleatoric effects. +Indeed, while the acquisition functions +for target value optimization in [Uhrenholt and Jensen, 2019] are not robust +against aleatoric effects, the few works that simultaneously try to optimize the +expected output and minimize the output variance due to variations in the envi- +ronmental variables either focus on maximization/minimization [Nguyen et al., +2018, Iwazaki et al., 2021] or fail to fully exploit the mathematical peculiarities +of target value optimization [Hoffer et al., 2022]. +Thus, the literature exhibits a striking and practically relevant gap that this +work seeks to fill. Specifically, we will derive acquisition functions for robust +Bayesian target value optimization, with the aim of selecting design variables +such that the black box function output is close to a target value in the sense of +an expected squared error. This aim not only requires that the expected output +of the black box function is close to the target, but that also its variation due +to aleatoric effects is small. Essentially, our approach is based on a separation +between aleatoric (for evaluating the expected squared error) and epistemic (for +formulating the acquisition function) uncertainties. +We set out with the assumptions that the aleatoric effects are Gaussian with +known variance function and that they can be quantified separately from epis- +temic effects (Section 2), and we show in our experiments in Section 5 that +they approximately hold for certain practically relevant models based on GPs. +Based on these assumptions, we derive acquisition functions for target value +optimization in Section 4. Specifically, we show that by measuring the quality +of the optimization by the squared error expected due to aleatoric effects, that +the resulting acquisition functions can be computed in closed form and are remi- +niscent of those for noise-free target vector optimization [Uhrenholt and Jensen, +2019]. Using both synthetic and real-world examples, we show in Section 5 in +which cases our proposed acquisition functions outperform classical BO even +when some of our assumptions are violated. We summarize the insights from +these experiments and discuss limitations of our work in Section 6. +2 + +2 +Problem Definition +We consider the optimization of a black box function f: RD+A → R that maps +a vector x of D controllable design variables and a vector η of A uncontrollable +environmental variables to a scalar output y, i.e., y = f(x, η). Our aim is to +select the design variables x such that the output y is as close as possible to a +target value y• in the sense of an expected squared error, where the expectation +is taken over the unknown environmental variables. +Mathematically, we are +interested in finding a solution to +arg min +x Eη +�� +y• − f(x, η) +�2� +=: arg min +x E(x) +(1) +where Eη (·) denotes expectation w.r.t. η. In this setting, the environmental +variables η correspond to aleatoric effects that cannot be controlled by opti- +mization. Further, the setup is general enough to cover measurement errors +(y = f(x) + η), uncertain inputs to a black box function (y = f(x + η)), and +more complicated settings. +To formulate this optimization problem within the framework of BO and to +approach it using GPs, we will introduce three simplifying assumptions. +Assumption 1. The mapping from the design variables x to the output y +is Gaussian, i.e., we assume that it is fully characterized by a mean function +m(x) := Eη (f(x, η)) and a variance function σ2 +a(x) := Eη +�� +f(x, η) − m(x) +�2� +. +Assumption 1 allows us to derive analytic expressions for the acquisition +functions. +Further, under this assumption the optimization objective in (1) +simplifies to +E(x) = +� +y• − m(x) +�2 + σ2 +a(x). +(2) +Assumption 2. For all x, we can take noise-free measurements of the mean +function m(x), i.e., we can build a dataset D = +�� +xi, m(xi) +�� +i=1,...,N. +Assumption 2 essentially requires that we have access to measurements of the +black box function for which aleatoric effects have been averaged out. This can +be achieved by i) taking sufficiently many repeated measurements or simulations +with the same design variable, approximating the expectation Eη (f(x, η)) by an +average, or by ii) measuring (in addition to x and y) and subsequently marginal- +izing the (uncontrollable, but measurable) environmental effects η. This latter +approach is common in the literature on robust BO, where GP models are cre- +ated for f(x, η) and marginalized over η [Oliveira et al., 2019, Iwazaki et al., +2021, Toscano-Palmerin and Frazier, 2022]. In particular, [Ankenman et al., +2010] found that this is a good approximation for separating ’intrinsic’ and ’ex- +trinsic’ variances even with very low numbers of repetitions. Also, e.g., [Girard, +2004] assumes, based on GPs, either noise-free learning or noise-free inference, +albeit mainly due to the mathematical complexity of considering noise in both +simultaneously. As a consequence of Assumption 2, we can use a GP-based +3 + +estimate ˆm of m: RD → R which coincides with m exactly at all values of the +design variable xi that have been measured so far. +A real-world example for such kind of data could be manufacturing chains, +where adjusting the design parameters can be very costly while several repeti- +tions for a fixed design parameter set can be cheap. Another real-world example +are data from computer simulations, where the design parameters may be re- +lated to geometry descriptions, which require elaborate mesh (re-)constructions, +or scale parameters in stochastic simulations, which require (re-)tuning of algo- +rithmic parameters such as in Markov Chain Monte Carlo methods. +Assumption 3. We have full knowledge of σ2 +a: RD → R+ +0 . +This last requirement of a fully known aleatoric variance function seems +restrictive and appears to limit the practical utility of our approach. However, +while we will derive our acquisition functions based on this assumption, our +experiments in Section 5 will show that even with crude (learned or computed) +estimates ˆσ2 +a of σ2 +a our approach can outperform current alternatives to Bayesian +target value optimization. In the real-world examples mentioned before, such +estimates can usually be obtained with reasonable effort. +3 +Related Work +A plethora of acquisition functions based has been suggested, see e.g. [Shahri- +ari et al., 2015, Frazier, 2018, Brochu et al., 2010, Loredo, 2004, Preuss and +von Toussaint, 2021, Ramachandran et al., 2018] [Zhan and Xing, 2020, Zhou +et al., 2020] for an overview. Also the need to account for aleatoric effects in BO +has been long recognized and is known under “noisy” or “robust BO”, predom- +inantly for maximization/minimization settings (rather than for target value +optimization). For improvement-based acquisition functions such as expected +improvement (EI) or probability of improvement (PoI), the inherent stochastic- +ity of the mapping makes it difficult to judge which input xt in the dataset D +is the previous “best”. To account for this in a minimization setting, the au- +thors of [Gramacy and Lee, 2011] set the previous best input to arg minx µ(x), +where µ(x) is obtained by training a GP on D. Computing and optimizing the +resulting EI in a similar way as in the noiseless case was shown to lead to slow +convergence to the minimum of the mapping, cf. [Letham et al., 2019, Sec. 2.1]. +[Huang et al., 2006] instead select xt as the “effective best solution” and augment +EI by a multiplicative factor that favors inputs with large epistemic uncertainty. +[Letham et al., 2019] also consider noisy observations (and noisy constraints) +when training GPs for BO, but compute and optimize EI on the predictive +distribution for the noiseless mapping f(x), rather than for the noisy mapping +y(x) = f(x) + η. The authors of [Picheny et al., 2010] replace the expected +value µ(x) by the quantile of the GP’s predictive distribution, and compute +EI based on the improvement of this quantile. None of these works, however, +accounts for variations of the output due to aleatoric effects. Further, most of +the mentioned studies focus on noise at the output of a deterministic mapping, +4 + +i.e., y(x) = f(x) + η. Considering deterministic mappings with noisy inputs, +i.e., y(x) = f(x + η), [Girard, 2004, Girard et al., 2003] derive approximations +for a GP’s predictive posterior mean and variance. These approximations were +used in [Nguyen et al., 2018] to separate epistemic effects and aleatoric effects +propagated through the mapping to derive an acquisition function for the up- +per confidence bound (UCB) to stabilize BO in the sense of minimizing output +variations due to noisy inputs. +Robust BO more generally considers optimization in uncertain environments, +where in addition to controllable design variables x, the black box function de- +pends also on uncontrollable environmental variables η, i.e., y = f(x, η). In this +line of research, [Toscano-Palmerin and Frazier, 2022] optimize the expectation +of y over η, i.e., it maximizes the mean function m(x). [Iwazaki et al., 2021] +extend beyond [Toscano-Palmerin and Frazier, 2022] by proposing confidence +bound-based approaches to simultaneously maximize the expectation m(x) and +minimize the variance σ2 +a(x), cf. [Iwazaki et al., 2021, Fig. 1 and Sec. 3.2], +while [Kirschner et al., 2020] remain optimizing m(x) but relax the assumption +that the distribution of η is known. [Daulton et al., 2022] maximize the (multi- +variate) value-at-risk, which implicitly balances maximizing the expectation and +minimizing the variance. [Fröhlich et al., 2020] aims to maximize the expecta- +tion m(x) of y = f(x + η) using a variant of entropy search. [Nogueira et al., +2016] proposed to use the expectation of the unscented expected improvement +(UEI) acquisition function with respect to the input noise. +[Oliveira et al., +2019] aim to maximize a black box function with noise at the input and the +output. While they also consider noisy measurements during surrogate model- +ing and optimization, [Fröhlich et al., 2020, Nogueira et al., 2016, Beland and +Nair, 2017, Bogunovic et al., 2018] assume exact knowledge of the function in- +put during optimization and only aim to achieve robustness during deployment. +Thus, these works make assumptions similar to our Assumption 2, but focus on +maximization/minimization and do not study the peculiarities of target value +optimization in the light of (aleatoric) input uncertainty. +For target vector optimization, one aims at minimizing a distance d(x) = +d(y(x), y•), e.g., the squared Euclidean distance ∥y − y•∥2 +2, between the map- +ping’s output y and a target value or target vector y•.1 In the Gaussian case +and for a k-dimensional output vector y(x), a non-central χ2-distribution was +shown to be an unbiased estimate for the posterior p(d(x)|x, D) in [Uhrenholt +and Jensen, 2019]. For this setting, the authors derived acquisition functions +for EI and lower confidence bound (LCB) for both standard GP and warped +GP regression, cf. [Snelson et al., 2004]. Target value optimization for y• = 0 +was considered in [Osborne et al., 2009] for a stochastic mapping and additional +stochasticity in the input, but this work does not distinguish between epistemic +and aleatoric effects. The authors of [Pandita et al., 2016] acknowledge the dis- +tinct nature of aleatoric and epistemic uncertainty and quantify the epistemic +uncertainty of the optimal location, but they do not quantify or exploit aleatoric +1An entirely different approach was proposed in [Beland and Nair, 2017], where a GP +prior is imposed directly onto an optimality criterion or loss function, in contrast to the more +widely-spread procedure of imposing the GP prior on the function which is to be optimized. +5 + +uncertainty in their procedure. Also [Jeong and Shin, 2021] considered vector- +valued outputs and targets and arrived at a similar result as [Uhrenholt and +Jensen, 2019], albeit approaching the problem from a multi-objective perspec- +tive. They allowed for different weightings of the different objectives in contrast +to [Uhrenholt and Jensen, 2019], and, in contrast to this work, did not consider +robustness with respect to aleatoric effects in stochastic environmental variables. +[Astudillo and Frazier, 2022, Astudillo and Frazier, 2019] leveraged prior knowl- +edge on composite structures of the objective function similarly as it appears in +[Uhrenholt and Jensen, 2019], albeit again without considering aleatoric input +noise. To the best of our knowledge, the only approach that performs target +vector optimization and that considers the variance of the black box function +output due to aleatoric effects is [Hoffer et al., 2022], where the authors utilize +the UCB acquisition function from [Nguyen et al., 2018] and where the aleatoric +uncertainty is given by the mean function of a separate GP. Their approach, +however, ignores the fact that the squaring operation ensures that the posterior +of d(x) given x and the dataset D has non-negative support. +4 +Robust Bayesian Target Value Optimization Us- +ing Gaussian Processes +Suppose that at optimization iteration t we have access to a dataset D = +�� +xi, m(xi) +�� +, i = 1, . . . , t of noise-free measurements of the mean function +m, and that we approximate this mean function by a GP. We denote the predic- +tive posterior mean function of this GP as µ to distinguish it from the real mean +function m; the predictive posterior variance function of this GP, which esti- +mates the epistemic uncertainty resulting from the finite dataset D, is denoted +as σ2 +e: RD → R+ +0 . Thus, our estimate ˆm of m is given by +ˆm(x) ∼ N +� +µ(x), σ2 +e(x) +� +(3) +with +µ(x) = k(x) +� +K + σ2I +�−1 mT +(4a) +σ2 +e(x) = k(x, x) − k(x) +� +K + σ2I +�−1 kT (x) +(4b) +where σ2 is measurement noise, k: R2 → R denotes a kernel function, x = +(x1, . . . , xt) and m = +� +m(x1), . . . , m(xt) +� +are the vectorized dataset D, [K]ij = +k(xi, xj) is the kernel matrix, and where k(x) = +� +k(x1, x), . . . , k(xt, x) +� +denotes +the vector of covariances between x in D and the candidate input x. +Utilizing Assumption 2, i.e., the fact of being noise-free in our design of the +GP, we can set the hyperparameter σ2 to zero. As a consequence, for all xi ∈ D, +we have that the epistemic uncertainty σ2 +e(xi) = 0 [Rasmussen and Williams, +2006, eq. (2.19)], and that ˆm(xi) ≡ µ(xi) = m(xi). For general x, our surrogate +ˆy(x) of the black box function output y(x) is Gaussian with mean µ(x) and +6 + +variance σ2 +a(x) + σ2 +e(x). Summarizing, we have +y(x) = f(x, η) ∼ N +� +m(x), σ2 +a(x) +� +Assump. 1 +ˆm(x) ∼ N +� +µ(x), σ2 +e(x) +� +eq. (3) +∀xi ∈ D: σ2 +e(xi) = 0 +Assump. 2 +∀xi ∈ D: µ(xi) = m(xi) +Assump. 2 +ˆy(x) ∼ N +� +ˆm(x), σ2 +a(x) +� +Assump. 3 +⇒ ˆy(x) ∼ N +� +µ(x), σ2 +a(x) + σ2 +e(x) +� +The first line in this summary should denote that the output y of the black box +function f is Gaussian for given x, subsuming the stochasticity of environmental +variables η in f. +We now propose acquisition functions to optimize this surrogate. Specifi- +cally, and connecting to (1), we aim to select x such that ˆy(x) is as close as +possible to a target value y•. Specifically, we aim to find an input x such that +the expected squared error +ˆE(x) := +� +ˆm(x) − y•�2 + σ2 +a(x) +(6) +is minimized. Note that due to the epistemic uncertainty resulting from the +finite dataset D, ˆE(x) is a random variable unless x ∈ D. Thus, while E(x) +from (1) is deterministic since it is an expected value, ˆE(x) is estimated from the +GP surrogate of the black box function and is hence random due to the inherent +epistemic uncertainty. Indeed, for x /∈ D the normalized expected squared error +e(x) follows a non-central χ2-distribution: +e(x) := +ˆE(x) − σ2 +a(x) +σ2e(x) +∼ NCχ2� +K = 1, λ(x) +� +, +(7a) +with K = 1 degree of freedom and non-centrality parameter +λ(x) = +� +µ(x) − y•�2 +σ2e(x) +. +(7b) +For xi ∈ D, we have µ(xi) = m(xi) and σ2 +e(xi) = 0, hence ˆE(xi) = E(xi). +On the one hand, this justifies optimizing with the surrogate instead of the +black box function, On the other hand, it follows that for these inputs, ˆE(xi) is +deterministic. Further, the normalized expected squared error either evaluates +to e(xi) = 0 if m(xi) = y•, or to e(xi) = ∞ otherwise. +To prevent numerical issues, in our experiments we set the hyperparameter +σ2 in (4) to a small, positive number, yielding σ2 +e(x) > 0 for all x. If σ2 ≪ σ2 +a(x), +one still has ˆE(xi) ≈ E(xi) for all xi ∈ D and, thus, a valid surrogate model for +optimization. In the remainder of this section, however, we will stick to σ2 = 0 +and to Assumption 2. +7 + +4.1 +Improvement-Based Acquisition Functions +Let without loss of generality xt = minxi∈D ˆE(xi) be the input that minimizes +the expected squared error over all previous inputs. By Assumption 2, Emin := +ˆE(xt) is deterministic and evaluates to +Emin = ˆE(xt) = E(xt) = +� +m(xt) − y•�2 + σ2 +a(xt). +(8) +Probability of Improvement. +We can select xt+1 such that the PoI over +Emin is maximized, i.e., we solve +xt+1 = arg max +x +P +� +ˆE(x) ≤ Emin − ζ +� +(9) +where ζ ≥ 0 is a parameter that prefers larger, improbable improvements over +more probable, but small improvements. +Given that e(x) has a non-central +χ2-distribution with cumulative distribution function (CDF) F1,λ(x), the PoI is +given by +P +� +ˆE(x) ≤ Emin − ζ +� += P +� +e(x) ≤ Emin − ζ − σ2 +a(x) +σ2e(x) +� += F1,λ(x) +�� +m(xt) − y•�2 + σ2 +a(xt) − ζ − σ2 +a(x) +σ2e(x) +� +(10) +where λ(x) is given in (7b) and Emin was substituted by (8). +Expected Improvement. +We can also select xt+1 such that the EI over Emin +is maximized, i.e., we solve +xt+1 = arg max +x +E +� +max +� +0, Emin − ˆE(x) +�� +(11) +where E (·) takes the expectation w.r.t. epistemic uncertainty. By the linearity +of expectation and with emin(x) = +� +Emin − σ2 +a(x) +� +/σ2 +e(x) this can be rewritten +as +E +� +max +� +0, Emin − ˆE(x) +�� += σ2 +e(x)E +� +max +� +0, emin(x) − e(x) +�� += σ2 +e(x) +� emin(x) +0 +(emin(x) − e) f1,λ(x)(e)de +(12) +where f1,λ(x) is the probability density function of the non-central χ2-distribution. +Similarly as in [Uhrenholt and Jensen, 2019, Sec. 3.2], this integral can be com- +puted in closed form as +E +� +max +� +0, Emin − ˆE(x) +�� +σ2e(x) += emin(x)F1,λ(x) +� +emin(x) +� +− F3,λ(x) +� +emin(x) +� ++ λ(x)F5,λ(x) +� +emin(x) +� +(13) +8 + +4.2 +Lower Confidence Bound +We can also select xt+1 such that the q-quantile of ˆE(x) is minimized. As before, +we have a non-central χ2-distribution for e(x). Thus, the q-quantile of the l.h.s. +of above equation is given by the inverse CDF, F −1 +1,λ(x)(q). As a consequence, +we obtain +xt+1 = arg min +x +� +σ2 +e(x)F −1 +1,λ(x)(q) + σ2 +a(x) +� +, +(14) +where the quantile q in F −1 +1,λ(x)(q) =: β might be considered a tuning parameter. +5 +Experiments +We evaluate our acquisition functions for robust Bayesian target value optimiza- +tion in three experimental settings, where in each we make specific and realistic +assumptions regarding the aleatoric uncertainty due to environmental variables +η. In experiment 1, η represents aleatoric uncertainty at the output of the black +box function, i.e., y(x) = f(x)+η. In experiment 2, η represents aleatoric effects +on the inputs, i.e., y(x) = f(x+η); further, we relax Assumption 3 by replacing +σ2 +a with an estimate ˆσ2 +a computed from the respective GP surrogate. Finally, in +experiment 3 both the GP surrogate and the estimate ˆσ2 +a are learned from noisy +data, i.e., Assumptions 2 and 3 are both relaxed. We compare the performance +of our acquisition functions to those suggested by classical BO (for target value +optimization), where aleatoric uncertainties are not treated separately. +5.1 +Experiment 1 +In the first experiment, we consider a synthetic stochastic mapping with noise +at its output. +Specifically, let y(x) = sin(x) + η, x ∈ [−π/2, π/2] where +η ∼ N(0, σ2 +a) represents the homoskedastic aleatoric uncertainty of the process +(see Fig. 1a-1c). To achieve the separation between aleatoric and epistemic un- +certainties required in our setting, we train a GP with RBF kernel on noise-free +data D = +�� +xi, sin(xi) +�� +. While we have σ2 +a(x) = σ2 +a, we obtain the mean func- +tion and the epistemic uncertainty from (4) for σ2 = 10−10, i.e., we use a small, +positive number to prevent numerical issues. We compare our approach with +the work of [Uhrenholt and Jensen, 2019], where we train a GP on noisy data +{(xi, yi)} using σ2 = σ2 +a in (4). Both GPs are initially trained on two randomly +drawn data points. To account for the randomness of the initial training data +points, the mean and standard deviation of all obtained experimental results +are calculated and reported in the evaluation, see Figure 1d - 1f. +Our aim is to find x such that y is as close to y• = 0 as possible. +We +utilize our EI acquisition functions (denoted as “robust NCχ2”) from Section 4 +and compare it to the EI acquisition functions from [Uhrenholt and Jensen, +2019] (denoted as “NCχ2”), for different values of σ2 +a. These acquisition func- +tions are evaluated on a fixed, evenly spaced grid of 100 candidate positions in +[−π/2, π/2]. Experiments were evaluated for σa ∈ {0.01, 0.1, 0.5}. +9 + +1 +0 +1 +x +1.0 +0.5 +0.0 +0.5 +1.0 +f / y +target +f +y +NC +2 +robust NC +2 +a σa = 0.01 +1 +0 +1 +x +1 +0 +1 +f / y +target +f +y +NC +2 +robust NC +2 +b σa = 0.1 +1 +0 +1 +x +1 +0 +1 +2 +f / y +target +f +y +NC +2 +robust NC +2 +c σa = 0.5 +0 +1 +2 +3 +4 +number of iterations +10 +4 +10 +3 +10 +2 +10 +1 +Emin +NC +2 +robust NC +2 +2 +a +d σa = 0.01 +0 +1 +2 +3 +4 +number of iterations +10 +2 +10 +1 +Emin +NC +2 +robust NC +2 +2 +a +e σa = 0.1 +0 +1 +2 +3 +4 +number of iterations +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +Emin +NC +2 +robust NC +2 +2 +a +f σa = 0.5 +Figure 1: Experiment 1 with EI (13). +(a)-(c) Scatterplots for a single run +with different levels of aleatoric variance σa. Solid lines represent the noise-free +function f, crosses the observations y, colored dots the sampled candidates for +x, and dash-dotted lines the target. (d)-(f) Emin (mean and standard deviation +over 10 folds) over the number of sampled candidate positions, i.e., number of +optimization iterations, for different levels of aleatoric variance σa. +10 + +We present the sampled data points of one cross-validation step in Fig. 1a-1c +and the minimum expected squared error Emin over the number of optimization +steps in Fig. 1d-1f. For low aleatoric uncertainty (σa = 0.01), our acquisition +function performs similarly as the one proposed in [Uhrenholt and Jensen, 2019], +see Fig. 1d. This is expected, as the influence of σa is rather small in Emin, and +both approaches coincide for σa → 0. For increasing σa, a clear advantage w.r.t. +the convergence of Emin towards σ2 +a can be observed, see Figs. 1e-1f. Indeed, our +approach samples inputs resulting in outputs close to the target much earlier +during optimization, see Figs. 1b-1c. Concretely, given the initial training data +points marked in red in Fig. 1c, our acquisition function immediately suggests +a position close to the zero crossing (and subsequent draws visible in Fig. 1c +simply reduce epistemic uncertainty without affecting Emin, cf. Fig. 1f). This is +as expected, since our GP is trained on noise-free data and since our acquisition +function makes explicit use of E(x) and Emin, respectively. +5.2 +Experiment 2 +In the second experiment, we consider a synthetic stochastic mapping with +no measurement noise, but with noisy inputs. Formally, y = f(x + η) with +u = x + η and η ∼ N(0, σ2 +u). This scenario constitutes a special case of a GP +with uncertain inputs, for which, e.g., [Girard et al., 2003, Nguyen et al., 2018, +McHutchon and Rasmussen, 2011] provided closed expressions for the aleatoric +σ2 +a(x) and the epistemic σ2 +e(x) contributions to the predictive variance. In other +words, [Girard et al., 2003] propagated, approximately, the input uncertainty, +defined by σ2 +u, through a GP to the output. In this experiment, the GP surrogate +and this approximation is used in order to estimate the aleatoric uncertainty at +the output ˆσ2 +a(x) ≈ σ2 +a(x) for a finite σ2 +u. Again we set σ2 = 10−10 in (4). +Again we aim at target value optimization, with a target value of y• = 0. +We design an illustrative test function f, and let f have a comparatively flat +region (i.e. small gradient) in the vicinity of the target, where the target itself +lies in (or in vicinity of) a region of a steep gradient +This is fulfilled by f(x) := q(x−2)3+p/((x−3)2+r2)+sx+t on x ∈ [1.8, 2.5] +with q = 50, p = −1, r = 0.1, s = 2, and t = −3.5. The test function is shown +in Fig. 2a. Note that the scale of “flat” and “steep” here is determined by the +variance of the input, σ2 +u. Thus we set the parameter as σu in units of ∆, where +∆ is the interval length of the domain of test function f. Results are compared +for various values of σu. Note that input uncertainties propagated to the output +of a GP can become large quickly, particularly when highly non-linear functions +are being modelled like here. This necessitates relatively low values for σu in +order to enable usable GP surrogates in the first place. For the training of the +GP we use noise-free data D = {xi, f(xi)}, as assumed previously, inputs for +prediction at new inputs remain noisy. Extensions to noisy training inputs ui +may be derived from the findings of [Girard, 2004]. +We use an initial data set of two random data points, and the experiment +is averaged over 1000 random repetitions. For the GP, we use an RBF kernel. +The acquisition function is evaluated on 100 equidistant candidate positions x in +11 + +the abovementioned intervals. I.e. the acquisition function is optimized with a +simple grid search. We compare the performance of: i) our EI acquisition func- +tion (13), which exploits separated aleatoric uncertainty explicitly for target +value optimization (robust NCχ2 EI), to ii) the standard (non-robust) EI ac- +quisition function, adapted for target value optimization but ignoring aleatoric +uncertainty altogether (NCχ2 EI, σa := 0) as proposed by [Uhrenholt and +Jensen, 2019]. We further compare iii) a (non-robust) variant of (ii) where the +GP surrogate is corrected for aleatoric uncertainty as in [Nguyen et al., 2018] +(NCχ2 EI, σa := ˆσa). In other words, variant (iii) is a naive combination of the +NCχ2-distribution for target value optimization from [Uhrenholt and Jensen, +2019] with stability-inducing terms from the Gaussian assumptions in [Nguyen +et al., 2018]. +The results for two different noise levels are shown in Fig. 2b and 2c. The re- +sults demonstrate that our acquisition function (13) (red), leveraging computed +estimates of the aleatoric variance, can perform better than equivalent proce- +dures that either do not distinguish aleatoric and epistemic uncertainty (black), +or neglect the aleatoric uncertainty (blue) in the first place. In our examples, +the state of the art only poorly manages the optimization task. +This could be attributed to a rather high output uncertainty of the un- +derlying GP due to even comparatively low input uncertainty. This becomes +particularly important in the vicinity of highly non-linear behaviour, and con- +siderations for and correction of the aleatoric input uncertainty become relevant. +In Fig. 2, both naive approaches behave similarly and get stuck in an apparent +local optimum that is close to the target, however in a region where the func- +tion derivative is large, leading to a large expected error due to input noise. In +contrast, our acquisition function converges towards an optimum that is more +stable - at the expense of moving slightly away from the original target value. +5.3 +Experiment 3 +In our third experiment, we utilize a use case in the field of manufacturing: +preforming a super alloy billet on a forging machine. In forging, outputs must +be within a pre-defined tolerance range, w.r.t. control measurements. If a forged +part is out of tolerance it has to be reworked, e.g., machined, or in extreme +cases, scrapped. Due to the fact that a process cannot be fully controlled, there +is aleatoric variance present, which has to be considered to be within tolerance. +The use case addresses the problem of finding ideal input variables to achieve +an output that minimizes Emin, w.r.t. a chosen output target. +The inputs x for this manufacturing process are process-related, such as set- +ting values θ1 and θ2 for the forging machine, and material-related, such as the +initial billet diameter d and height h, and the billet temperature T. We treat +these inputs jointly, i.e., x = (θ1, θ2, d, h, T). +The maximum billet diameter +y(x) = dmax(x) is the output of the process and is subject to heteroskedastic +aleatoric uncertainty with unknown variance σ2 +a(x) due to variation in environ- +mental variables η. +We utilize a GP for the mean m(x) = Eη (dmax(x)), represented by mean +12 + +a +b +c +Figure 2: Experiment 2 with EI. Panel (a) shows the test function f(x) at it- +eration 10. Light blue: true test function. Black diamonds: data probed in an +exemplary optimization loop. Red diamond: Current best value acc. to (1). +Blue diamond: Latest proposed next query point. Black dashed: target. Red +line and shade: GP mean and uncertainty. Note that the optimum (1) differs +from the target due to stochastic inputs. Panels (b) and (c) show the conver- +gence for f (from panel (a)) over the number of sampled candidate positions, +i.e., number of optimization iterations, for different levels of aleatoric variance +in environmental inputs, σu. Red corresponds to our acquisition function (13). +Dark blue and black denote the reference methods, [Uhrenholt and Jensen, 2019] +and [Nguyen et al., 2018], respectively, both detailed in Sec. 5.2. +13 + +2 +2 +1.8 +2 +2.2 +2.4 +X=0.0001.△ +100 +robustNCx2EI +-NCx2 EI α := 0 +.. NCx? EI oa ~ o a +10-1 +min +E +10-2 +10° +3 +0 +2 +4 +6 +8 +10 12 14 16 18 20 +number of iterations=0.0005.△ +100 +robustNCx2EI +--NCx2 EI α := 0 +. NCx? EI oa ~ a +10-1 +min +E +10-2 +0 +2 +4 +6 +8 +10 12 14 16 18 20 +numberof iterationsµ(x) and epistemic variance σ2 +e(x), and a kernel ridge regression (KRR) model +for its empirical variance, representing ˆσ2 +a(x). GP and KRR are initially trained +with two data points, and data points are selected, s.t. +the distance in the +feature space is maximized, i.e., the first data point holds the lowest possible +input values and the second data point the highest possible input values w.r.t. +input x = (θ1, θ2, d, h, T). We have found that this selection of initial training +data counteracts falling into local optima. We chose two targets d• +max from the +pool dataset, s.t. one is in a location of low, and one in a location of greater +aleatoric variance, cf. Figures 3a and 3b. That means that optimal solutions +in terms of Emin are either dominated by the error between target and mean +prediction or by the aleatoric variance. +We selected different acquisition functions for our target value BO approach: +i) an LCB computed from a Gaussian distribution (Gaussian LCB), where +aleatoric and epistemic uncertainties are considered jointly [Hoffer et al., 2022], +ii) an LCB from a Gaussian distribution that maximizes epistemic and mini- +mizes aleatoric uncertainties [Hoffer et al., 2022] and is motivated by [Nguyen +et al., 2018] (robust Gaussian LCB), iii) the LCB computed from a non-central +χ2 distribution [Uhrenholt and Jensen, 2019] ignoring aleatoric uncertainty +(NCχ2 LCB), iv) our LCB (14) (robust NCχ2 LCB), v) the EI computed from +a non-central χ2 distribution [Uhrenholt and Jensen, 2019] ignoring aleatoric un- +certainty (NCχ2 EI), and vi) our EI (13) (robust NCχ2 EI). Note that ii) +simultaneously optimizes the expected output w.r.t. the target and minimizes +the output variance due to aleatoric effects, while iii) and v) explicitly model +the distribution of the squared error. Only iv) and vi) both consider the output +variance due to aleatoric effects and use the non-central χ2 distribution to model +the error. We use the data from [Hoffer et al., 2022]. +We present optimization results for each iteration in Figure 3a and 3b. The +number of iterations is equal to the number of pool data, s.t. all approaches +converge to the optimal solution. In Figure 3a, we chose a target in a location +of lower aleatoric uncertainty in the feature space, s.t. the overall best solution +Emin is also low, compared to Figure 3b, where we chose the target near high +aleatoric uncertainty regions, s.t. Emin is greater. Comparison of Figure 3a and +3b shows that our approach exhibits superior performance w.r.t. convergence +in both scenarios. However, in a setting where aleatoric uncertainty is more +dominant, see Figure 3b, the benefit of distinguishing aleatoric and epistemic +uncertainties is more substantial. +Furthermore, evaluation is based on a pool dataset, s.t. in each optimization +iteration a data point is drawn from the pool dataset and added to training +dataset without laying back. +For each iteration E(xi) is calculated for the +actual training set and minimal values, i.e. Emin, are used for plotting, see +Figure 3. +Further, by comparing Emin and squared error values of optimization results +for targets affected by high aleatoric variance (Figure 3b), one can discover that +approaches, which consider aleatoric effects prefer solutions that minimize Emin, +independent if the squared error is increased. For example, our robust NCχ2 EI +procedure shows a squared error of about 83.7 at the beginning, which is the +14 + +0 +100 +200 +300 +400 +number of iterations +101 +102 +Emin +Gaussian LCB +robust Gaussian LCB +NC +2 LCB +robust NC +2 LCB +NC +2 EI +robust NC +2 EI +a target affected by low aleatoric variance +0 +100 +200 +300 +400 +number of iterations +103 +6 × 102 +2 × 103 +Emin +Gaussian LCB +robust Gaussian LCB +NC +2 LCB +robust NC +2 LCB +NC +2 EI +robust NC +2 EI +b target affected by high aleatoric variance +Figure 3: Experiment 3: Emin achieved by different methods as a function of +the number of sampled data points, i.e., number of optimization steps. The +robust Gaussian LCB method models the variance of the squared error as +Gaussian and differentiates between aleatoric and epistemic uncertainty, where +a baseline Gaussian LCB method neglects the differentiation of variances. The +remaining methods estimate variance of squared errors by NCχ2 distribution +and robust methods additionally take advantage of aleatoric variance. In (a) +we choose a target in a region with low aleatoric variance and in (b) we choose +a target in a region with high aleatoric variance. All data points from the pool +data set are drawn, therefore, all approaches converge to the overall minimum. +15 + +same as that of the standard NCχ2 EI procedure. However, after the third +optimization iteration, the robust NCχ2 EI prefers a data point with a higher +squared error (about 526.7) but a lower aleatoric variance to minimize Emin +overall. Neglecting aleatoric effects, the standard NCχ2 EI keeps its optimiza- +tion result at 83.7 until iteration 155, where a better optimization result is +found. +We observed that methods that do not use aleatoric uncertainty in the ac- +quisition function spend long time near the selected target y•, i.e., d• +max as the +seemingly ’best’ optimization result until the actual optimum is found randomly +by further drawing data points. +6 +Discussion and Limitations +We proposed acquisition functions for robust BO with the aim that the output +of a black box function is close to a target value in the sense of an expected +squared error and under the assumption that aleatoric uncertainty due to en- +vironmental effects is known or can be learned. We show in our experiments +that this assumption is at least approximately compatible with a large set of +scenarios, including standard GPs with noisy measurements, GPs with noisy in- +puts (including cascades of GPs such as deep [Damianou and Lawrence, 2013] or +stacked GPs [Abdelfatah et al., 2016, Neumann et al., 2009], in which aleatoric +uncertainties can be propagated), and machine learning approaches in which +aleatoric and epistemic uncertainties are learned separately. +While our results show that our acquisition functions outperform classical +approaches in the considered task, our approach and Assumptions 2 and 3 imply +certain limitations. In the case of GPs, for example, Assumption 2 requires that +training and re-training (after obtaining a new data point) relies on noise-free +data. When measurement noise is included in the dataset, then the variance of +the predictive posterior would include mixed, inseparable components from both +epistemic and aleatoric uncertainties, making the separation necessary for our +derivations impossible. Similarly, if epistemic and aleatoric uncertainties are +represented by other machine learning models, these models must be trained +on data that facilitates such a separation. For example, if the mean and the +aleatoric uncertainty of the mapping are modeled by a GP and a nonlinear +regression model, respectively (as in Section 5.3), we need noise-free and un- +biased estimates of the mean for training the GP, and noise-free and unbiased +estimates of the aleatoric variance for training the regression model. This, in +turn, requires taking sufficiently many measurements at each position x, such +that the mean and the variance of the resulting measurement can be estimated +with little error. The advantage of faster convergence of the optimization prob- +lem thus has to be traded against the requirement to take multiple (simulation +or experimental) measurements. A possible remedy for this limitation could be +to allow finite measurement noise up to a magnitude that is small compared +to the aleatoric uncertainty, or to include a separate mixing term representing +inferential uncertainty in the predictive posterior. Finally, [Ankenman et al., +16 + +2010] showed that estimates for the aleatoric variance from even very small +sample sizes allow for good approximations in the predictive model. +In this work, we measured utility by the expected squared error between the +output of the mapping and a given target value, where the expectation is taken +over aleatoric effects. Modeling aleatoric and epistemic uncertainties with Gaus- +sian distributions, this operational goal allowed us to derive acquisition functions +in closed form. Future research shall extend our work to different practically +relevant operational goals. For example, replacing the expected squared error +by the probability for an excess error leads to the aim of finding an x that +maximizes P (|y − y•| < ε), where ε defines the tolerance level and where the +probability is evaluated w.r.t. the aleatoric uncertainty. Such a setting may be +useful in applications where certain tolerance bands must not be violated. +Until now, we focused on optimization of individual stochastic mappings. +However, if one aims to optimize entire manufacturing chains, GP surrogate +models can be stacked [Neumann et al., 2009]. In stacked GPs, the output of a +previous GP is (a part of) the input for the following GP, and uncertainties are +propagated through the entire chain [Abdelfatah et al., 2016]. Future work shall +investigate how aleatoric and epistemic uncertainties can be propagated sepa- +rately through the stacked GPs, such that our proposed acquisition functions +can be utilized for the optimization of entire process chains. +Finally, Bayesian Optimization has also been investigated with respect to +scalability. Scalable BO algorithms have been derived for large data sets [Snoek +et al., 2015, Eriksson et al., 2019], large input dimensions [Wang et al., 2016, +Daulton et al., 2021], many objectives [Martín and Garrido-Merchán, 2021] +and large output dimensions or many tasks [Hakhamaneshi et al., 2021]. +A +possible future direction is under which circumstances our proposed robustness +towards stochastic environmental variables can be extended to these scaling +variants. Note however that we made no strong assumptions on the number of +environmental variables. +7 +Conclusion +In this work, we derived a set of acquisition functions for Bayesian target value +optimization that is robust against stochastic environmental variables, based +on a common Gaussian process surrogate. In contrast to the usual Gaussian +distributions of simple minimization/maximization, this leads to non-central chi- +square probability density functions for the sought-for optimization objective. +This optimization problem was then considered in the presence of aleatoric +effects in environmental (non-controllable) variables. We find that knowledge +of this aleatoric uncertainty can be leveraged advantageously towards optima +that are robust against such stochastic environmental variables. For this, we +demonstrate experimentally that estimates or learned models of the aleatoric +variance can be sufficient, and that the approach is of particular advantage if +aleatoric variance is indeed large. +Based on the good performance in an alloy billet forging problem, it is spec- +17 + +ulated that the approach might be useful for broader applications in manufac- +turing and industrial engineering. Aleatoric uncertainy is, after all, present in +many data sets and hence a large class of machine learning or optimization +problems. +Acknowledgements +J. G. Hoffer and B. C. Geiger were supported by the project BrAIN - Brownfield +Artificial Intelligence Network for Forging of High Quality Aerospace Compo- +nents (FFG Grant No. 881039). The project is funded in the framework of the +program ’TAKE OFF’, which is a research and technology program of the Aus- +trian Federal Ministry of Transport, Innovation and Technology. S. 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Conf. on Artificial Intelligence and +Statistics (AISTATS), volume 89 of PMLR, pages 2661–2670. +[Wang et al., 2016] Wang, Z., Hutter, F., Zoghi, M., Matheson, D., and +De Feitas, N. (2016). Bayesian optimization in a billion dimensions via ran- +dom embeddings. Journal of Artificial Intelligence Research, 55:361–387. +[Zhan and Xing, 2020] Zhan, D. and Xing, H. (2020). +Expected improve- +ment for expensive optimization: a review. Journal of Global Optimization, +78(3):507–544. +[Zhou et al., 2020] Zhou, D., Li, L., and Gu, Q. (2020). +Neural contextual +bandits with ucb-based exploration. In International Conference on Machine +Learning, pages 11492–11502. PMLR. +22 + +A +Appendix A: Practical Aspects of the Acqui- +sition Functions +Maximizing (10) over x is complicated, since the CDF of a non-central χ2- +distribution is not available in closed form. However, it was shown in [Sankaran, +1959] that using a non-linear transform, the distribution of e ∼ NCχ2(K, λ) +can be transformed into an approximately Gaussian distribution. Specifically, +by setting +z = +� +e +K + λ +�ℓ +(15) +with ℓ = 1−r1r3/3r2 +2, rs = 2s−1(s−1)!(K+sλ), we get that z has approximately +the distribution N +� +α, ρ2� +with +α = 1 + ℓ(ℓ − 1) +� r2 +2r2 +1 +− (2 − ℓ)(1 − 3ℓ) r2 +2 +8r4 +1 +� +(16a) +ρ = ℓr2 +2 +r1 +� +1 − (1 − ℓ)(1 − 3ℓ) +4r2 +1 +r2 +� +. +(16b) +By this approximation, we obtain a closed-form approximation of the CDF +of, e.g., e(x) as +F1,λ(x)(e(x)) ≈ Φ +�e(x) − α +ρ +� +. +(17) +With this, we circumvent the straightforward approach of modelling directly +pN (e|x, D) = N(e|µ(x), σ2(x)), where p(e|x) is imprecise, because it would be +symmetric and with negative support. +B +Additional Figures for Experiment 1 +For experiment 1, we did in addition an evaluation by using LCB acquisi- +tion function, see Figure 4. Similar to evaluation with EI acquisition function, +we can show that by increasing aleatoric uncertainty σa (Figure 4b - 4c) our +robust NCχ2 outperforms, see Figure 4e - 4f. In a setting where aleatoric uncer- +tainty is low (Figure 4a) our robust NCχ2 acquisition function performs similar +to the approach of [Uhrenholt and Jensen, 2019] as expected, as the influence +of σa is rather small in Emin. +23 + +1 +0 +1 +x +1.0 +0.5 +0.0 +0.5 +1.0 +f / y +target +f +y +NC +2 +robust NC +2 +a σa = 0.01 +1 +0 +1 +x +1.0 +0.5 +0.0 +0.5 +1.0 +f / y +target +f +y +NC +2 +robust NC +2 +b σa = 0.1 +1 +0 +1 +x +2 +1 +0 +1 +f / y +target +f +y +NC +2 +robust NC +2 +c σa = 0.5 +0 +1 +2 +3 +4 +number of iterations +10 +4 +10 +3 +10 +2 +10 +1 +Emin +NC +2 +robust NC +2 +2 +a +d σa = 0.01 +0 +1 +2 +3 +4 +number of iterations +10 +2 +10 +1 +Emin +NC +2 +robust NC +2 +2 +a +e σa = 0.1 +0 +1 +2 +3 +4 +number of iterations +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +Emin +NC +2 +robust NC +2 +2 +a +f σa = 0.5 +Figure 4: Experiment 1 with LCB (14). (a)-(c) Scatterplots of the used data +with different levels of aleatoric variance σa. Solid lines represent the noise-free +function f, crosses the observations y, dots the sampled candidate positions, +and dash-dotted lines the target. +(d)-(f) Emin over the number of sampled +candidate positions, i.e., number of optimization iterations, for different levels +of aleatoric variance σa. +24 + diff --git a/kNE3T4oBgHgl3EQfJglg/content/tmp_files/load_file.txt b/kNE3T4oBgHgl3EQfJglg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9551c98c2b1950fad2d028293beae34fb99c7cef --- /dev/null +++ b/kNE3T4oBgHgl3EQfJglg/content/tmp_files/load_file.txt @@ -0,0 +1,1018 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf,len=1017 +page_content='Robust Bayesian Target Value Optimization Johannes G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Hoffer∗, Sascha Ranftl†, Bernhard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Geiger‡ January 12, 2023 Abstract We consider the problem of finding an input to a stochastic black box function such that the scalar output of the black box function is as close as possible to a target value in the sense of the expected squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' While the optimization of stochastic black boxes is classic in (ro- bust) Bayesian optimization, the current approaches based on Gaussian processes predominantly focus either on i) maximization/minimization rather than target value optimization or ii) on the expectation, but not the variance of the output, ignoring output variations due to stochasticity in uncontrollable environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In this work, we fill this gap and derive acquisition functions for common criteria such as the expected improvement, the probability of improvement, and the lower confidence bound, assuming that aleatoric effects are Gaussian with known variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Our experiments illustrate that this setting is compatible with certain extensions of Gaussian processes, and show that the thus derived acquisi- tion functions can outperform classical Bayesian optimization even if the latter assumptions are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' An industrial use case in billet forging is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1 Introduction Inverse problems, where one aims to find parameters of a system either explain- ing or guaranteeing certain behavior, are ubiquitous in science and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Consider, for example, process control in manufacturing, where a part or mate- rial undergoes a specific manufacturing process characterized by tuneable design variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' These design variables should be optimized such that the output of the manufacturing process is as close as possible to a defined target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, manufacturing processes are often influenced by uncertainties of different kinds, such as material imperfections, variation within process tolerances, seasonal ef- fects, or limited accuracy for controlling process variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' These uncertainties, ∗voestalpine Böhler Aerospace GmbH & Co KG, Mariazellerstraße 25, Kapfenberg, Austria †Institute of Theoretical Physics-Computational Physics, Graz University of Technology, 8010 Graz, Austria ‡Know-Center GmbH, Inffeldgasse 13, Graz, Austria 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='04344v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='LG] 11 Jan 2023 which are often summarized as environmental effects, need to be taken into account when solving inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Standard Gaussian processes (GPs) are capable of solving inverse problems under uncertainties, and may comprise distinct kinds of uncertainties [Ranftl and von der Linden, 2021, Ranftl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', aleatoric and epistemic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Indeed, several acquisition functions have been proposed for “noisy” [Gramacy and Lee, 2011, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2006, Letham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019, Picheny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2010], and “robust” [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020, Bogunovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] Bayesian optimization (BO), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, the majority of the previ- ous works on GP-based BO does not distinguish between epistemic and aleatoric uncertainties, which arise from finiteness of training data and stochasticity in the relationship between input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Those works that do either focus on maximization/minimization settings rather than on target value optimization, or their understanding of robustness against aleatoric effects is limited to optimizing the expected output of the black box function, ignoring its variance due to aleatoric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Indeed, while the acquisition functions for target value optimization in [Uhrenholt and Jensen, 2019] are not robust against aleatoric effects, the few works that simultaneously try to optimize the expected output and minimize the output variance due to variations in the envi- ronmental variables either focus on maximization/minimization [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018, Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021] or fail to fully exploit the mathematical peculiarities of target value optimization [Hoffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus, the literature exhibits a striking and practically relevant gap that this work seeks to fill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifically, we will derive acquisition functions for robust Bayesian target value optimization, with the aim of selecting design variables such that the black box function output is close to a target value in the sense of an expected squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This aim not only requires that the expected output of the black box function is close to the target, but that also its variation due to aleatoric effects is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Essentially, our approach is based on a separation between aleatoric (for evaluating the expected squared error) and epistemic (for formulating the acquisition function) uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We set out with the assumptions that the aleatoric effects are Gaussian with known variance function and that they can be quantified separately from epis- temic effects (Section 2), and we show in our experiments in Section 5 that they approximately hold for certain practically relevant models based on GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Based on these assumptions, we derive acquisition functions for target value optimization in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifically, we show that by measuring the quality of the optimization by the squared error expected due to aleatoric effects, that the resulting acquisition functions can be computed in closed form and are remi- niscent of those for noise-free target vector optimization [Uhrenholt and Jensen, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Using both synthetic and real-world examples, we show in Section 5 in which cases our proposed acquisition functions outperform classical BO even when some of our assumptions are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We summarize the insights from these experiments and discuss limitations of our work in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2 2 Problem Definition We consider the optimization of a black box function f: RD+A → R that maps a vector x of D controllable design variables and a vector η of A uncontrollable environmental variables to a scalar output y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y = f(x, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Our aim is to select the design variables x such that the output y is as close as possible to a target value y• in the sense of an expected squared error, where the expectation is taken over the unknown environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Mathematically, we are interested in finding a solution to arg min x Eη �� y• − f(x, η) �2� =: arg min x E(x) (1) where Eη (·) denotes expectation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In this setting, the environmental variables η correspond to aleatoric effects that cannot be controlled by opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Further, the setup is general enough to cover measurement errors (y = f(x) + η), uncertain inputs to a black box function (y = f(x + η)), and more complicated settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To formulate this optimization problem within the framework of BO and to approach it using GPs, we will introduce three simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The mapping from the design variables x to the output y is Gaussian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', we assume that it is fully characterized by a mean function m(x) := Eη (f(x, η)) and a variance function σ2 a(x) := Eη �� f(x, η) − m(x) �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Assumption 1 allows us to derive analytic expressions for the acquisition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Further, under this assumption the optimization objective in (1) simplifies to E(x) = � y• − m(x) �2 + σ2 a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (2) Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For all x, we can take noise-free measurements of the mean function m(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', we can build a dataset D = �� xi, m(xi) �� i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=',N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Assumption 2 essentially requires that we have access to measurements of the black box function for which aleatoric effects have been averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This can be achieved by i) taking sufficiently many repeated measurements or simulations with the same design variable, approximating the expectation Eη (f(x, η)) by an average, or by ii) measuring (in addition to x and y) and subsequently marginal- izing the (uncontrollable, but measurable) environmental effects η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This latter approach is common in the literature on robust BO, where GP models are cre- ated for f(x, η) and marginalized over η [Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019, Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021, Toscano-Palmerin and Frazier, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In particular, [Ankenman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2010] found that this is a good approximation for separating ’intrinsic’ and ’ex- trinsic’ variances even with very low numbers of repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Also, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', [Girard, 2004] assumes, based on GPs, either noise-free learning or noise-free inference, albeit mainly due to the mathematical complexity of considering noise in both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' As a consequence of Assumption 2, we can use a GP-based 3 estimate ˆm of m: RD → R which coincides with m exactly at all values of the design variable xi that have been measured so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' A real-world example for such kind of data could be manufacturing chains, where adjusting the design parameters can be very costly while several repeti- tions for a fixed design parameter set can be cheap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Another real-world example are data from computer simulations, where the design parameters may be re- lated to geometry descriptions, which require elaborate mesh (re-)constructions, or scale parameters in stochastic simulations, which require (re-)tuning of algo- rithmic parameters such as in Markov Chain Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We have full knowledge of σ2 a: RD → R+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This last requirement of a fully known aleatoric variance function seems restrictive and appears to limit the practical utility of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, while we will derive our acquisition functions based on this assumption, our experiments in Section 5 will show that even with crude (learned or computed) estimates ˆσ2 a of σ2 a our approach can outperform current alternatives to Bayesian target value optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In the real-world examples mentioned before, such estimates can usually be obtained with reasonable effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 3 Related Work A plethora of acquisition functions based has been suggested, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Shahri- ari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2015, Frazier, 2018, Brochu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2010, Loredo, 2004, Preuss and von Toussaint, 2021, Ramachandran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] [Zhan and Xing, 2020, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020] for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Also the need to account for aleatoric effects in BO has been long recognized and is known under “noisy” or “robust BO”, predom- inantly for maximization/minimization settings (rather than for target value optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For improvement-based acquisition functions such as expected improvement (EI) or probability of improvement (PoI), the inherent stochastic- ity of the mapping makes it difficult to judge which input xt in the dataset D is the previous “best”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To account for this in a minimization setting, the au- thors of [Gramacy and Lee, 2011] set the previous best input to arg minx µ(x), where µ(x) is obtained by training a GP on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Computing and optimizing the resulting EI in a similar way as in the noiseless case was shown to lead to slow convergence to the minimum of the mapping, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Letham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2006] instead select xt as the “effective best solution” and augment EI by a multiplicative factor that favors inputs with large epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Letham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019] also consider noisy observations (and noisy constraints) when training GPs for BO, but compute and optimize EI on the predictive distribution for the noiseless mapping f(x), rather than for the noisy mapping y(x) = f(x) + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The authors of [Picheny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2010] replace the expected value µ(x) by the quantile of the GP’s predictive distribution, and compute EI based on the improvement of this quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' None of these works, however, accounts for variations of the output due to aleatoric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Further, most of the mentioned studies focus on noise at the output of a deterministic mapping, 4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y(x) = f(x) + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Considering deterministic mappings with noisy inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y(x) = f(x + η), [Girard, 2004, Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2003] derive approximations for a GP’s predictive posterior mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' These approximations were used in [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] to separate epistemic effects and aleatoric effects propagated through the mapping to derive an acquisition function for the up- per confidence bound (UCB) to stabilize BO in the sense of minimizing output variations due to noisy inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Robust BO more generally considers optimization in uncertain environments, where in addition to controllable design variables x, the black box function de- pends also on uncontrollable environmental variables η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y = f(x, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In this line of research, [Toscano-Palmerin and Frazier, 2022] optimize the expectation of y over η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', it maximizes the mean function m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021] extend beyond [Toscano-Palmerin and Frazier, 2022] by proposing confidence bound-based approaches to simultaneously maximize the expectation m(x) and minimize the variance σ2 a(x), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Iwazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2], while [Kirschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020] remain optimizing m(x) but relax the assumption that the distribution of η is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Daulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022] maximize the (multi- variate) value-at-risk, which implicitly balances maximizing the expectation and minimizing the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020] aims to maximize the expecta- tion m(x) of y = f(x + η) using a variant of entropy search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016] proposed to use the expectation of the unscented expected improvement (UEI) acquisition function with respect to the input noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019] aim to maximize a black box function with noise at the input and the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' While they also consider noisy measurements during surrogate model- ing and optimization, [Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020, Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016, Beland and Nair, 2017, Bogunovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] assume exact knowledge of the function in- put during optimization and only aim to achieve robustness during deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus, these works make assumptions similar to our Assumption 2, but focus on maximization/minimization and do not study the peculiarities of target value optimization in the light of (aleatoric) input uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For target vector optimization, one aims at minimizing a distance d(x) = d(y(x), y•), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', the squared Euclidean distance ∥y − y•∥2 2, between the map- ping’s output y and a target value or target vector y•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 In the Gaussian case and for a k-dimensional output vector y(x), a non-central χ2-distribution was shown to be an unbiased estimate for the posterior p(d(x)|x, D) in [Uhrenholt and Jensen, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For this setting, the authors derived acquisition functions for EI and lower confidence bound (LCB) for both standard GP and warped GP regression, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Snelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Target value optimization for y• = 0 was considered in [Osborne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2009] for a stochastic mapping and additional stochasticity in the input, but this work does not distinguish between epistemic and aleatoric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The authors of [Pandita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016] acknowledge the dis- tinct nature of aleatoric and epistemic uncertainty and quantify the epistemic uncertainty of the optimal location, but they do not quantify or exploit aleatoric 1An entirely different approach was proposed in [Beland and Nair, 2017], where a GP prior is imposed directly onto an optimality criterion or loss function, in contrast to the more widely-spread procedure of imposing the GP prior on the function which is to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5 uncertainty in their procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Also [Jeong and Shin, 2021] considered vector- valued outputs and targets and arrived at a similar result as [Uhrenholt and Jensen, 2019], albeit approaching the problem from a multi-objective perspec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' They allowed for different weightings of the different objectives in contrast to [Uhrenholt and Jensen, 2019], and, in contrast to this work, did not consider robustness with respect to aleatoric effects in stochastic environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Astudillo and Frazier, 2022, Astudillo and Frazier, 2019] leveraged prior knowl- edge on composite structures of the objective function similarly as it appears in [Uhrenholt and Jensen, 2019], albeit again without considering aleatoric input noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To the best of our knowledge, the only approach that performs target vector optimization and that considers the variance of the black box function output due to aleatoric effects is [Hoffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022], where the authors utilize the UCB acquisition function from [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] and where the aleatoric uncertainty is given by the mean function of a separate GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Their approach, however, ignores the fact that the squaring operation ensures that the posterior of d(x) given x and the dataset D has non-negative support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 4 Robust Bayesian Target Value Optimization Us- ing Gaussian Processes Suppose that at optimization iteration t we have access to a dataset D = �� xi, m(xi) �� , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' , t of noise-free measurements of the mean function m, and that we approximate this mean function by a GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We denote the predic- tive posterior mean function of this GP as µ to distinguish it from the real mean function m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the predictive posterior variance function of this GP, which esti- mates the epistemic uncertainty resulting from the finite dataset D, is denoted as σ2 e: RD → R+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus, our estimate ˆm of m is given by ˆm(x) ∼ N � µ(x), σ2 e(x) � (3) with µ(x) = k(x) � K + σ2I �−1 mT (4a) σ2 e(x) = k(x, x) − k(x) � K + σ2I �−1 kT (x) (4b) where σ2 is measurement noise, k: R2 → R denotes a kernel function, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' , xt) and m = � m(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' , m(xt) � are the vectorized dataset D, [K]ij = k(xi, xj) is the kernel matrix, and where k(x) = � k(x1, x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' , k(xt, x) � denotes the vector of covariances between x in D and the candidate input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Utilizing Assumption 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', the fact of being noise-free in our design of the GP, we can set the hyperparameter σ2 to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' As a consequence, for all xi ∈ D, we have that the epistemic uncertainty σ2 e(xi) = 0 [Rasmussen and Williams, 2006, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='19)], and that ˆm(xi) ≡ µ(xi) = m(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For general x, our surrogate ˆy(x) of the black box function output y(x) is Gaussian with mean µ(x) and 6 variance σ2 a(x) + σ2 e(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Summarizing, we have y(x) = f(x, η) ∼ N � m(x), σ2 a(x) � Assump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1 ˆm(x) ∼ N � µ(x), σ2 e(x) � eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (3) ∀xi ∈ D: σ2 e(xi) = 0 Assump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2 ∀xi ∈ D: µ(xi) = m(xi) Assump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2 ˆy(x) ∼ N � ˆm(x), σ2 a(x) � Assump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 3 ⇒ ˆy(x) ∼ N � µ(x), σ2 a(x) + σ2 e(x) � The first line in this summary should denote that the output y of the black box function f is Gaussian for given x, subsuming the stochasticity of environmental variables η in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We now propose acquisition functions to optimize this surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifi- cally, and connecting to (1), we aim to select x such that ˆy(x) is as close as possible to a target value y•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifically, we aim to find an input x such that the expected squared error ˆE(x) := � ˆm(x) − y•�2 + σ2 a(x) (6) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note that due to the epistemic uncertainty resulting from the finite dataset D, ˆE(x) is a random variable unless x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus, while E(x) from (1) is deterministic since it is an expected value, ˆE(x) is estimated from the GP surrogate of the black box function and is hence random due to the inherent epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Indeed, for x /∈ D the normalized expected squared error e(x) follows a non-central χ2-distribution: e(x) := ˆE(x) − σ2 a(x) σ2e(x) ∼ NCχ2� K = 1, λ(x) � , (7a) with K = 1 degree of freedom and non-centrality parameter λ(x) = � µ(x) − y•�2 σ2e(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (7b) For xi ∈ D, we have µ(xi) = m(xi) and σ2 e(xi) = 0, hence ˆE(xi) = E(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' On the one hand, this justifies optimizing with the surrogate instead of the black box function, On the other hand, it follows that for these inputs, ˆE(xi) is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Further, the normalized expected squared error either evaluates to e(xi) = 0 if m(xi) = y•, or to e(xi) = ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To prevent numerical issues, in our experiments we set the hyperparameter σ2 in (4) to a small, positive number, yielding σ2 e(x) > 0 for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' If σ2 ≪ σ2 a(x), one still has ˆE(xi) ≈ E(xi) for all xi ∈ D and, thus, a valid surrogate model for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In the remainder of this section, however, we will stick to σ2 = 0 and to Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 Improvement-Based Acquisition Functions Let without loss of generality xt = minxi∈D ˆE(xi) be the input that minimizes the expected squared error over all previous inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' By Assumption 2, Emin := ˆE(xt) is deterministic and evaluates to Emin = ˆE(xt) = E(xt) = � m(xt) − y•�2 + σ2 a(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (8) Probability of Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We can select xt+1 such that the PoI over Emin is maximized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', we solve xt+1 = arg max x P � ˆE(x) ≤ Emin − ζ � (9) where ζ ≥ 0 is a parameter that prefers larger, improbable improvements over more probable, but small improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Given that e(x) has a non-central χ2-distribution with cumulative distribution function (CDF) F1,λ(x), the PoI is given by P � ˆE(x) ≤ Emin − ζ � = P � e(x) ≤ Emin − ζ − σ2 a(x) σ2e(x) � = F1,λ(x) �� m(xt) − y•�2 + σ2 a(xt) − ζ − σ2 a(x) σ2e(x) � (10) where λ(x) is given in (7b) and Emin was substituted by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Expected Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We can also select xt+1 such that the EI over Emin is maximized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', we solve xt+1 = arg max x E � max � 0, Emin − ˆE(x) �� (11) where E (·) takes the expectation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' By the linearity of expectation and with emin(x) = � Emin − σ2 a(x) � /σ2 e(x) this can be rewritten as E � max � 0, Emin − ˆE(x) �� = σ2 e(x)E � max � 0, emin(x) − e(x) �� = σ2 e(x) � emin(x) 0 (emin(x) − e) f1,λ(x)(e)de (12) where f1,λ(x) is the probability density function of the non-central χ2-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Similarly as in [Uhrenholt and Jensen, 2019, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2], this integral can be com- puted in closed form as E � max � 0, Emin − ˆE(x) �� σ2e(x) = emin(x)F1,λ(x) � emin(x) � − F3,λ(x) � emin(x) � + λ(x)F5,λ(x) � emin(x) � (13) 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 Lower Confidence Bound We can also select xt+1 such that the q-quantile of ˆE(x) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' As before, we have a non-central χ2-distribution for e(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus, the q-quantile of the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' of above equation is given by the inverse CDF, F −1 1,λ(x)(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' As a consequence, we obtain xt+1 = arg min x � σ2 e(x)F −1 1,λ(x)(q) + σ2 a(x) � , (14) where the quantile q in F −1 1,λ(x)(q) =: β might be considered a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5 Experiments We evaluate our acquisition functions for robust Bayesian target value optimiza- tion in three experimental settings, where in each we make specific and realistic assumptions regarding the aleatoric uncertainty due to environmental variables η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In experiment 1, η represents aleatoric uncertainty at the output of the black box function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y(x) = f(x)+η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In experiment 2, η represents aleatoric effects on the inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', y(x) = f(x+η);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' further, we relax Assumption 3 by replacing σ2 a with an estimate ˆσ2 a computed from the respective GP surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Finally, in experiment 3 both the GP surrogate and the estimate ˆσ2 a are learned from noisy data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', Assumptions 2 and 3 are both relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We compare the performance of our acquisition functions to those suggested by classical BO (for target value optimization), where aleatoric uncertainties are not treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 Experiment 1 In the first experiment, we consider a synthetic stochastic mapping with noise at its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifically, let y(x) = sin(x) + η, x ∈ [−π/2, π/2] where η ∼ N(0, σ2 a) represents the homoskedastic aleatoric uncertainty of the process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1a-1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To achieve the separation between aleatoric and epistemic un- certainties required in our setting, we train a GP with RBF kernel on noise-free data D = �� xi, sin(xi) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' While we have σ2 a(x) = σ2 a, we obtain the mean func- tion and the epistemic uncertainty from (4) for σ2 = 10−10, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', we use a small, positive number to prevent numerical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We compare our approach with the work of [Uhrenholt and Jensen, 2019], where we train a GP on noisy data {(xi, yi)} using σ2 = σ2 a in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Both GPs are initially trained on two randomly drawn data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' To account for the randomness of the initial training data points, the mean and standard deviation of all obtained experimental results are calculated and reported in the evaluation, see Figure 1d - 1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Our aim is to find x such that y is as close to y• = 0 as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We utilize our EI acquisition functions (denoted as “robust NCχ2”) from Section 4 and compare it to the EI acquisition functions from [Uhrenholt and Jensen, 2019] (denoted as “NCχ2”), for different values of σ2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' These acquisition func- tions are evaluated on a fixed, evenly spaced grid of 100 candidate positions in [−π/2, π/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Experiments were evaluated for σa ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 9 1 0 1 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 f / y target f y NC 2 robust NC 2 a σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01 1 0 1 x 1 0 1 f / y target f y NC 2 robust NC 2 b σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 1 0 1 x 1 0 1 2 f / y target f y NC 2 robust NC 2 c σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 0 1 2 3 4 number of iterations 10 4 10 3 10 2 10 1 Emin NC 2 robust NC 2 2 a d σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01 0 1 2 3 4 number of iterations 10 2 10 1 Emin NC 2 robust NC 2 2 a e σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 0 1 2 3 4 number of iterations 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 Emin NC 2 robust NC 2 2 a f σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 Figure 1: Experiment 1 with EI (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (a)-(c) Scatterplots for a single run with different levels of aleatoric variance σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Solid lines represent the noise-free function f, crosses the observations y, colored dots the sampled candidates for x, and dash-dotted lines the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (d)-(f) Emin (mean and standard deviation over 10 folds) over the number of sampled candidate positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', number of optimization iterations, for different levels of aleatoric variance σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 10 We present the sampled data points of one cross-validation step in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1a-1c and the minimum expected squared error Emin over the number of optimization steps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1d-1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For low aleatoric uncertainty (σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01), our acquisition function performs similarly as the one proposed in [Uhrenholt and Jensen, 2019], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This is expected, as the influence of σa is rather small in Emin, and both approaches coincide for σa → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For increasing σa, a clear advantage w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the convergence of Emin towards σ2 a can be observed, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1e-1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Indeed, our approach samples inputs resulting in outputs close to the target much earlier during optimization, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1b-1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Concretely, given the initial training data points marked in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1c, our acquisition function immediately suggests a position close to the zero crossing (and subsequent draws visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1c simply reduce epistemic uncertainty without affecting Emin, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This is as expected, since our GP is trained on noise-free data and since our acquisition function makes explicit use of E(x) and Emin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 Experiment 2 In the second experiment, we consider a synthetic stochastic mapping with no measurement noise, but with noisy inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Formally, y = f(x + η) with u = x + η and η ∼ N(0, σ2 u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This scenario constitutes a special case of a GP with uncertain inputs, for which, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', [Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2003, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018, McHutchon and Rasmussen, 2011] provided closed expressions for the aleatoric σ2 a(x) and the epistemic σ2 e(x) contributions to the predictive variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In other words, [Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2003] propagated, approximately, the input uncertainty, defined by σ2 u, through a GP to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In this experiment, the GP surrogate and this approximation is used in order to estimate the aleatoric uncertainty at the output ˆσ2 a(x) ≈ σ2 a(x) for a finite σ2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Again we set σ2 = 10−10 in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Again we aim at target value optimization, with a target value of y• = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We design an illustrative test function f, and let f have a comparatively flat region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' small gradient) in the vicinity of the target, where the target itself lies in (or in vicinity of) a region of a steep gradient This is fulfilled by f(x) := q(x−2)3+p/((x−3)2+r2)+sx+t on x ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5] with q = 50, p = −1, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1, s = 2, and t = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The test function is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note that the scale of “flat” and “steep” here is determined by the variance of the input, σ2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Thus we set the parameter as σu in units of ∆, where ∆ is the interval length of the domain of test function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Results are compared for various values of σu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note that input uncertainties propagated to the output of a GP can become large quickly, particularly when highly non-linear functions are being modelled like here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This necessitates relatively low values for σu in order to enable usable GP surrogates in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For the training of the GP we use noise-free data D = {xi, f(xi)}, as assumed previously, inputs for prediction at new inputs remain noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Extensions to noisy training inputs ui may be derived from the findings of [Girard, 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We use an initial data set of two random data points, and the experiment is averaged over 1000 random repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For the GP, we use an RBF kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The acquisition function is evaluated on 100 equidistant candidate positions x in 11 the abovementioned intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the acquisition function is optimized with a simple grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We compare the performance of: i) our EI acquisition func- tion (13), which exploits separated aleatoric uncertainty explicitly for target value optimization (robust NCχ2 EI), to ii) the standard (non-robust) EI ac- quisition function, adapted for target value optimization but ignoring aleatoric uncertainty altogether (NCχ2 EI, σa := 0) as proposed by [Uhrenholt and Jensen, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We further compare iii) a (non-robust) variant of (ii) where the GP surrogate is corrected for aleatoric uncertainty as in [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] (NCχ2 EI, σa := ˆσa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In other words, variant (iii) is a naive combination of the NCχ2-distribution for target value optimization from [Uhrenholt and Jensen, 2019] with stability-inducing terms from the Gaussian assumptions in [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The results for two different noise levels are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2b and 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The re- sults demonstrate that our acquisition function (13) (red), leveraging computed estimates of the aleatoric variance, can perform better than equivalent proce- dures that either do not distinguish aleatoric and epistemic uncertainty (black), or neglect the aleatoric uncertainty (blue) in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In our examples, the state of the art only poorly manages the optimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This could be attributed to a rather high output uncertainty of the un- derlying GP due to even comparatively low input uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This becomes particularly important in the vicinity of highly non-linear behaviour, and con- siderations for and correction of the aleatoric input uncertainty become relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 2, both naive approaches behave similarly and get stuck in an apparent local optimum that is close to the target, however in a region where the func- tion derivative is large, leading to a large expected error due to input noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In contrast, our acquisition function converges towards an optimum that is more stable - at the expense of moving slightly away from the original target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='3 Experiment 3 In our third experiment, we utilize a use case in the field of manufacturing: preforming a super alloy billet on a forging machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In forging, outputs must be within a pre-defined tolerance range, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' control measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' If a forged part is out of tolerance it has to be reworked, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', machined, or in extreme cases, scrapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Due to the fact that a process cannot be fully controlled, there is aleatoric variance present, which has to be considered to be within tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The use case addresses the problem of finding ideal input variables to achieve an output that minimizes Emin, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' a chosen output target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The inputs x for this manufacturing process are process-related, such as set- ting values θ1 and θ2 for the forging machine, and material-related, such as the initial billet diameter d and height h, and the billet temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We treat these inputs jointly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', x = (θ1, θ2, d, h, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The maximum billet diameter y(x) = dmax(x) is the output of the process and is subject to heteroskedastic aleatoric uncertainty with unknown variance σ2 a(x) due to variation in environ- mental variables η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We utilize a GP for the mean m(x) = Eη (dmax(x)), represented by mean 12 a b c Figure 2: Experiment 2 with EI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Panel (a) shows the test function f(x) at it- eration 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Light blue: true test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Black diamonds: data probed in an exemplary optimization loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Red diamond: Current best value acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Blue diamond: Latest proposed next query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Black dashed: target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Red line and shade: GP mean and uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note that the optimum (1) differs from the target due to stochastic inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Panels (b) and (c) show the conver- gence for f (from panel (a)) over the number of sampled candidate positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', number of optimization iterations, for different levels of aleatoric variance in environmental inputs, σu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Red corresponds to our acquisition function (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Dark blue and black denote the reference methods, [Uhrenholt and Jensen, 2019] and [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018], respectively, both detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 13 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='4 X=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='△ 100 robustNCx2EI NCx2 EI α := 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='. NCx?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' EI oa ~ o a 10-1 min E 10-2 10° 3 0 2 4 6 8 10 12 14 16 18 20 number of iterations=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='△ 100 robustNCx2EI --NCx2 EI α := 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' NCx?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' EI oa ~ a 10-1 min E 10-2 0 2 4 6 8 10 12 14 16 18 20 numberof iterationsµ(x) and epistemic variance σ2 e(x), and a kernel ridge regression (KRR) model for its empirical variance, representing ˆσ2 a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' GP and KRR are initially trained with two data points, and data points are selected, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the distance in the feature space is maximized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', the first data point holds the lowest possible input values and the second data point the highest possible input values w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' input x = (θ1, θ2, d, h, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We have found that this selection of initial training data counteracts falling into local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We chose two targets d• max from the pool dataset, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' one is in a location of low, and one in a location of greater aleatoric variance, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Figures 3a and 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' That means that optimal solutions in terms of Emin are either dominated by the error between target and mean prediction or by the aleatoric variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We selected different acquisition functions for our target value BO approach: i) an LCB computed from a Gaussian distribution (Gaussian LCB), where aleatoric and epistemic uncertainties are considered jointly [Hoffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022], ii) an LCB from a Gaussian distribution that maximizes epistemic and mini- mizes aleatoric uncertainties [Hoffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022] and is motivated by [Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2018] (robust Gaussian LCB), iii) the LCB computed from a non-central χ2 distribution [Uhrenholt and Jensen, 2019] ignoring aleatoric uncertainty (NCχ2 LCB), iv) our LCB (14) (robust NCχ2 LCB), v) the EI computed from a non-central χ2 distribution [Uhrenholt and Jensen, 2019] ignoring aleatoric un- certainty (NCχ2 EI), and vi) our EI (13) (robust NCχ2 EI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note that ii) simultaneously optimizes the expected output w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the target and minimizes the output variance due to aleatoric effects, while iii) and v) explicitly model the distribution of the squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Only iv) and vi) both consider the output variance due to aleatoric effects and use the non-central χ2 distribution to model the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We use the data from [Hoffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We present optimization results for each iteration in Figure 3a and 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The number of iterations is equal to the number of pool data, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' all approaches converge to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In Figure 3a, we chose a target in a location of lower aleatoric uncertainty in the feature space, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the overall best solution Emin is also low, compared to Figure 3b, where we chose the target near high aleatoric uncertainty regions, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Emin is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Comparison of Figure 3a and 3b shows that our approach exhibits superior performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' convergence in both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, in a setting where aleatoric uncertainty is more dominant, see Figure 3b, the benefit of distinguishing aleatoric and epistemic uncertainties is more substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Furthermore, evaluation is based on a pool dataset, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' in each optimization iteration a data point is drawn from the pool dataset and added to training dataset without laying back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For each iteration E(xi) is calculated for the actual training set and minimal values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Emin, are used for plotting, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Further, by comparing Emin and squared error values of optimization results for targets affected by high aleatoric variance (Figure 3b), one can discover that approaches, which consider aleatoric effects prefer solutions that minimize Emin, independent if the squared error is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For example, our robust NCχ2 EI procedure shows a squared error of about 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='7 at the beginning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' which is the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='number of iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='Emin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='Gaussian LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust Gaussian LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 EI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 EI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='a target affected by low aleatoric variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='number of iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='6 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 × 103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='Emin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='Gaussian LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust Gaussian LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 LCB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 EI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='robust NC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='2 EI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='b target affected by high aleatoric variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='Figure 3: Experiment 3: Emin achieved by different methods as a function of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='the number of sampled data points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', number of optimization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The robust Gaussian LCB method models the variance of the squared error as Gaussian and differentiates between aleatoric and epistemic uncertainty, where a baseline Gaussian LCB method neglects the differentiation of variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The remaining methods estimate variance of squared errors by NCχ2 distribution and robust methods additionally take advantage of aleatoric variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In (a) we choose a target in a region with low aleatoric variance and in (b) we choose a target in a region with high aleatoric variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' All data points from the pool data set are drawn, therefore, all approaches converge to the overall minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 15 same as that of the standard NCχ2 EI procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, after the third optimization iteration, the robust NCχ2 EI prefers a data point with a higher squared error (about 526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='7) but a lower aleatoric variance to minimize Emin overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Neglecting aleatoric effects, the standard NCχ2 EI keeps its optimiza- tion result at 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='7 until iteration 155, where a better optimization result is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We observed that methods that do not use aleatoric uncertainty in the ac- quisition function spend long time near the selected target y•, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', d• max as the seemingly ’best’ optimization result until the actual optimum is found randomly by further drawing data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 6 Discussion and Limitations We proposed acquisition functions for robust BO with the aim that the output of a black box function is close to a target value in the sense of an expected squared error and under the assumption that aleatoric uncertainty due to en- vironmental effects is known or can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We show in our experiments that this assumption is at least approximately compatible with a large set of scenarios, including standard GPs with noisy measurements, GPs with noisy in- puts (including cascades of GPs such as deep [Damianou and Lawrence, 2013] or stacked GPs [Abdelfatah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016, Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2009], in which aleatoric uncertainties can be propagated), and machine learning approaches in which aleatoric and epistemic uncertainties are learned separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' While our results show that our acquisition functions outperform classical approaches in the considered task, our approach and Assumptions 2 and 3 imply certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In the case of GPs, for example, Assumption 2 requires that training and re-training (after obtaining a new data point) relies on noise-free data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' When measurement noise is included in the dataset, then the variance of the predictive posterior would include mixed, inseparable components from both epistemic and aleatoric uncertainties, making the separation necessary for our derivations impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Similarly, if epistemic and aleatoric uncertainties are represented by other machine learning models, these models must be trained on data that facilitates such a separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For example, if the mean and the aleatoric uncertainty of the mapping are modeled by a GP and a nonlinear regression model, respectively (as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='3), we need noise-free and un- biased estimates of the mean for training the GP, and noise-free and unbiased estimates of the aleatoric variance for training the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This, in turn, requires taking sufficiently many measurements at each position x, such that the mean and the variance of the resulting measurement can be estimated with little error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The advantage of faster convergence of the optimization prob- lem thus has to be traded against the requirement to take multiple (simulation or experimental) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' A possible remedy for this limitation could be to allow finite measurement noise up to a magnitude that is small compared to the aleatoric uncertainty, or to include a separate mixing term representing inferential uncertainty in the predictive posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Finally, [Ankenman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 16 2010] showed that estimates for the aleatoric variance from even very small sample sizes allow for good approximations in the predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In this work, we measured utility by the expected squared error between the output of the mapping and a given target value, where the expectation is taken over aleatoric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Modeling aleatoric and epistemic uncertainties with Gaus- sian distributions, this operational goal allowed us to derive acquisition functions in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Future research shall extend our work to different practically relevant operational goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For example, replacing the expected squared error by the probability for an excess error leads to the aim of finding an x that maximizes P (|y − y•| < ε), where ε defines the tolerance level and where the probability is evaluated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' the aleatoric uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Such a setting may be useful in applications where certain tolerance bands must not be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Until now, we focused on optimization of individual stochastic mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, if one aims to optimize entire manufacturing chains, GP surrogate models can be stacked [Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In stacked GPs, the output of a previous GP is (a part of) the input for the following GP, and uncertainties are propagated through the entire chain [Abdelfatah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Future work shall investigate how aleatoric and epistemic uncertainties can be propagated sepa- rately through the stacked GPs, such that our proposed acquisition functions can be utilized for the optimization of entire process chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Finally, Bayesian Optimization has also been investigated with respect to scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Scalable BO algorithms have been derived for large data sets [Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2015, Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2019], large input dimensions [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2016, Daulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021], many objectives [Martín and Garrido-Merchán, 2021] and large output dimensions or many tasks [Hakhamaneshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' A possible future direction is under which circumstances our proposed robustness towards stochastic environmental variables can be extended to these scaling variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Note however that we made no strong assumptions on the number of environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 7 Conclusion In this work, we derived a set of acquisition functions for Bayesian target value optimization that is robust against stochastic environmental variables, based on a common Gaussian process surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In contrast to the usual Gaussian distributions of simple minimization/maximization, this leads to non-central chi- square probability density functions for the sought-for optimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' This optimization problem was then considered in the presence of aleatoric effects in environmental (non-controllable) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' We find that knowledge of this aleatoric uncertainty can be leveraged advantageously towards optima that are robust against such stochastic environmental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' For this, we demonstrate experimentally that estimates or learned models of the aleatoric variance can be sufficient, and that the approach is of particular advantage if aleatoric variance is indeed large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Based on the good performance in an alloy billet forging problem, it is spec- 17 ulated that the approach might be useful for broader applications in manufac- turing and industrial engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Aleatoric uncertainy is, after all, present in many data sets and hence a large class of machine learning or optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Acknowledgements J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Hoffer and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Geiger were supported by the project BrAIN - Brownfield Artificial Intelligence Network for Forging of High Quality Aerospace Compo- nents (FFG Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 881039).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' The project is funded in the framework of the program ’TAKE OFF’, which is a research and technology program of the Aus- trian Federal Ministry of Transport, Innovation and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Ranftl was supported by University of Technology’s LEAD Project ’Mechanics, Modeling and Simulation of Aortic Dissection’.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Zhan and Xing, 2020] Zhan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' and Xing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Expected improve- ment for expensive optimization: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Journal of Global Optimization, 78(3):507–544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', 2020] Zhou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', and Gu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Neural contextual bandits with ucb-based exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 11492–11502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 22 A Appendix A: Practical Aspects of the Acqui- sition Functions Maximizing (10) over x is complicated, since the CDF of a non-central χ2- distribution is not available in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' However, it was shown in [Sankaran, 1959] that using a non-linear transform, the distribution of e ∼ NCχ2(K, λ) can be transformed into an approximately Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Specifically, by setting z = � e K + λ �ℓ (15) with ℓ = 1−r1r3/3r2 2, rs = 2s−1(s−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (K+sλ), we get that z has approximately the distribution N � α, ρ2� with α = 1 + ℓ(ℓ − 1) � r2 2r2 1 − (2 − ℓ)(1 − 3ℓ) r2 2 8r4 1 � (16a) ρ = ℓr2 2 r1 � 1 − (1 − ℓ)(1 − 3ℓ) 4r2 1 r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (16b) By this approximation, we obtain a closed-form approximation of the CDF of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', e(x) as F1,λ(x)(e(x)) ≈ Φ �e(x) − α ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (17) With this, we circumvent the straightforward approach of modelling directly pN (e|x, D) = N(e|µ(x), σ2(x)), where p(e|x) is imprecise, because it would be symmetric and with negative support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' B Additional Figures for Experiment 1 For experiment 1, we did in addition an evaluation by using LCB acquisi- tion function, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Similar to evaluation with EI acquisition function, we can show that by increasing aleatoric uncertainty σa (Figure 4b - 4c) our robust NCχ2 outperforms, see Figure 4e - 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' In a setting where aleatoric uncer- tainty is low (Figure 4a) our robust NCχ2 acquisition function performs similar to the approach of [Uhrenholt and Jensen, 2019] as expected, as the influence of σa is rather small in Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 23 1 0 1 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 f / y target f y NC 2 robust NC 2 a σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01 1 0 1 x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='0 f / y target f y NC 2 robust NC 2 b σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 1 0 1 x 2 1 0 1 f / y target f y NC 2 robust NC 2 c σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 0 1 2 3 4 number of iterations 10 4 10 3 10 2 10 1 Emin NC 2 robust NC 2 2 a d σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='01 0 1 2 3 4 number of iterations 10 2 10 1 Emin NC 2 robust NC 2 2 a e σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='1 0 1 2 3 4 number of iterations 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 Emin NC 2 robust NC 2 2 a f σa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='5 Figure 4: Experiment 1 with LCB (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (a)-(c) Scatterplots of the used data with different levels of aleatoric variance σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' Solid lines represent the noise-free function f, crosses the observations y, dots the sampled candidate positions, and dash-dotted lines the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' (d)-(f) Emin over the number of sampled candidate positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=', number of optimization iterations, for different levels of aleatoric variance σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE3T4oBgHgl3EQfJglg/content/2301.04344v1.pdf'} diff --git a/kb_46/content/tmp_files/kb_46.pdf.txt b/kb_46/content/tmp_files/kb_46.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..164eca57107007f8876ee4aac70bf2508e8b4800 --- /dev/null +++ b/kb_46/content/tmp_files/kb_46.pdf.txt @@ -0,0 +1,774 @@ +ANYmal - A Highly Mobile and Dynamic Quadrupedal Robot* +Marco Hutter1, Christian Gehring2, Dominic Jud1, Andreas Lauber1, C. Dario Bellicoso1, Vassilios Tsounis1, +Jemin Hwangbo1, Karen Bodie 1, Peter Fankhauser1, Michael Bloesch2, Remo Diethelm2, Samuel Bachmann2, +Amir Melzer 1, Mark Hoepflinger 1 +Abstract— This paper introduces ANYmal, a quadrupedal +robot that features outstanding mobility and dynamic motion +capability. Thanks to novel, compliant joint modules with +integrated electronics, the 30 kg, 0.5 m tall robotic dog is torque +controllable and very robust against impulsive loads during +running or jumping. The presented machine was designed +with a focus on outdoor suitability, simple maintenance, and +user-friendly handling to enable future operation in real world +scenarios. Performance tests with the joint actuators indicated a +torque control bandwidth of more than 70 Hz, high disturbance +rejection capability, as well as impact robustness when moving +with maximal velocity. It is demonstrated in a series of experi- +ments that ANYmal can execute walking gaits, dynamically trot +at moderate speed, and is able to perform special maneuvers +to stand up or crawl very steep stairs. Detailed measurements +unveil that even full-speed running requires less than 280 W, +resulting in an autonomy of more than 2 h. +I. INTRODUCTION +Legged robotics has potential advantages in terms of +mobility and versatility as compared to tracked or wheeled +vehicles. So far, the technological complexity to build and +control such vehicles has prevented these systems from being +applied in real world scenarios and only few teams managed +to develop machines that work beyond laboratory test-bench +settings. With major advances over the recent years, pushed +by various large scale research programs or investment +from industry, our community is about to overcome the last +technical hurdles and make legged robots available for real +world applications. Most prominently, the DARPA Robotics +Challenge (DRC) brought together some of the best research +groups in the field of humanoid robots to successfully use +such machines in a disaster mitigation scenario [1]. Since +the scenario is very close to reality, all teams were forced +to massively invest in hardware development to improve +not only versatility but also reliability and ruggedness of +the robots. These developments resulted in many high- +performance machines like ATLAS[2], Valkyrie [3], DRC +Hubo [4], HRP2+ [5], Walkman and others, most of them +based on earlier robot versions. This new generation of +humanoid robots commonly feature some sort of force or +torque control - either by integrated load cells in the joints +or at the end-effector, or by a series elasticity in every +*This work was supported in part by the Swiss National Science +Foundation (SNF) through the National Centre of Competence in Research +Robotics, by the EC’s 7th Famework ECHORD++ Project Module, and by +TOTAL SA through the ARGOS Challenge. +1 Authors are with the Robotic Systems Lab, ETH Zurich, Switzerland, +mahutter@ethz.ch +2 Authors are with the Autonomous Systems Lab, ETH Zurich, Switzer- +land. +Fig. 1. +ANYmal, an autonomous quadrupedal robot for rough terrain +operation +actuator. This allows them to properly control interaction +forces with the environment and hence to balance the system +or manipulate the environment. +Despite all these advances, the locomotory performance +of the human-like robots is still far behind the natural +counterparts. All these robots are relatively slow, require a +lot of power, and can only negotiate small terrain obstacles. +Better locomotion performance in terms of speed, ener- +getic efficiency, and obstacle negotiation skills, is achieved +with multi-legged systems. Paramount example is Boston +Dynamics’ Spot robot, a direct successor of Big Dog [6], of +which unfortunately no scientific publications are available. +Similar locomotion performance, demonstrated in various +highly dynamic gaits and maneuvers, was also achieved +by research groups around IIT’s hydraulic HyQ [7] and its +follower HyQ2max [8], MIT’s directly electrically actuated +cheetah [9], or ETH’s serial elastic robot StarlETH [10]. +All these robots have demonstrated dynamic running on +different grounds or to dynamically overcome obstacles - +however, none of these machines has been used in a real +world application. +This paper presents ANYmal (Fig. 1), a highly mobile +2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) +Daejeon Convention Center +October 9-14, 2016, Daejeon, Korea +978-1-5090-3762-9/16/$31.00 ©2016 IEEE +38 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +Fig. 2. +Main components of ANYmal +and rugged quadrupedal platform developed for autonomous +operation in challenging environments. ANYmal was de- +signed to combine outstanding mobility with dynamic motion +capability that enables it to climb large obstacles as well as +to dynamically run. This completely autonomous machine +paves the road for real world applications. It is in use for +the NCCR Search and Rescue grand challenge1 as well as +in the ARGOS oil and gas site inspection challenge2 - both +scenarios with very harsh and demanding environments. In +the following, we present the underlying mechanics and +actuation concept, illustrate the electronics and software +setup, sketch out the applied locomotion control algorithms +with appropriate references to their implementation, and +finally summarize the paper with a series of experiments3 +highlighting the overall system performance. +II. SYSTEM DESCRIPTION +ANYmal was specifically built for long endurance au- +tonomous operation in harsh environments. Focus was put +on large mobility, fast and dynamic locomotion skills, high +robustness, simple maintenance, and safe handling by a +single operator. +A. Overview +The presented quadrupedal robot, with the main compo- +nents depicted in Fig. 2, features three actuated joints per leg +with point feet. With an approximate link length of 250 mm +for thigh and shank, and a total weight of slightly less than +30 kg, it resembles a medium-sized dog. To achieve this +lightweight design, the main body and the leg segments are +built from aluminum and carbon fiber. Onboard batteries of +about 650 Wh energy and 3 kg weight provide power for +more than 2 h autonomous operation. A protection frame +and pads at the legs prevent the system from damage when +falling and allows for handy transportation and deployment. +1http://www.argos-challenge.com +2http://www.nccr-robotics.ch/RescueRobots +3for illustration, see http://www.rsl.ethz.ch/robots-media/anymal +Fig. 3. +Range of motion of a single leg of ANYmal +Optofoce sensors are used as tactile feet and rotating Hokuyo +UTM-30LX sensors provide 3D perception of the environ- +ment. To make ANYmal applicable for different scenarios, +a modular pan-tilt head with variable sensory payload can +be mounted. For example, in the setup for the ARGOS +challenge, the sensory head includes an optical zoom and +thermal camera for visual inspection, a gas detection sensor, +microphones for sound identification, as well as artificial +lighting. +B. Modular joint setup +Key to simultaneously achieving the design goals are +the robotic joint units described in Sec. III. This enabled +the creation of a very simple mechanical topology with +three equal joint units per leg that are linked by rigid +mechanical segments and interconnected with a power and +communication bus. Since the encapsulated and sealed joint +units integrate drive electronics and sensing, as well as the +joint axle bearing, the robot does not require any additional +bearings, transmission, proprioceptive sensors, or electronics +in its legs. Such a setup combines several advantages: Given +the drive units, the robot is simple to manufacture, assemble, +and maintain. In case of failure, a complete joint can be +quickly exchanged. Furthermore, design variations to build +different robots requires only to change the mechanical links. +The joint arrangement of ANYmal is chosen mammalian +with successive hip abduction/adduction (HAA), hip flex- +ion/extension (HFE), and knee flexion/extension (KFE). In +contrast to its predecessor StarlETH [10], the MIT cheetah +[9], IIT HyQ [7], Big Dog [6] or other legged systems, the +leg links of ANYmal are built with an offset such that all +joints can be fully rotated. So far, this was typically only +done in walking machines like JPL’s robosimian [11] that +moves in a quasi static manner. As depicted in Fig. 3, the +joint offset enables a huge range of motion which is key to +high mobility. With this, ANYmal is able to use its feet high +above ground for tasks like opening a door or surmounting +large obstacles, it can be folded for transport or deployment, +and can change its leg configuration (Fig. 4). +C. Main body package +Computers, batteries, network devices, the power manage- +ment system, and basic navigation sensors are integrated in +a single box-shaped and ingress protected main body. Three +39 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +JFig. 4. +Full rotation in all joints of ANYmal allow for various configura- +tions. +intel NUC PCs connected over an internal gigabit network +form the removable brain of ANYmal that is accessible +via WiFi link from any operator machine. To get proper +heat dissipation from the sealed main body, all components +are thermally coupled to the main body which is used as +heat sink. The main body is controlled from a small touch +screen on the back of the robot which allows to individually +enable PCs and sensors. A rotating Hokuyo UTM-30lx laser +sensor and an Xsens MTi-100 IMU are fixedly installed for +localization, navigation, and environment mapping. +D. Software architecture +The three PCs share the work load of the locomotion, +navigation and inspection tasks as illustrated in Fig. 5. The +data is transferred over the network by the Robot Operating +System (ROS) running on a low-latency patched Ubuntu +14.04. The ROS master, which manages the connections +between the different processes, runs on the locomotion +PC. The real-time critical whole-body controller and state +estimator are timed by the CAN driver that communicates +with the actuator units at 400 Hz. The readings and com- +mands are exchanged through shared memory and published +through ROS to less time-critical workers like the foothold +planner. The localization and mapping tasks are outsourced +to the navigation PC that is responsible for the laser-based +localization and mapping of the environment. High-level +navigation tasks are coordinated by a mission planner and +executed by a path planner that sends velocity commands +to the locomotion controller. Optionally, a third application +specific PC can be activated to handle for example the +computationally extensive video processing for inspection. +III. ANYDRIVE - MODULAR JOINT UNITS +Dynamic locomotion imposes very demanding require- +ments on the actuation system, namely: +• High impact robustness +• Fast motion tracking +• Low impedance force controllability +Furthermore, actuators must be lightweight and energetically +as efficient as possible. +Fig. 5. +The software architecture with clear real-time priority ranked +separation on different PCs. +A. A brief review on existing actuation concepts +The classical approach of electric motors with high- +reduction mechanical gears as employed in almost all in- +dustrial robot arms does not satisfy the first requirement. +Legged robots using such actuation approach are limited to +slow and static locomotion in order to prevent the actuators +from impulsive forces. For dynamic locomotion, three major +concepts have established as adequate actuation technology. +1) Hydraulic actuation: Hydraulic actuators as used in +machines like HyQ [7], BigDog [6] or Atlas [2] are naturally +robust against impulsive loads and provide extremely high +power and force density. Thanks to very fast valve units +in combination with load cells for force measurement or +pressure based force estimation [12], hydraulic actuators +provide also high performance torque control. +On the negative side, hydraulic systems tend to be energet- +ically inefficient, in particular when operated with constant +pressure. For this reason, many systems used in research +still rely on off-board supply. At the cost of increased +system complexity, this can be overcome to certain extent +by sophisticated pumps and variable pressure levels. Another +problem is scalability which makes hydraulic legged systems +rather large and heavy. +2) Pseudo-direct-drive systems: When using gearing sys- +tems of very low reduction and high efficiency, electric ac- +tuators can become very transparent and the reflected inertia +of the actuation compared to the output becomes small. As a +result, motor current control, which can be done at very high +bandwidth, is equivalent to regulation of the output force +[13]. These benefits have been exploited for many years in +rehabilitation engineering and for haptic devices. Thanks to +recent advances in actuator development, pseudo-direct-drive +concepts find application in high-dynamic legged robots as +in the example of MIT-cheetah [9], which is able to run and +jump at high speeds. +Unfortunately, while electric motors have extremely high +power, their torque is limited. Therefore, direct actuation +without any gear is not possible with existing technology. +Furthermore, motors of large diameter must be used to create +high forces, which largely limits flexibility in system design. +3) Series elastic actuation: Inspired from biology and +along the seminal work of Pratt [14], the third common actu- +ation approach for legged robots are series elastic actuators. +40 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +Fig. 6. +ANYdrive: Compact, compliant joint units for advanced interaction + + +Position Control +Torque Control +PID +θ j +des +τ +des +i +des +PID +1 +N K T ++ ++ +e τ ++ +- +τ +Friction +Compensation ++ ++ +eθ +SEA +˙θ j +d +dt +θ j +θg ++ +- +k +- ++ +Fig. 7. +Block diagram of the cascaded joint position and torque control +loop. The SEA block represents the physical actuator unit including field +oriented control (FOC) to apply the desired current ides +By integrating a carefully selected mechanical compliance +between the gearbox output and joint, classical geared motors +can be adapted for applications with dynamic interactions. +Several state of the art robots like the humanoid Valkyrie +[15] or the quadruped StarlETH [10] showed how to use +such actuators not only for precise output force regulation, +but even to temporarily store energy during locomotion and +hence to increase locomotion efficiency [16]. In order to +simplify the use of such actuators, different groups target +the development of modular units [17], [18]. +The mechanical compliance in the system is not only a +low pass filter (and hence protection) for the impact loads +at the output, but additionally limits control bandwidth and +requires careful design of the joint level control structure. +B. Overview +ANYdrive (Fig. 6), the joint units of ANYmal, is a +highly integrated series elastic actuator. It is built upon high +torque motors and harmonic drive gears in series with a +rotational spring. Joint output position and spring deflection +are measured using absolute position sensors providing a po- +sition accuracy of 0.025◦and a torque resolution of 0.08 Nm. +Thanks to integrated custom motor control electronics, the +joint torque, position, and impedance can be directly regu- +lated without any additional components. The corresponding +command values are sent over CAN bus using CANopen +standard. With a nominal voltage of 48 V, the joint reaches +a speed of 12 rad/s and a maximal torque of 40 Nm. +C. Control structure +Joint torque, position and impedance control is realized as +a cascaded structure that considers the motor as torque source +(c.f. [19]) as illustrated in Fig. 7. Similar to the work by Paine +[20], which is also the basis of the control of Valkyrie [15], +we realized a simple PID torque feedback loop with feedback +friction compensation. The position PID control builds upon +the torque controller as an additional cascade. +The torque controller tracks a desired torque τ des by +measuring the actual output torque τ and setting the desired +current ides accordingly. The spring deflection is calculated +from the difference in the joint position θj and the gear +position θg. The output torque τ is then calculated using +the spring constant k. The torque controller consists of three +elements, i.e. a PID controller, a feed forward term and a +feedback friction compensation. The feed forward term is +determined from the inverse of the gear ratio N and the +motor constant KT , both typically provided in by data sheets. +The friction compensation +icomp( ˙θj) = ibasSign( ˙θj, ˙θband) + µ ˙θj +(1) +takes two effects into account, namely stiction and viscous +friction. Firstly the break-away current iba is modeled as +Coulomb friction. To prevent undesired switching around the +zero velocity point, it is implemented as simple smooth sign +function +sSign(x, xb) = +� +� +� +� +� +−1, +if x < xb +1, +if x > xb +−1 + 2( x+xb +2xb )2(2 − x +xb ), +otherwise +(2) +Secondly, the joint velocity dependent viscous friction is +linearly modeled with the friction coefficient µ. All these +parameters can be experimentally identified from very few +measurements. +The position controller is a PID controller that tracks +a desired joint position θdes +j +by setting a desired torque +τ des. An important note is that the position gains are highly +depending on the output load since there is no knowledge +about the joint load in the control architecture. +D. Performance evaluation +The performance of ANYdrive with respect to torque and +position reference tracking as well as impulsive disturbance +rejection is evaluated on a single axis test bench. As illus- +trated in Fig. 8, the bandwidth for low amplitudes is as high +as 70 Hz. Due to motor saturation effects, the bandwidth +gradually decreases to 24 Hz for 10 Nm amplitude. These +performance values are substantially higher than what was +achieved with our previous system [21] and about the same +as in Valkyrie [15]. Interestingly, this high performance was +achievable without a disturbance observer as used in [22]. +As illustrated in Fig. 9, the controller is very reactive +showing a 90% settling time of 13 ms for a step of 10 Nm +and 35 ms for a step of 40 Nm with only small overshoot. +Disturbance rejection to impulsive loads is evaluated in +a collision test. To this end, a pendulum is mounted at the +output and the actuator is requested to produce zero torque. +The free swinging pendulum is crashed with high velocity +into a hard wall and brought to instantaneous rest (ideal +plastic collision with a restitution coefficient of zero). Despite +41 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +CFig. 8. Experimentally identified torque control transfer function indicating +a bandwidth of 70 Hz. +Fig. 9. +Torque step responses show a quick response time and low +overshoot. +high motor speed before the collision, the motor produces +only little torque during the impact (Fig. 10). In fact, already +2 ms after the collision, the motor maximally decelerates to +keep the torque in the spring as small as possible. Due to +the motor and gearbox inertia, it takes about 10 ms to bring +the motor to a complete rest. If the pendulum collides with +the maximal motor velocity, the peak force is smaller than +7 Nm. This implies that, whatever collision a system that is +built from these joint units experiences, forces occurring at +the gear never exceed the peak loads it is rated for. In other +words, the drive is ”perfectly robust” against self inflicted +collisions. +As final performance evaluation experiment, the actuator +was again commanded to produce zero torque while the out- +put is randomly moved by hand (Fig. 11). Despite very large +disturbances (2 rad amplitude and about 4 Hz motion), the +output torque can be kept at less than 0.2 Nm. A qualitative +comparison to Valkyrie [3] indicates a significantly better +disturbance rejection performance. +IV. LOCOMOTION CONTROL +Since ANYmal is fully torque controllable and of simi- +lar geometry as its predecessor StarlETH [10], locomotion +control could be transferred relatively directly. A detailed +description would go beyond the scope of this paper, we +Fig. 10. +Joint torque during impulsive collision. The motor velocity is +scaled with the gear ratio for plotting purposes. +Fig. 11. Zero torque tracking error (blue) when the output joint is randomly +moved by hand (red). +refer the interested reader to the related work introduced in +the following. +ANYmal features a purely proprioceptive state estimation +based on fusion of IMU, leg kinematics, and ground contact +measurements [23], [24]. For static walking gaits, a ZMP +planner [25] is implemented to plan a smooth main body +trajectory while applying a standard crawling gait [26]. +Foothold placement during dynamic gaits is based on simpli- +fied inverted pendulum models [27]. To balance the system, +we build upon whole body control techniques that accounts +for the complete system kinematics and dynamics [28], [29]. +The optimal actuator commands are found at every time step +by solving a constrained optimization problem of prioritized +tasks and constraints on joint torques, contact forces, and +body motion. +V. EXPERIMENTS +The performance of ANYmal was tested in different ma- +neuvers and locomotion experiments illustrated in Fig. 124. +In order to ensure fast and stable locomotion, particular +attention was paid to accurate swing leg position and stance +leg force tracking, as well as good following behavior of the +target base motion. +A. Walking +ANYmal is able to perform a very smooth walking gait, +whereby a single leg is moved at the time and the base is +shifted in order to maintain balance. As illustrated in Fig. 13, +joint torques and positions are followed very accurately +during the entire gait cycle and hence also the base position +can be accurately moved according to the preplanned tra- +jectory. It is important to know that the latter follows only +from virtual model control (task space control) at the base +4For videos, see http://www.rsl.ethz.ch/robots-media/anymal +42 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +(a) Walking +(b) Trotting +(c) Stair Climbing +Fig. 12. +ANYmal was tested in different gaits like walking, trotting, or stair climbing +Fig. 13. +Torque and position tracking while walking. +and without any joint position or impedance regulation. By +applying a classical ZMP planner [25], forward locomotion +results in a very smooth and almost straight line of the base +as illustrated in the movie. In such gait, the robot moves with +approximately 0.3 m/s. Thanks to the full rotation capability, +the motion planner does not have to account for complex +geometric collision constraints but only for limited abduction +freedom due to the main body. Furthermore, ANYmal can +take fairly big steps. +B. Trotting +ANYmal is able to trot on different grounds and under +large external disturbances. Similar to the walking gait, +already the first experiments unveiled large advantages of the +big range of motion as the legs can be moved relatively far +in all directions. Using a 50% duty cycle gait, the machine +achieves a speed of about 0.8 m/s. Key to robust trotting +is fast and accurate position tracking. For a typical joint +motion of a fast gait (Fig. 14), joint positions and velocities +are followed accurately despite the joint compliance. +A +thorough evaluation of the overall energy consumption at +the onboard battery indicated a relatively low consumption +even during dynamic trotting gait. As depicted in Fig. 15, +ANYmal requires in average about 290 W with about 5% +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +time [s] +knee joint position [rad] + + +measured +desired +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +time [s] +knee joint velocity [rad/s] + + +measured +desired +Fig. 14. +Tracking performance of the position and velocity of the knee +joint. +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +0 +2 +4 +6 +8 +10 +current [A] +50 +100 +150 +200 +250 +300 +350 +power [W] +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +47 +47.5 +48 +48.5 +49 +voltage [V] +time [s] +trotting +Fig. 15. +Power consumption during trotting. +fluctuation when trotting, about 100 W is consumed while +idling in standing configuration. The total power consump- +tion corresponds to a cost of transport of about 1.2. These +measurements are comparable to our previous results with +StarlETH [30] and enables the machine to autonomously run +for more than 2 h with its current batteries. +C. Stair Climbing +As a proof of high mobility, ANYmal was tested for the +ability to get up an industrial ladder of about 50◦. To do +this in a save manner and to prevent falling by all possible +means, a turtle like crawling gait was implemented. The main +body lies on the ground, the legs are moved to find the +next stable contact holds, and the machine is subsequently +pulled upwards (see Fig. 12(c)). Due to ANYmal’s large +range of motion, the legs can be literally turned overhead +to prevent collision with the ground or side rails. This +maneuver was inspired by our work with ALoF, a kinematic +quadrupedal robot that was developed for the ESA Lunar +43 +Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore. Restrictions apply. + +ETHzurich +Legged Robotics +Machines for locomotion over +challenging terrain. +slywalk,climb +HYMER +www.asLethz.ch +DMAVTRobotic Challenge [31]. This machine successfully exhibited +such gait to reliably overcome steep inclinations with loose +sand during a moon testing scenario on a volcano. +VI. CONCLUSION/FUTURE WORK +ANYmal is considered a step towards unification of high +mobility with dynamic locomotion capability. +From the beginning of the design phase, special attention +was put on a rugged and simple to maintain system. This +was achieved with the modular joint units ANYdrive that +allow to very simply create robots of different kinematic +structure. In case of failure, these modules can be easily +and quickly exchanged without special knowledge. These +actuators are based on a series elastic concept as already +implemented on StarlETH, where we did not have a single +gearbox failure in 4 years of almost daily operation with +high-dynamic maneuvers. The presented experiments support +the claim of robustness since even completely plastic and +unexpected output collisions do not lead to higher gearbox +loads than during nominal operation. +Beside the highly improved protection, the biggest advan- +tage of ANYmal is clearly the outstanding range of motion +in all joints. This enables a large variety of maneuvers to +overcome obstacles or to get up after falling. Furthermore, it +simplifies motion planning as there are less internal system +constraints. The initial objectives of creating a dynamic +and highly mobile autonomous walking machine could be +confirmed in preliminary experiments including careful stair +climbing, ZMP-based walking and dynamic trotting. The +present development shall enable deployment of legged +robots in real world scenarios such as for search and rescue +or industrial inspection. +REFERENCES +[1] DRC, +“DARPA +Robotics +Challenge +(DRC) +, +http://www.theroboticschallenge.org/.” +[2] Boston +Dynamics, +“ATLAS +Robot +- +http://www.bostondynamics.com/robot Atlas.html.” +[3] N. A. Radford et al., “Valkyrie: NASA’s First Bipedal Humanoid +Robot,” Journal of Field Robotics, vol. 32, no. 3, pp. 397–419, 2015. +[4] H. Wang, Y. F. Zheng, Y. Jun, and P. Oh, “DRC-hubo walking on +rough terrains,” in IEEE International Conference on Technologies +for Practical Robot Applications (TePRA), 2014. +[5] K. Kaneko et al., “Humanoid robot HRP-2Kai Improvement of HRP-2 +towards disaster response tasks,” in 2015 IEEE-RAS 15th International +Conference on Humanoid Robots (Humanoids), pp. 132–139, 2015. +[6] M. Raibert, K. Blankespoor, G. Nelson, and R. 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Restrictions apply. + diff --git a/kb_46/content/tmp_files/load_file.txt b/kb_46/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8006a3be688766e901ed8242bb5b9e2d7b5a0f0d --- /dev/null +++ b/kb_46/content/tmp_files/load_file.txt @@ -0,0 +1,541 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf,len=540 +page_content='ANYmal - A Highly Mobile and Dynamic Quadrupedal Robot* Marco Hutter1, Christian Gehring2, Dominic Jud1, Andreas Lauber1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Dario Bellicoso1, Vassilios Tsounis1, Jemin Hwangbo1, Karen Bodie 1, Peter Fankhauser1, Michael Bloesch2, Remo Diethelm2, Samuel Bachmann2, Amir Melzer 1, Mark Hoepflinger 1 Abstract— This paper introduces ANYmal, a quadrupedal robot that features outstanding mobility and dynamic motion capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Thanks to novel, compliant joint modules with integrated electronics, the 30 kg, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 m tall robotic dog is torque controllable and very robust against impulsive loads during running or jumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The presented machine was designed with a focus on outdoor suitability, simple maintenance, and user-friendly handling to enable future operation in real world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Performance tests with the joint actuators indicated a torque control bandwidth of more than 70 Hz, high disturbance rejection capability, as well as impact robustness when moving with maximal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' It is demonstrated in a series of experi- ments that ANYmal can execute walking gaits, dynamically trot at moderate speed, and is able to perform special maneuvers to stand up or crawl very steep stairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Detailed measurements unveil that even full-speed running requires less than 280 W, resulting in an autonomy of more than 2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' INTRODUCTION Legged robotics has potential advantages in terms of mobility and versatility as compared to tracked or wheeled vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' So far, the technological complexity to build and control such vehicles has prevented these systems from being applied in real world scenarios and only few teams managed to develop machines that work beyond laboratory test-bench settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' With major advances over the recent years, pushed by various large scale research programs or investment from industry, our community is about to overcome the last technical hurdles and make legged robots available for real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Most prominently, the DARPA Robotics Challenge (DRC) brought together some of the best research groups in the field of humanoid robots to successfully use such machines in a disaster mitigation scenario .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Since the scenario is very close to reality, all teams were forced to massively invest in hardware development to improve not only versatility but also reliability and ruggedness of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' These developments resulted in many high- performance machines like ATLAS, Valkyrie , DRC Hubo , HRP2+ , Walkman and others, most of them based on earlier robot versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This new generation of humanoid robots commonly feature some sort of force or torque control - either by integrated load cells in the joints or at the end-effector, or by a series elasticity in every This work was supported in part by the Swiss National Science Foundation (SNF) through the National Centre of Competence in Research Robotics, by the EC’s 7th Famework ECHORD++ Project Module, and by TOTAL SA through the ARGOS Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 1 Authors are with the Robotic Systems Lab, ETH Zurich, Switzerland, mahutter@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ch 2 Authors are with the Autonomous Systems Lab, ETH Zurich, Switzer- land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYmal, an autonomous quadrupedal robot for rough terrain operation actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This allows them to properly control interaction forces with the environment and hence to balance the system or manipulate the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Despite all these advances, the locomotory performance of the human-like robots is still far behind the natural counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' All these robots are relatively slow, require a lot of power, and can only negotiate small terrain obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Better locomotion performance in terms of speed, ener- getic efficiency, and obstacle negotiation skills, is achieved with multi-legged systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Paramount example is Boston Dynamics’ Spot robot, a direct successor of Big Dog , of which unfortunately no scientific publications are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Similar locomotion performance, demonstrated in various highly dynamic gaits and maneuvers, was also achieved by research groups around IIT’s hydraulic HyQ and its follower HyQ2max , MIT’s directly electrically actuated cheetah , or ETH’s serial elastic robot StarlETH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' All these robots have demonstrated dynamic running on different grounds or to dynamically overcome obstacles - however, none of these machines has been used in a real world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This paper presents ANYmal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 1), a highly mobile 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Daejeon Convention Center October 9-14, 2016, Daejeon, Korea 978-1-5090-3762-9/16/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='00 ©2016 IEEE 38 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Main components of ANYmal and rugged quadrupedal platform developed for autonomous operation in challenging environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYmal was de- signed to combine outstanding mobility with dynamic motion capability that enables it to climb large obstacles as well as to dynamically run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This completely autonomous machine paves the road for real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' It is in use for the NCCR Search and Rescue grand challenge1 as well as in the ARGOS oil and gas site inspection challenge2 - both scenarios with very harsh and demanding environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In the following, we present the underlying mechanics and actuation concept, illustrate the electronics and software setup, sketch out the applied locomotion control algorithms with appropriate references to their implementation, and finally summarize the paper with a series of experiments3 highlighting the overall system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' SYSTEM DESCRIPTION ANYmal was specifically built for long endurance au- tonomous operation in harsh environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Focus was put on large mobility, fast and dynamic locomotion skills, high robustness, simple maintenance, and safe handling by a single operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Overview The presented quadrupedal robot, with the main compo- nents depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 2, features three actuated joints per leg with point feet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' With an approximate link length of 250 mm for thigh and shank, and a total weight of slightly less than 30 kg, it resembles a medium-sized dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To achieve this lightweight design, the main body and the leg segments are built from aluminum and carbon fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Onboard batteries of about 650 Wh energy and 3 kg weight provide power for more than 2 h autonomous operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A protection frame and pads at the legs prevent the system from damage when falling and allows for handy transportation and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='argos-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='com 2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='nccr-robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ch/RescueRobots 3for illustration, see http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='rsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ch/robots-media/anymal Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Range of motion of a single leg of ANYmal Optofoce sensors are used as tactile feet and rotating Hokuyo UTM-30LX sensors provide 3D perception of the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To make ANYmal applicable for different scenarios, a modular pan-tilt head with variable sensory payload can be mounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' For example, in the setup for the ARGOS challenge, the sensory head includes an optical zoom and thermal camera for visual inspection, a gas detection sensor, microphones for sound identification, as well as artificial lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Modular joint setup Key to simultaneously achieving the design goals are the robotic joint units described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This enabled the creation of a very simple mechanical topology with three equal joint units per leg that are linked by rigid mechanical segments and interconnected with a power and communication bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Since the encapsulated and sealed joint units integrate drive electronics and sensing, as well as the joint axle bearing, the robot does not require any additional bearings, transmission, proprioceptive sensors, or electronics in its legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Such a setup combines several advantages: Given the drive units, the robot is simple to manufacture, assemble, and maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In case of failure, a complete joint can be quickly exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Furthermore, design variations to build different robots requires only to change the mechanical links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The joint arrangement of ANYmal is chosen mammalian with successive hip abduction/adduction (HAA), hip flex- ion/extension (HFE), and knee flexion/extension (KFE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In contrast to its predecessor StarlETH , the MIT cheetah , IIT HyQ , Big Dog or other legged systems, the leg links of ANYmal are built with an offset such that all joints can be fully rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' So far, this was typically only done in walking machines like JPL’s robosimian that moves in a quasi static manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 3, the joint offset enables a huge range of motion which is key to high mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' With this, ANYmal is able to use its feet high above ground for tasks like opening a door or surmounting large obstacles, it can be folded for transport or deployment, and can change its leg configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Main body package Computers, batteries, network devices, the power manage- ment system, and basic navigation sensors are integrated in a single box-shaped and ingress protected main body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Three 39 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' JFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Full rotation in all joints of ANYmal allow for various configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' intel NUC PCs connected over an internal gigabit network form the removable brain of ANYmal that is accessible via WiFi link from any operator machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To get proper heat dissipation from the sealed main body, all components are thermally coupled to the main body which is used as heat sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The main body is controlled from a small touch screen on the back of the robot which allows to individually enable PCs and sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A rotating Hokuyo UTM-30lx laser sensor and an Xsens MTi-100 IMU are fixedly installed for localization, navigation, and environment mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Software architecture The three PCs share the work load of the locomotion, navigation and inspection tasks as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The data is transferred over the network by the Robot Operating System (ROS) running on a low-latency patched Ubuntu 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The ROS master, which manages the connections between the different processes, runs on the locomotion PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The real-time critical whole-body controller and state estimator are timed by the CAN driver that communicates with the actuator units at 400 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The readings and com- mands are exchanged through shared memory and published through ROS to less time-critical workers like the foothold planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The localization and mapping tasks are outsourced to the navigation PC that is responsible for the laser-based localization and mapping of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' High-level navigation tasks are coordinated by a mission planner and executed by a path planner that sends velocity commands to the locomotion controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Optionally, a third application specific PC can be activated to handle for example the computationally extensive video processing for inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYDRIVE - MODULAR JOINT UNITS Dynamic locomotion imposes very demanding require- ments on the actuation system, namely: High impact robustness Fast motion tracking Low impedance force controllability Furthermore, actuators must be lightweight and energetically as efficient as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The software architecture with clear real-time priority ranked separation on different PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A brief review on existing actuation concepts The classical approach of electric motors with high- reduction mechanical gears as employed in almost all in- dustrial robot arms does not satisfy the first requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Legged robots using such actuation approach are limited to slow and static locomotion in order to prevent the actuators from impulsive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' For dynamic locomotion, three major concepts have established as adequate actuation technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 1) Hydraulic actuation: Hydraulic actuators as used in machines like HyQ , BigDog or Atlas are naturally robust against impulsive loads and provide extremely high power and force density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Thanks to very fast valve units in combination with load cells for force measurement or pressure based force estimation , hydraulic actuators provide also high performance torque control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' On the negative side, hydraulic systems tend to be energet- ically inefficient, in particular when operated with constant pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' For this reason, many systems used in research still rely on off-board supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' At the cost of increased system complexity, this can be overcome to certain extent by sophisticated pumps and variable pressure levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Another problem is scalability which makes hydraulic legged systems rather large and heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 2) Pseudo-direct-drive systems: When using gearing sys- tems of very low reduction and high efficiency, electric ac- tuators can become very transparent and the reflected inertia of the actuation compared to the output becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As a result, motor current control, which can be done at very high bandwidth, is equivalent to regulation of the output force .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' These benefits have been exploited for many years in rehabilitation engineering and for haptic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Thanks to recent advances in actuator development, pseudo-direct-drive concepts find application in high-dynamic legged robots as in the example of MIT-cheetah , which is able to run and jump at high speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Unfortunately, while electric motors have extremely high power, their torque is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Therefore, direct actuation without any gear is not possible with existing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Furthermore, motors of large diameter must be used to create high forces, which largely limits flexibility in system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 3) Series elastic actuation: Inspired from biology and along the seminal work of Pratt , the third common actu- ation approach for legged robots are series elastic actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 40 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYdrive: Compact, compliant joint units for advanced interaction Position Control Torque Control PID θ j des τ des i des PID 1 N K T + + e τ + τ Friction Compensation + + eθ SEA ˙θ j d dt θ j θg + k + Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Block diagram of the cascaded joint position and torque control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The SEA block represents the physical actuator unit including field oriented control (FOC) to apply the desired current ides By integrating a carefully selected mechanical compliance between the gearbox output and joint, classical geared motors can be adapted for applications with dynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Several state of the art robots like the humanoid Valkyrie or the quadruped StarlETH showed how to use such actuators not only for precise output force regulation, but even to temporarily store energy during locomotion and hence to increase locomotion efficiency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In order to simplify the use of such actuators, different groups target the development of modular units , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The mechanical compliance in the system is not only a low pass filter (and hence protection) for the impact loads at the output, but additionally limits control bandwidth and requires careful design of the joint level control structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Overview ANYdrive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 6), the joint units of ANYmal, is a highly integrated series elastic actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' It is built upon high torque motors and harmonic drive gears in series with a rotational spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Joint output position and spring deflection are measured using absolute position sensors providing a po- sition accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='025◦and a torque resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='08 Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Thanks to integrated custom motor control electronics, the joint torque, position, and impedance can be directly regu- lated without any additional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The corresponding command values are sent over CAN bus using CANopen standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' With a nominal voltage of 48 V, the joint reaches a speed of 12 rad/s and a maximal torque of 40 Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Control structure Joint torque, position and impedance control is realized as a cascaded structure that considers the motor as torque source (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ) as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Similar to the work by Paine , which is also the basis of the control of Valkyrie , we realized a simple PID torque feedback loop with feedback friction compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The position PID control builds upon the torque controller as an additional cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The torque controller tracks a desired torque τ des by measuring the actual output torque τ and setting the desired current ides accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The spring deflection is calculated from the difference in the joint position θj and the gear position θg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The output torque τ is then calculated using the spring constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The torque controller consists of three elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' a PID controller, a feed forward term and a feedback friction compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The feed forward term is determined from the inverse of the gear ratio N and the motor constant KT , both typically provided in by data sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The friction compensation icomp( ˙θj) = ibasSign( ˙θj, ˙θband) + µ ˙θj (1) takes two effects into account, namely stiction and viscous friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Firstly the break-away current iba is modeled as Coulomb friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To prevent undesired switching around the zero velocity point, it is implemented as simple smooth sign function sSign(x, xb) = � � � � � −1, if x < xb 1, if x > xb −1 + 2( x+xb 2xb )2(2 − x xb ), otherwise (2) Secondly, the joint velocity dependent viscous friction is linearly modeled with the friction coefficient µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' All these parameters can be experimentally identified from very few measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The position controller is a PID controller that tracks a desired joint position θdes j by setting a desired torque τ des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' An important note is that the position gains are highly depending on the output load since there is no knowledge about the joint load in the control architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Performance evaluation The performance of ANYdrive with respect to torque and position reference tracking as well as impulsive disturbance rejection is evaluated on a single axis test bench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 8, the bandwidth for low amplitudes is as high as 70 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Due to motor saturation effects, the bandwidth gradually decreases to 24 Hz for 10 Nm amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' These performance values are substantially higher than what was achieved with our previous system and about the same as in Valkyrie .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Interestingly, this high performance was achievable without a disturbance observer as used in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 9, the controller is very reactive showing a 90% settling time of 13 ms for a step of 10 Nm and 35 ms for a step of 40 Nm with only small overshoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Disturbance rejection to impulsive loads is evaluated in a collision test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To this end, a pendulum is mounted at the output and the actuator is requested to produce zero torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The free swinging pendulum is crashed with high velocity into a hard wall and brought to instantaneous rest (ideal plastic collision with a restitution coefficient of zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Despite 41 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' CFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Experimentally identified torque control transfer function indicating a bandwidth of 70 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Torque step responses show a quick response time and low overshoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' high motor speed before the collision, the motor produces only little torque during the impact (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In fact, already 2 ms after the collision, the motor maximally decelerates to keep the torque in the spring as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Due to the motor and gearbox inertia, it takes about 10 ms to bring the motor to a complete rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' If the pendulum collides with the maximal motor velocity, the peak force is smaller than 7 Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This implies that, whatever collision a system that is built from these joint units experiences, forces occurring at the gear never exceed the peak loads it is rated for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In other words, the drive is ”perfectly robust” against self inflicted collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As final performance evaluation experiment, the actuator was again commanded to produce zero torque while the out- put is randomly moved by hand (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Despite very large disturbances (2 rad amplitude and about 4 Hz motion), the output torque can be kept at less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='2 Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A qualitative comparison to Valkyrie indicates a significantly better disturbance rejection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' LOCOMOTION CONTROL Since ANYmal is fully torque controllable and of simi- lar geometry as its predecessor StarlETH , locomotion control could be transferred relatively directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A detailed description would go beyond the scope of this paper, we Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Joint torque during impulsive collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The motor velocity is scaled with the gear ratio for plotting purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Zero torque tracking error (blue) when the output joint is randomly moved by hand (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' refer the interested reader to the related work introduced in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYmal features a purely proprioceptive state estimation based on fusion of IMU, leg kinematics, and ground contact measurements , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' For static walking gaits, a ZMP planner is implemented to plan a smooth main body trajectory while applying a standard crawling gait .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Foothold placement during dynamic gaits is based on simpli- fied inverted pendulum models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To balance the system, we build upon whole body control techniques that accounts for the complete system kinematics and dynamics , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The optimal actuator commands are found at every time step by solving a constrained optimization problem of prioritized tasks and constraints on joint torques, contact forces, and body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' EXPERIMENTS The performance of ANYmal was tested in different ma- neuvers and locomotion experiments illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In order to ensure fast and stable locomotion, particular attention was paid to accurate swing leg position and stance leg force tracking, as well as good following behavior of the target base motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Walking ANYmal is able to perform a very smooth walking gait, whereby a single leg is moved at the time and the base is shifted in order to maintain balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 13, joint torques and positions are followed very accurately during the entire gait cycle and hence also the base position can be accurately moved according to the preplanned tra- jectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' It is important to know that the latter follows only from virtual model control (task space control) at the base 4For videos, see http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='rsl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ch/robots-media/anymal 42 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' (a) Walking (b) Trotting (c) Stair Climbing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ANYmal was tested in different gaits like walking, trotting, or stair climbing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Torque and position tracking while walking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' and without any joint position or impedance regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' By applying a classical ZMP planner , forward locomotion results in a very smooth and almost straight line of the base as illustrated in the movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In such gait, the robot moves with approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='3 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Thanks to the full rotation capability, the motion planner does not have to account for complex geometric collision constraints but only for limited abduction freedom due to the main body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Furthermore, ANYmal can take fairly big steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Trotting ANYmal is able to trot on different grounds and under large external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Similar to the walking gait, already the first experiments unveiled large advantages of the big range of motion as the legs can be moved relatively far in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Using a 50% duty cycle gait, the machine achieves a speed of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='8 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Key to robust trotting is fast and accurate position tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' For a typical joint motion of a fast gait (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 14), joint positions and velocities are followed accurately despite the joint compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' A thorough evaluation of the overall energy consumption at the onboard battery indicated a relatively low consumption even during dynamic trotting gait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 15, ANYmal requires in average about 290 W with about 5% 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='4 time [s] knee joint position [rad] measured desired 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='6 8 6 4 2 0 2 4 6 8 time [s] knee joint velocity [rad/s] measured desired Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Tracking performance of the position and velocity of the knee joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 current [A] 50 100 150 200 250 300 350 power [W] 0 50 100 150 200 250 300 350 400 450 500 47 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 48 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='5 49 voltage [V] time [s] trotting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Power consumption during trotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' fluctuation when trotting, about 100 W is consumed while idling in standing configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The total power consump- tion corresponds to a cost of transport of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' These measurements are comparable to our previous results with StarlETH and enables the machine to autonomously run for more than 2 h with its current batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Stair Climbing As a proof of high mobility, ANYmal was tested for the ability to get up an industrial ladder of about 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' To do this in a save manner and to prevent falling by all possible means, a turtle like crawling gait was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The main body lies on the ground, the legs are moved to find the next stable contact holds, and the machine is subsequently pulled upwards (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' 12(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Due to ANYmal’s large range of motion, the legs can be literally turned overhead to prevent collision with the ground or side rails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This maneuver was inspired by our work with ALoF, a kinematic quadrupedal robot that was developed for the ESA Lunar 43 Authorized licensed use limited to: Harbin Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' ETHzurich Legged Robotics Machines for locomotion over challenging terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' slywalk,climb HYMER www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='asLethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='ch DMAVTRobotic Challenge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This machine successfully exhibited such gait to reliably overcome steep inclinations with loose sand during a moon testing scenario on a volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' CONCLUSION/FUTURE WORK ANYmal is considered a step towards unification of high mobility with dynamic locomotion capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' From the beginning of the design phase, special attention was put on a rugged and simple to maintain system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This was achieved with the modular joint units ANYdrive that allow to very simply create robots of different kinematic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' In case of failure, these modules can be easily and quickly exchanged without special knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' These actuators are based on a series elastic concept as already implemented on StarlETH, where we did not have a single gearbox failure in 4 years of almost daily operation with high-dynamic maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The presented experiments support the claim of robustness since even completely plastic and unexpected output collisions do not lead to higher gearbox loads than during nominal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Beside the highly improved protection, the biggest advan- tage of ANYmal is clearly the outstanding range of motion in all joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' This enables a large variety of maneuvers to overcome obstacles or to get up after falling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Furthermore, it simplifies motion planning as there are less internal system constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The initial objectives of creating a dynamic and highly mobile autonomous walking machine could be confirmed in preliminary experiments including careful stair climbing, ZMP-based walking and dynamic trotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' The present development shall enable deployment of legged robots in real world scenarios such as for search and rescue or industrial inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' REFERENCES DRC, “DARPA Robotics Challenge (DRC) , http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='theroboticschallenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='org/.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Boston Dynamics, “ATLAS Robot http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content='bostondynamics.' metadata={'source': 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+page_content=' Downloaded on May 09,2023 at 08:52:18 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} +page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_46/content/kb_46.pdf'} diff --git a/mdE1T4oBgHgl3EQfhATk/content/tmp_files/2301.03237v1.pdf.txt b/mdE1T4oBgHgl3EQfhATk/content/tmp_files/2301.03237v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..659a411d96982e1eb4c47ad4c0806bd555edab0d --- /dev/null +++ b/mdE1T4oBgHgl3EQfhATk/content/tmp_files/2301.03237v1.pdf.txt @@ -0,0 +1,810 @@ +arXiv:2301.03237v1 [nlin.PS] 9 Jan 2023 +Emergent soliton-like solutions in the parametrically +driven 1-D nonlinear Schr¨odinger equation +K Dileep and S Murugesh +Department of Physics, Indian Institute of Space Science and Technology, +Thiruvananthapuram 695 547, India +E-mail: dileepk.17@res.iist.ac.in +Abstract. +We numerically investigate the long time dynamics of spatially periodic +breather solutions of the 1-D nonlinear Schr¨odinger equation under parametric forcing +of the form f(x) = f0 exp(iKx) along with dissipation. In the absence of dissipation, +robust soliton-like excitations are observed that travel with constant amplitude and +velocity. With dissipation, these solitons lose energy (and amplitude) yet gain speed +- a characteristic not observed in an ordinary soliton. Moreover, these novel solitons +are found to be stable against random perturbations. +1. Introduction +The one-dimensional nonlinear Schr¨odinger equation (NLSE) is a nonlinear dispersive +wave +equation +frequently +used +to +describe +wave +propagation +in +optics +and +hydrodynamics [1, 2]. Besides, the NLSE also naturally arises in the study of several +other physical systems, such as in the dynamics of the condensate wave function in +BEC, as a model describing kinematics of vortices in liquid Helium, and macromagnetic +excitations in ferromagnets, to name a few [3, 4, 5]. Further, it is a completely integrable +model with soliton solutions [6]. NLSE remains one of the well investigated models in the +subject of solitons, and nonlinear dynamics in general, which also adds to its pedagogical +significance [7]. Nevertheless, in spite of the rather elaborate literature on the subject, +the NLSE continues to be a rich source for unanticipated phenomena. For instance, +rogue wave behavior in NLSE has been a major subject of curiosity in its own right +since theoretical results were first reported in 1983 [8]. Since then the phenomenon has +been the subject matter of several investigations in diverse areas from water waves to +optics, and is predicted to occur in BEC [9, 10, 11]. It is now understood that the rogue +is a special case of the more general breather mode, witnessed either as a spatially or +temporally periodic localized excitation [12, 13, 14]. The NLSE has also been studied for +its of modulation instability (MI), where weak periodic perturbations on a continuous +wave background undergo growth-decay cycles [15, 16]. This nonlinear process is closely + +2 +related to Fermi-Pasta-Ulam (FPU) recurrence [17, 18]. +MI can also be described +analytically by various breather solutions of 1-D NLSE, owing to its integrability [14]. +While the integrable limit has a significance of its own, actual experimental systems +are dissipative. Such a dissipative system can exhibit nontrivial localized dynamical +structures, similar to solitons of the integrable NLSE, when an external driving is +added to compensate for the loss. These self-organized dynamical objects are commonly +called dissipative solitons [19]. They have been realized experimentally in optical fiber +cavities [20] and microresonators [21, 22]. +Lately, the formation of dissipative Kerr +solitons in optical microresonators were identified as states in Kerr-frequency combs thus +making them useful for practical applications [22]. The underlying physical mechanism +responsible for the formation of soliton pulses in these systems is four-wave mixing. +Mathematically, such a system can be described by the Lugiato-Lefever equation (LLE) +which is a driven, damped NLSE [23, 24]. The equation was originally introduced to +study spatially localized structures in driven nonlinear optical systems [23]. +Another frequently studied model for self-organization phenomenon in nonlinear +dissipative systems is the parametrically driven, +damped NLSE. It has several +applications. +For instance, the equation models the parametric excitation of spin +waves in ferromagnets and dynamics of small amplitude breathers in a long Josephson +junction [25, 26]. +Furthermore, under conditions when driving and dissipation are +balanced, the equation can also exhibit solitary wave solutions [26, 27]. These parametric +excitations are observed in water tanks when the oscillations are driven by periodically +varying a parameter of the system at twice its frequency [28, 29]. +In optics, they +come under the class of dissipative solitons and are known to occur in microresonators +where the parametric driving originates from the second order term in nonlinear +polarization [30]. Although the existence of dissipative solitons in these systems require +an external driving to counteract dissipation, there exist wide classes of traveling solitons +for the undamped parametrically driven NLSE [31]. +In all the aforementioned physical situations, the solutions were either a stationary +or a moving localized soliton. Even though breather solutions were obtained in the +numerical simulation of LLE [32], the effect of driving on periodic breather solutions +of NLSE and their stability have not been completely understood. In particular, the +parametric driving of breather solutions remains largely unexplored. +Hence, in this +work, we numerically study the evolution of a spatially periodic breaher solution of +NLSE - the Akhmediev breather - under parametric driving. We show that, for certain +range of parameter values, the initial breather profile travels with a constant speed +without decreasing its amplitude, curiously like a soliton. Moreover, we observe that +the dynamics show noticeable differences with a 1-soliton of NLSE when dissipation +is included. Finally, we also discuss the stability of these solutions under a variety of +random perturbations. + +3 +(a) +(b) +Figure 1: (a) Akhmediev breather for the parameter ξ = 0.5. (b) Peregrine ‘rogue’ +soliton +2. Dynamics of small-amplitude Akhmediev breathers under parametric +driving +In dimensionless form, the 1-D NLSE is given by +iψt + ψxx + 2|ψ|2ψ = 0 +(1) +where ψ(x, t) is a complex field and the variables x and t refer to dimensionless space +and time respectively. +In the context of propagation of light through optical fibers, +ψ(x, t) is the complex amplitude of electric field and the variables t and x correspond +to the propagation distance and time respectively. Equation (1) when supplemented +with initial and boundary conditions can be solved for soliton solutions by any of the +standard methods, such as the Darboux transformation. For example, the first order +periodic solution +ψAB(x, t) = ei2t +� +1 + ξ 2 cos(qx) − 2ξ cosh(Ωt) + iq sinh(Ωt) +cosh(Ωt) − ξ cos(qx) +� +, +Ω = q +� +4 − q2, +q = 2 +� +1 − ξ2 +(2) +could be generated from an initial seed plane wave solution, ei2t. +For 0 < ξ < 1, +equation (2) is the Akhmediev breather (AB) which is a train of localized pulses in +the intensity profile, |ψ|2, that are located periodically along the x−axis with period +π +√ +1−ξ2 (figure (1a)). The value of the parameter ξ determines the degree of localization. +When ξ = 0, ψAB reduces to the continuous wave (cw) solution, ei2t, and the limit ξ → 1 +produces the Peregrine soliton (figure (1b)) - a localized solution in both space (x) and +time (t). The AB can be viewed as the result of an instability of the cw solution, thus +providing an analytic description for MI. +In this paper, we discuss the evolution of AB in the parametrically driven NLSE +iψt + ψxx + 2|ψ|2ψ = f(x, t)ψ∗ − iβψ +(3) + +8 +2 +4105 +0 +X +-52 +5 +0 +-10 +-5 +0 +5 +t +104 +2 +25 +0 +X +-52 +4 +2 +2 +0 +-10 +-5 +0 +5 +t +104 +(a) +0 +20 +40 +60 +80 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(b) +0 +100 +200 +300 +400 +t +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +| +|2 +max +(c) +0 +100 +200 +300 +400 +t +-60 +-40 +-20 +0 +xmax +(d) +Figure 2: +(a) Soliton-like behaviour observed in the evolution of AB under +parametric driving and zero dissipation. +(b) Contour plot of (a). +(c) Evolution +of the intensity maximum. +(d) Position of the intensity maxima vs time. +The +forcing has a magnitude f0 = 0.05. The external driving has the same periodicity +as that of the lattice. Parameters used are κ = 1 and ξ = 0.1. +where f(x, t) = f0 exp(iKx) is a driving force, f0 and K are constants, and β > 0 is the +dissipation. It may be noted that the transformation +ψ(x, t) = Ψ(X, t) eiKx/2 +(4) +to a moving frame X = x − Kt leads to the equation +iΨt + ΨXX + 2|Ψ|2Ψ = f0Ψ∗ − iβΨ + K2 +4 Ψ +(5) +with an additional detuning term. +Equation (5) naturally arises in an optical +microresonator containing a Kerr medium, where the parametric driving is realised +using a nonlinear χ(2) medium [30]. +2.1. The nondissipative case +In this section, we describe the evolution of AB in the parametrically driven NLSE with +zero damping, i.e., β = 0. To this end, we numerically intergate (3) using the initial +condition +ψ0 ≡ ψAB(x, 0) = +� +1 + ξ 2 cos(qx) − 2ξ +1 − ξ cos(qx) +� +(6) + +1.4 +1.2 +2 +10.8 +2 +1 +0 +x +-1 +-21.5 +2 +0.5 +0 +20 +40 +60 +t +80 +1005 +Figure 3: Intensity plot when K = 10 and q = 1.99 (or ξ = 0.1). Here, unlike in +the K = q case, the dynamics is characterized by the appearance of maxima that +locally resemble breather excitations. The other parameters are as in figure (2). +0.04 +0.08 +0.12 +0.16 +0.2 +f0 +-0.4 +-0.3 +-0.2 +-0.1 +v +Figure 4: Velocity vs f0 for ξ = 0.1. We observe that the speed of the solution +increases with f0 until a threshold value of f0,max = 0.2. The lower limit of f0, to +observe soliton solution, for this particular initial condition is found to be 0.025. +and periodic boundary condition +ψ(x + L, t) = ψ(x, t). +(7) +The numerical simulations are performed using the split-step Fourier method +(SSFM) [33]. Here, the nonlinear equation is split into two parts +ψt = iψxx +(8) +ψt = i(2|ψ|2ψ − f(x, t)ψ∗) +(9) +wherein, the solution is advanced from t to t+δt in two steps. In the first step, we solve + +1.4 +1.2 +21 +0.8 +2 +1 +0 +-11.5 +2 +1 +0.5 +0 +20 +40 +60 +80 +t +-2 +1006 +Figure 5: Velocity as a function of maximum of initial amplitude and magnitude of +forcing. In the blue region where the velocity is small, the evolution is soliton-like. +Here, the initial amplitude retains its shape, making the initial waveform a very good +approximation to the soliton profile at later time t. For large values of f0, we see +that the velocity decreases with increase in the initial amplitude (red region). This +apparent anomaly in the velocity is due to the onset of a breather-like behaviour. +(8) in the Fourier domain with the initial condition ψ(x, t). The resulting solution is +used as the initial value to solve (9), yielding the final solution, ψ(x, t + δt), at t + δt. +In the second step, we employ the fourth order Runge-Kutta method. All simulations +are performed using a spatial lattice consisting of 512 points and a time step δt = 10−4. +As described in the beginning of this section, the Akhmediev breather is a periodic +and localized excitation. +It is naturally expected that this behavior prevails in the +driven damped case, for a suitable choice of the driving force and damping balancing +each other. +In the absence of the balancing damping term, the external forcing is +expected to render the breather mode unstable. Yet interestingly, we observe a new +localized waveform that travels like a usual 1-soliton solution. This is in contrast to the +unperturbed breather dynamics where localizations eventually decay to the continuous +wave background. Figure (2a) shows the intensity plot obtained by choosing the initial +condition (6). Here, the initial breather profile travels with a constant velocity without +much reduction in its amplitude. The velocity of the solution can be obtained by plotting +the position of maxima versus time. In figure (2d), the locations of maxima are plotted +against time for the parameter ξ = 0.1 and f0 = 0.05. We define the average velocity +of the solution as the slope of this line. It is noted that this soliton-like behavior is +only observed when the frequency of the driving force is equal to the wavenumber of +the initial breather profile (K = q). The case K ̸= q is described in figure(3). A few +remarks about the new soliton structure are in order. Firstly, the dynamics depicted + +0.6 +0.4 +0.20.6- +以0.15 +0.1 +05 +F +0Veloci +0.4 +0.2 - +0 +1.3 +1.25 +1.2 +1.15 +0 +1.1 +[(t=0)I +max7 +in figure (2a) shows that the initial waveform retains its shape during evolution. Thus +the resemblance to a 1-soliton solution is only qualitative and cannot be regarded as +a breather to soliton conversion as in [34]. This is because a soliton with energy of +the order of the breather is a more localized solution with higher amplitude. In other +words, the ratio of peak to full width at half maximum (FWHM) for a 1-soliton is higher +compared to a breather of the same energy. +Second, the occurrence of soliton-like solution is restricted to small amplitude +breathers which is determined by the parameter ξ. +Figure (2) shows the results of +numerical simulation when the parameter ξ = 0.1. As we increase the amplitude of the +initial breather, a large number of peaks with varying intensity appears in the intensity +profile at random locations. Our numerical simulations could not confirm any stable +pattern in the long time dynamics of higher amplitude ABs. +Finally, the magnitude of the forcing also plays an important role in the dynamics. +For small values of ξ, the initial breather profile travels with a constant velocity when +the magnitude of forcing f0 is in a certain range (determined numerically). The velocity +can be positive or negative depending on the sign of f0 ; when f0 is positive (negative) +the breather travels in the negative (positive) x-direction. As we increase the value of +f0, the speed of the solution increases until a threshold f0,max is reached beyond which +the pattern is destroyed. Figure (4) shows the variation of velocity with the magnitude +of forcing for the initial breather defined by the parameter ξ = 0.1. We also mention, in +figure (5), the region in (f0, |ψ(t = 0)|max) plane where a soliton-like solution is observed. +2.2. Effect of dissipation on the parametrically driven Akhmediev breather (PDAB) +For our model to apply to real physical situations, we must invariably take into account +dissipation. There are two different ways dissipation affects the solutions of a dynamical +system: +(i) It can cause stable solutions to exist such that the solution disappears when +dissipation is turned off. An example is the dissipative solitons of driven, damped +NLSE. Here, dissipation plays a principal role as the formation of dissipative solitons +crucially depend on the balance between dissipation and driving aside from a +balance of nonlinearity and dispersion. +(ii) It can cause energy losses to an otherwise stable solution. +For instance, when +dissipation is included in NLSE, the amplitude of a soliton solution decreases as +it propagates. Since the amplitude and velocity of solitons are linearly related, we +can also observe a decrease in the velocity of the soliton. +In this section, we indicate the effects of dissipation on the paramterically driven +Akhmediev breather (PDAB). We first note that, the PDAB is a stable solution in the +absence of dissipation as confirmed in our numerical experiments. Hence we expect the +solution to behave similar to the conservative NLSE soliton when dissipation is present. +In fact, the amplitude of the PDAB solution decreases over time as shown in figure +(6c). However, in contrast to the conservative soliton, the velocity of the PDAB solution + +8 +(a) +0 +20 +40 +60 +80 +100 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(b) +0 +100 +200 +300 +400 +t +1 +1.1 +1.2 +1.3 +1.4 +1.5 +| +|2 +max +(c) +0 +100 +200 +300 +400 +t +-160 +-120 +-80 +-40 +0 +xmax +(d) +Figure 6: (a) Effect of dissipation on the soliton-like solutions of figure(2). +(b) +Contour plot of (a). (c) Evolution of the intensity maximum. (d) Position of the +intensity maxima vs time. Here, β = 10−4. The other parameters are as in figure +(2). +increases (fig (6d)). We remark that such a solution wherein an increase in velocity along +with a decrease in amplitude is counterintuitive and has not been observed before. The +observation also brings about the following distinction; for a conventional conservative +soliton, amplitude and velocity are linearly related, whereas the new soliton structure +has an inverse relation between the two, for a range of parameter values. This intriguing +characteristic emphasizes that the newly observed soliton-like solution is fundamentally +different from the conservative soliton. +3. Stability of the parametrically driven Akhmediev breather +In reality, a dynamical system is always subjected to random perturbations. +Such +fluctuations could arise due to inhomogeneities in the system e.g. fluctuations in the +refractive index of an optical medium [35]. These perturbations, however small, can +make a soliton unstable and therefore pose a challenge in using solitons for practical +applications. Hence it is worthwhile to investigate the robustness of solitons against +random perturbations. Accordingly, in this section, we study the stability of the PDAB +solution in the presence of noise. In particular, we consider the evolution equation +iψt + ψxx + 2|ψ|2ψ = f0eiKxψ∗ + ǫ N (ψ; x, t) +(10) + +1.4 +1.2 +2 +10.8 +2 +1 +0 +X +-11.5 +2 +0.5 +0 +20 +40 +60 +80 +-2 +t +1009 +(a) +(b) +0 +20 +40 +60 +80 +100 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(c) +0 +20 +40 +60 +80 +100 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(d) +0 +100 +200 +300 +400 +t +1.1 +1.2 +1.3 +1.4 +1.5 +| +|2 +max +(e) +0 +50 +100 +150 +200 +250 +300 +350 +400 +t +1 +1.5 +2 +2.5 +3 +| +|2 +max +(f) +Figure 7: Effect of additive white noise on the PDAB solution. +(a),(c),(e) The +value of the parameter ǫ = 0.01. +The PDAB solution is stable as observed in +the intensity plot (a), and peak value of intensity (e). (b),(d),(f) The case when +ǫ = 0.05. Intensity plot in (b) shows occurence of localizations at different locations +with varying intensities. +where N (ψ; x, t) is a stochastic noise term and ǫ is a small parameter. We consider two +types of noise as described in the following sections. +3.1. Additive noise +For the additive noise, the function +N = η(x, t) +(11) +such that +⟨η(x, t)⟩ = 0 +⟨η(x, t)η(x′, t′)⟩ = δ(t − t′)δ(x − x′) +(12) + +2.5 +1.5 N0.5 +2 +1 +0 +X +-12. +2 +0 +0 +20 +40 +60 +t +80 +-2 +1001.4 +1.2 +2 +10.8 +2 +1 +0 +X +-11.5 +2 +2 +1 +0.5 +0 +20 +40 +60 +t +80 +-2 +10010 +(a) +(b) +0 +20 +40 +60 +80 +100 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(c) +0 +20 +40 +60 +80 +100 +t +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +x +(d) +0 +100 +200 +300 +400 +t +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +| +|2 +max +(e) +0 +100 +200 +300 +400 +t +1 +2 +3 +4 +| +|2 +max +(f) +Figure 8: Effect of multiplicative white noise on the PDAB solution. (a),(c),(e) The +value of the parameter ǫ = 0.05. The PDAB solution is stable as observed in the +intensity plot (a), and peak value of intensity (e). (b),(d),(f) The case when ǫ = 5. +The solution is more stable in the presence of multiplicative noise. +where ⟨· · ·⟩ denotes ensemble average. The results obtained by using (11) are shown +in figure (7). When ǫ = 0.01, the PDAB is stable except for small local variations in +the intensity (figure(7a)). As ǫ increases, fluctuations in the intensity also increases and +when ǫ is comparable to the magnitude of the parametric forcing, the soliton structure +is destroyed. Further increasing ǫ leads to isolated localizations in the intensity plot +(figure(7b)). These intensity maxima are similar to breather excitaions, with a moderate +increase in intensity (figure(7f)). +Furthermore, we note that the recurrences do not +exhibit any periodic behaviour. + +3 +2 +2 +12 +1 +0 +-12 +2 +0 +0 +20 +40 +60 +t +80 +-2 +1001.4 +1.2 +2 +10.8 +2 +1 +0 +X +-11.5 +2 +0.5 +0 +20 +40 +60 +t +80 +-2 +10011 +3.2. Multiplicative noise +Multiplicative noise, in the form +N (ψ; x, t) = η(x, t)ψ(x, t) +(13) +where η(x, t) is as in (12), is a type of noise that appears in physical situations such +as in light propagation through an optical medium with random linear fluctuations in +the refractive index, and in the dynamics of Bose-Einstein condensate in disordered +potentials [35, 36]. The effect of multiplicative noise on the PDAB is depicted in figure +(8). It is apparent from figure (8a) that, for small values of noise, the intensity profile of +the PDAB is almost indistinguishable from its noiseless dynamics. In fact, fluctuations +in the maximum value of the intensity show similar pattern as in the zero noise case +(figure(8e)). At higher values of noise, the soliton is replaced by breather-like excitations +with aperiodic recurrence. Figure (8b) shows the intensity plot when ǫ = 5. We note +that, as in the additive case, the maximum value of the intensity increases with time +(figure(8f)). +Although both additive and multiplicative noise eventually lead to instabilities in +the soliton structure, the range of noise intensities for which the soliton remains stable, +is different. Specifically, the PDAB is observed to be more stable against multiplicative +random noise in comparison to the additive noise. +4. Conclusion +In conclusion, we have investigated the dynamics of Akhmediev breather under +parametric driving. We have observed through numerical simulations that, for certain +range of parameters of the system and initial conditions, the initial breather travels +like a soliton whose amplitude and velocity are constants. The speed of the soliton +is determined by the magnitude of forcing (f0) and its direction is dictated by the +sign of f0. Although these new solutions are structurally similar to the conventional +solitons, they are characteristically different from the ordinary solitons. Specifically, +we notice that when dissipation is introduced, the amplitude of the soliton decreases +while its velocity increases. We remark that, such an unconventional behaviour also +opens up the possibility for new soliton solutions whose amplitude and velocity are +inversely related. Furthermore, we have studied the stability of these solutions under +random perturbations and found that the soliton is stable for sufficiently large values of +perturbations. +References +[1] Agrawal G P 2001 Nonlinear Fiber Optics (San Diego: Academic) +[2] Chabchoub A, Kibler B, Finot C, Millot G, Onorato M, Dudley J and Babanin A 2015 Annals of +Physics 361 490–500 +[3] Salasnich L, Parola A and Reatto L 2002 Physical Review A 65 043614 +[4] Hasimoto H 1972 Journal of Fluid Mechanics 51 477–485 + +12 +[5] Lakshmanan M 1977 Physics Letters A 61 53–54 +[6] Shabat A and Zakharov V 1972 Sov. Phys. 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf,len=340 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='03237v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='PS] 9 Jan 2023 Emergent soliton-like solutions in the parametrically driven 1-D nonlinear Schr¨odinger equation K Dileep and S Murugesh Department of Physics, Indian Institute of Space Science and Technology, Thiruvananthapuram 695 547, India E-mail: dileepk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='17@res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='iist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='in Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We numerically investigate the long time dynamics of spatially periodic breather solutions of the 1-D nonlinear Schr¨odinger equation under parametric forcing of the form f(x) = f0 exp(iKx) along with dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the absence of dissipation, robust soliton-like excitations are observed that travel with constant amplitude and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' With dissipation, these solitons lose energy (and amplitude) yet gain speed a characteristic not observed in an ordinary soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Moreover, these novel solitons are found to be stable against random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Introduction The one-dimensional nonlinear Schr¨odinger equation (NLSE) is a nonlinear dispersive wave equation frequently used to describe wave propagation in optics and hydrodynamics [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Besides, the NLSE also naturally arises in the study of several other physical systems, such as in the dynamics of the condensate wave function in BEC, as a model describing kinematics of vortices in liquid Helium, and macromagnetic excitations in ferromagnets, to name a few [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Further, it is a completely integrable model with soliton solutions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' NLSE remains one of the well investigated models in the subject of solitons, and nonlinear dynamics in general, which also adds to its pedagogical significance [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Nevertheless, in spite of the rather elaborate literature on the subject, the NLSE continues to be a rich source for unanticipated phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For instance, rogue wave behavior in NLSE has been a major subject of curiosity in its own right since theoretical results were first reported in 1983 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Since then the phenomenon has been the subject matter of several investigations in diverse areas from water waves to optics, and is predicted to occur in BEC [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It is now understood that the rogue is a special case of the more general breather mode, witnessed either as a spatially or temporally periodic localized excitation [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The NLSE has also been studied for its of modulation instability (MI), where weak periodic perturbations on a continuous wave background undergo growth-decay cycles [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' This nonlinear process is closely 2 related to Fermi-Pasta-Ulam (FPU) recurrence [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' MI can also be described analytically by various breather solutions of 1-D NLSE, owing to its integrability [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' While the integrable limit has a significance of its own, actual experimental systems are dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Such a dissipative system can exhibit nontrivial localized dynamical structures, similar to solitons of the integrable NLSE, when an external driving is added to compensate for the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' These self-organized dynamical objects are commonly called dissipative solitons [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' They have been realized experimentally in optical fiber cavities [20] and microresonators [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Lately, the formation of dissipative Kerr solitons in optical microresonators were identified as states in Kerr-frequency combs thus making them useful for practical applications [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The underlying physical mechanism responsible for the formation of soliton pulses in these systems is four-wave mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Mathematically, such a system can be described by the Lugiato-Lefever equation (LLE) which is a driven, damped NLSE [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The equation was originally introduced to study spatially localized structures in driven nonlinear optical systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Another frequently studied model for self-organization phenomenon in nonlinear dissipative systems is the parametrically driven, damped NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It has several applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For instance, the equation models the parametric excitation of spin waves in ferromagnets and dynamics of small amplitude breathers in a long Josephson junction [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Furthermore, under conditions when driving and dissipation are balanced, the equation can also exhibit solitary wave solutions [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' These parametric excitations are observed in water tanks when the oscillations are driven by periodically varying a parameter of the system at twice its frequency [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In optics, they come under the class of dissipative solitons and are known to occur in microresonators where the parametric driving originates from the second order term in nonlinear polarization [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Although the existence of dissipative solitons in these systems require an external driving to counteract dissipation, there exist wide classes of traveling solitons for the undamped parametrically driven NLSE [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In all the aforementioned physical situations, the solutions were either a stationary or a moving localized soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Even though breather solutions were obtained in the numerical simulation of LLE [32], the effect of driving on periodic breather solutions of NLSE and their stability have not been completely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In particular, the parametric driving of breather solutions remains largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Hence, in this work, we numerically study the evolution of a spatially periodic breaher solution of NLSE - the Akhmediev breather - under parametric driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We show that, for certain range of parameter values, the initial breather profile travels with a constant speed without decreasing its amplitude, curiously like a soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Moreover, we observe that the dynamics show noticeable differences with a 1-soliton of NLSE when dissipation is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Finally, we also discuss the stability of these solutions under a variety of random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 3 (a) (b) Figure 1: (a) Akhmediev breather for the parameter ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (b) Peregrine ‘rogue’ soliton 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Dynamics of small-amplitude Akhmediev breathers under parametric driving In dimensionless form, the 1-D NLSE is given by iψt + ψxx + 2|ψ|2ψ = 0 (1) where ψ(x, t) is a complex field and the variables x and t refer to dimensionless space and time respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the context of propagation of light through optical fibers, ψ(x, t) is the complex amplitude of electric field and the variables t and x correspond to the propagation distance and time respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Equation (1) when supplemented with initial and boundary conditions can be solved for soliton solutions by any of the standard methods, such as the Darboux transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For example, the first order periodic solution ψAB(x, t) = ei2t � 1 + ξ 2 cos(qx) − 2ξ cosh(Ωt) + iq sinh(Ωt) cosh(Ωt) − ξ cos(qx) � , Ω = q � 4 − q2, q = 2 � 1 − ξ2 (2) could be generated from an initial seed plane wave solution, ei2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For 0 < ξ < 1, equation (2) is the Akhmediev breather (AB) which is a train of localized pulses in the intensity profile, |ψ|2, that are located periodically along the x−axis with period π √ 1−ξ2 (figure (1a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The value of the parameter ξ determines the degree of localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' When ξ = 0, ψAB reduces to the continuous wave (cw) solution, ei2t, and the limit ξ → 1 produces the Peregrine soliton (figure (1b)) - a localized solution in both space (x) and time (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The AB can be viewed as the result of an instability of the cw solution, thus providing an analytic description for MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In this paper, we discuss the evolution of AB in the parametrically driven NLSE iψt + ψxx + 2|ψ|2ψ = f(x, t)ψ∗ − iβψ (3) 8 2 4105 0 X 52 5 0 10 5 0 5 t 104 2 25 0 X 52 4 2 2 0 10 5 0 5 t 104 (a) 0 20 40 60 80 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (b) 0 100 200 300 400 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='6 | |2 max (c) 0 100 200 300 400 t 60 40 20 0 xmax (d) Figure 2: (a) Soliton-like behaviour observed in the evolution of AB under parametric driving and zero dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (b) Contour plot of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (c) Evolution of the intensity maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (d) Position of the intensity maxima vs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The forcing has a magnitude f0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The external driving has the same periodicity as that of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Parameters used are κ = 1 and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' where f(x, t) = f0 exp(iKx) is a driving force, f0 and K are constants, and β > 0 is the dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It may be noted that the transformation ψ(x, t) = Ψ(X, t) eiKx/2 (4) to a moving frame X = x − Kt leads to the equation iΨt + ΨXX + 2|Ψ|2Ψ = f0Ψ∗ − iβΨ + K2 4 Ψ (5) with an additional detuning term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Equation (5) naturally arises in an optical microresonator containing a Kerr medium, where the parametric driving is realised using a nonlinear χ(2) medium [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The nondissipative case In this section, we describe the evolution of AB in the parametrically driven NLSE with zero damping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=', β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' To this end, we numerically intergate (3) using the initial condition ψ0 ≡ ψAB(x, 0) = � 1 + ξ 2 cos(qx) − 2ξ 1 − ξ cos(qx) � (6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='8 2 1 0 x 1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 20 40 60 t 80 1005 Figure 3: Intensity plot when K = 10 and q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='99 (or ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, unlike in the K = q case, the dynamics is characterized by the appearance of maxima that locally resemble breather excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The other parameters are as in figure (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 v Figure 4: Velocity vs f0 for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We observe that the speed of the solution increases with f0 until a threshold value of f0,max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The lower limit of f0, to observe soliton solution, for this particular initial condition is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' and periodic boundary condition ψ(x + L, t) = ψ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (7) The numerical simulations are performed using the split-step Fourier method (SSFM) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, the nonlinear equation is split into two parts ψt = iψxx (8) ψt = i(2|ψ|2ψ − f(x, t)ψ∗) (9) wherein, the solution is advanced from t to t+δt in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the first step, we solve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='8 2 1 0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 20 40 60 80 t 2 1006 Figure 5: Velocity as a function of maximum of initial amplitude and magnitude of forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the blue region where the velocity is small, the evolution is soliton-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, the initial amplitude retains its shape, making the initial waveform a very good approximation to the soliton profile at later time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For large values of f0, we see that the velocity decreases with increase in the initial amplitude (red region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' This apparent anomaly in the velocity is due to the onset of a breather-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (8) in the Fourier domain with the initial condition ψ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The resulting solution is used as the initial value to solve (9), yielding the final solution, ψ(x, t + δt), at t + δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the second step, we employ the fourth order Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' All simulations are performed using a spatial lattice consisting of 512 points and a time step δt = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' As described in the beginning of this section, the Akhmediev breather is a periodic and localized excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It is naturally expected that this behavior prevails in the driven damped case, for a suitable choice of the driving force and damping balancing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In the absence of the balancing damping term, the external forcing is expected to render the breather mode unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Yet interestingly, we observe a new localized waveform that travels like a usual 1-soliton solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' This is in contrast to the unperturbed breather dynamics where localizations eventually decay to the continuous wave background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Figure (2a) shows the intensity plot obtained by choosing the initial condition (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, the initial breather profile travels with a constant velocity without much reduction in its amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The velocity of the solution can be obtained by plotting the position of maxima versus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In figure (2d), the locations of maxima are plotted against time for the parameter ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 and f0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We define the average velocity of the solution as the slope of this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It is noted that this soliton-like behavior is only observed when the frequency of the driving force is equal to the wavenumber of the initial breather profile (K = q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The case K ̸= q is described in figure(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' A few remarks about the new soliton structure are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Firstly, the dynamics depicted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='6- 以0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 05 F 0Veloci 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 - 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='15 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 [(t=0)I max7 in figure (2a) shows that the initial waveform retains its shape during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Thus the resemblance to a 1-soliton solution is only qualitative and cannot be regarded as a breather to soliton conversion as in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' This is because a soliton with energy of the order of the breather is a more localized solution with higher amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In other words, the ratio of peak to full width at half maximum (FWHM) for a 1-soliton is higher compared to a breather of the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Second, the occurrence of soliton-like solution is restricted to small amplitude breathers which is determined by the parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Figure (2) shows the results of numerical simulation when the parameter ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' As we increase the amplitude of the initial breather, a large number of peaks with varying intensity appears in the intensity profile at random locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Our numerical simulations could not confirm any stable pattern in the long time dynamics of higher amplitude ABs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Finally, the magnitude of the forcing also plays an important role in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For small values of ξ, the initial breather profile travels with a constant velocity when the magnitude of forcing f0 is in a certain range (determined numerically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The velocity can be positive or negative depending on the sign of f0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' when f0 is positive (negative) the breather travels in the negative (positive) x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' As we increase the value of f0, the speed of the solution increases until a threshold f0,max is reached beyond which the pattern is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Figure (4) shows the variation of velocity with the magnitude of forcing for the initial breather defined by the parameter ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We also mention, in figure (5), the region in (f0, |ψ(t = 0)|max) plane where a soliton-like solution is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Effect of dissipation on the parametrically driven Akhmediev breather (PDAB) For our model to apply to real physical situations, we must invariably take into account dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' There are two different ways dissipation affects the solutions of a dynamical system: (i) It can cause stable solutions to exist such that the solution disappears when dissipation is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' An example is the dissipative solitons of driven, damped NLSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, dissipation plays a principal role as the formation of dissipative solitons crucially depend on the balance between dissipation and driving aside from a balance of nonlinearity and dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (ii) It can cause energy losses to an otherwise stable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' For instance, when dissipation is included in NLSE, the amplitude of a soliton solution decreases as it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Since the amplitude and velocity of solitons are linearly related, we can also observe a decrease in the velocity of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In this section, we indicate the effects of dissipation on the paramterically driven Akhmediev breather (PDAB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We first note that, the PDAB is a stable solution in the absence of dissipation as confirmed in our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Hence we expect the solution to behave similar to the conservative NLSE soliton when dissipation is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In fact, the amplitude of the PDAB solution decreases over time as shown in figure (6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' However, in contrast to the conservative soliton, the velocity of the PDAB solution 8 (a) 0 20 40 60 80 100 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (b) 0 100 200 300 400 t 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 | |2 max (c) 0 100 200 300 400 t 160 120 80 40 0 xmax (d) Figure 6: (a) Effect of dissipation on the soliton-like solutions of figure(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (b) Contour plot of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (c) Evolution of the intensity maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (d) Position of the intensity maxima vs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Here, β = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The other parameters are as in figure (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' increases (fig (6d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We remark that such a solution wherein an increase in velocity along with a decrease in amplitude is counterintuitive and has not been observed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The observation also brings about the following distinction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' for a conventional conservative soliton, amplitude and velocity are linearly related, whereas the new soliton structure has an inverse relation between the two, for a range of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' This intriguing characteristic emphasizes that the newly observed soliton-like solution is fundamentally different from the conservative soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Stability of the parametrically driven Akhmediev breather In reality, a dynamical system is always subjected to random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Such fluctuations could arise due to inhomogeneities in the system e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' fluctuations in the refractive index of an optical medium [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' These perturbations, however small, can make a soliton unstable and therefore pose a challenge in using solitons for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Hence it is worthwhile to investigate the robustness of solitons against random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Accordingly, in this section, we study the stability of the PDAB solution in the presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In particular, we consider the evolution equation iψt + ψxx + 2|ψ|2ψ = f0eiKxψ∗ + ǫ N (ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' x, t) (10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='8 2 1 0 X 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 20 40 60 80 2 t 1009 (a) (b) 0 20 40 60 80 100 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (c) 0 20 40 60 80 100 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (d) 0 100 200 300 400 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 | |2 max (e) 0 50 100 150 200 250 300 350 400 t 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 3 | |2 max (f) Figure 7: Effect of additive white noise on the PDAB solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (a),(c),(e) The value of the parameter ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The PDAB solution is stable as observed in the intensity plot (a), and peak value of intensity (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (b),(d),(f) The case when ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Intensity plot in (b) shows occurence of localizations at different locations with varying intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' where N (ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' x, t) is a stochastic noise term and ǫ is a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We consider two types of noise as described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Additive noise For the additive noise, the function N = η(x, t) (11) such that ⟨η(x, t)⟩ = 0 ⟨η(x, t)η(x′, t′)⟩ = δ(t − t′)δ(x − x′) (12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 1 0 X 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 2 0 0 20 40 60 t 80 2 1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='8 2 1 0 X 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 20 40 60 t 80 2 10010 (a) (b) 0 20 40 60 80 100 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (c) 0 20 40 60 80 100 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 x (d) 0 100 200 300 400 t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='6 | |2 max (e) 0 100 200 300 400 t 1 2 3 4 | |2 max (f) Figure 8: Effect of multiplicative white noise on the PDAB solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (a),(c),(e) The value of the parameter ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The PDAB solution is stable as observed in the intensity plot (a), and peak value of intensity (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' (b),(d),(f) The case when ǫ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The solution is more stable in the presence of multiplicative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' where ⟨· · ·⟩ denotes ensemble average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The results obtained by using (11) are shown in figure (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' When ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='01, the PDAB is stable except for small local variations in the intensity (figure(7a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' As ǫ increases, fluctuations in the intensity also increases and when ǫ is comparable to the magnitude of the parametric forcing, the soliton structure is destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Further increasing ǫ leads to isolated localizations in the intensity plot (figure(7b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' These intensity maxima are similar to breather excitaions, with a moderate increase in intensity (figure(7f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Furthermore, we note that the recurrences do not exhibit any periodic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 3 2 2 12 1 0 12 2 0 0 20 40 60 t 80 2 1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='8 2 1 0 X 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='5 0 20 40 60 t 80 2 10011 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Multiplicative noise Multiplicative noise, in the form N (ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' x, t) = η(x, t)ψ(x, t) (13) where η(x, t) is as in (12), is a type of noise that appears in physical situations such as in light propagation through an optical medium with random linear fluctuations in the refractive index, and in the dynamics of Bose-Einstein condensate in disordered potentials [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The effect of multiplicative noise on the PDAB is depicted in figure (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' It is apparent from figure (8a) that, for small values of noise, the intensity profile of the PDAB is almost indistinguishable from its noiseless dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' In fact, fluctuations in the maximum value of the intensity show similar pattern as in the zero noise case (figure(8e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' At higher values of noise, the soliton is replaced by breather-like excitations with aperiodic recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Figure (8b) shows the intensity plot when ǫ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We note that, as in the additive case, the maximum value of the intensity increases with time (figure(8f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Although both additive and multiplicative noise eventually lead to instabilities in the soliton structure, the range of noise intensities for which the soliton remains stable, is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Specifically, the PDAB is observed to be more stable against multiplicative random noise in comparison to the additive noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Conclusion In conclusion, we have investigated the dynamics of Akhmediev breather under parametric driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We have observed through numerical simulations that, for certain range of parameters of the system and initial conditions, the initial breather travels like a soliton whose amplitude and velocity are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' The speed of the soliton is determined by the magnitude of forcing (f0) and its direction is dictated by the sign of f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Although these new solutions are structurally similar to the conventional solitons, they are characteristically different from the ordinary solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Specifically, we notice that when dissipation is introduced, the amplitude of the soliton decreases while its velocity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' We remark that, such an unconventional behaviour also opens up the possibility for new soliton solutions whose amplitude and velocity are inversely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE1T4oBgHgl3EQfhATk/content/2301.03237v1.pdf'} +page_content=' Furthermore, we have studied the stability of 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b/ntE3T4oBgHgl3EQfjAo0/content/tmp_files/2301.04584v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d75d989b9d6812931f80f0f6ae0134b5922ad5e --- /dev/null +++ b/ntE3T4oBgHgl3EQfjAo0/content/tmp_files/2301.04584v1.pdf.txt @@ -0,0 +1,1337 @@ +Continual Few-Shot Learning Using HyperTransformers +Max Vladymyrov, Andrey Zhmoginov, Mark Sandler +Google Research +{mxv,azhmogin,sandler}@google.com +Abstract +We focus on the problem of learning without forgetting from multiple tasks arriving se- +quentially, where each task is defined using a few-shot episode of novel or already seen +classes. +We approach this problem using the recently published HyperTransformer +(HT), a Transformer-based hypernetwork that generates a specialized task-specific CNN +weights directly from the support set. In order to learn from a continual sequence of task, +we propose to recursively re-use the generated weights as input to the HT for the next +task. This way, the generated CNN weights themselves act as a representation of previ- +ously learned tasks, and the HT is trained to update these weights so that the new task +can be learned without forgetting past tasks. This approach is different from most con- +tinual learning algorithms that typically rely on using replay buffers, weight regularization +or task-dependent architectural changes. We demonstrate that our proposed Continual +HyperTransformer method equipped with a prototypical loss is capable of learning and +retaining knowledge about past tasks for a variety of scenarios, including learning from +mini-batches, and task-incremental and class-incremental learning scenarios. +1 +Introduction +Continual few-shot learning involves learning from a continuous stream of tasks described by a small number +of examples without forgetting previously learned information. This type of learning closely resembles how +humans and other biological systems acquire new information, as we can continually learn novel concepts +with a small amount of information and retain that knowledge for an extended period of time. Algorithms for +few-shot continual learning can be useful in many real-world applications where there is a need to classify a +large number of classes in a dynamic environment with limited observations. Some practical applications can +include enabling robots to continually adapt to changing environments based on an incoming stream of sparse +demonstrations or allowing for privacy-preserving learning, where the model can be trained sequentially on +private data sharing only the weights without ever exposing the data. +To tackle this problem, we propose using HyperTransformer (HT; Zhmoginov et al. 2022), a recently +published few-shot learning method that utilizes a large hypernetwork (Ha et al., 2016) to meta-learn from +episodes sampled from a large set of few-shot learning tasks. The HT is trained to directly generate weights +of a much smaller specialized Convolutional Neural Network (CNN) model using only few labeled examples. +This works by decoupling the domain knowledge model (represented by a Transformer; Vaswani et al. 2017) +from the learner itself (a CNN), generated to solve only a given specific few-shot learning problem. +We present a modification to HT method, called Continual HyperTransformer (CHT), that is aimed +at exploring the capability of the HT to sequentially update the CNN weights with the information from a +new task, while retaining the knowledge about the tasks that were already learned. In other words, given +the CNN weights θt−1 generated after seeing some previous tasks 0, . . . , t − 1 and a description of the new +task t, the CHT generates the weights θt that are suited for all the tasks 0, . . . , t. +In order for the CHT to be able to absorb a continual stream of tasks, we modified the loss function from +a cross-entropy that was used in the HT to a more flexible prototypical loss (Snell et al., 2017), that uses +prototypes as a learned representation of every class from all the tasks. As the tasks come along, we maintain +1 +arXiv:2301.04584v1 [cs.LG] 11 Jan 2023 + +Figure 1: In few-shot continual learning, the model learns from T tasks sequentially. For the first task (task +0), the CNN weights θ0 are generated using only the support set S(0). For each subsequent task t, the +Continual HyperTransformer (CHT) uses the support set S(t) and the previously generated weights +θt−1 to generate the weights θt. To update the weights ψ of the CHT, the loss is calculated by summing +the individual losses computed for each generated weight θt when evaluated on the query set of all the prior +tasks (Q(τ))T +τ=0. +and update a set of prototypes in the embedding space. The prototypes are then used to predict the class +and task attributes for a given input sample. +We evaluate CHT in three realistic scenarios where a continual few-shot learning model like ours might be +used: the mini-batch version, where every task consists of the same classes; the lifelong learning version, +where classes for all the tasks are drawn from the same overall distribution; and the heterogeneous task +semantic version, where every task has its own unique distribution of classes. +We also test CHT in two different continual learning scenarios: task-incremental learning (predicting class +attributes using the task information) and class-incremental learning (predicting class attributes without +access to task information; also known as lifelong learning). Moreover, we show empirically that a model +trained for class-incremental learning can also perform well in task-incremental learning, similar to a model +specifically trained for task-incremental learning. +Our approach has several advantages. First, as a hypernetwork, the CHT is able to generate and update +the weights of the CNN on the fly with no training required. A trained Transformer holds the domain +world-knowledge and can generalize from limited few-shot observations. There is also evidence to suggest +that similar functions are performed by the prefrontal cortex in the human brain (Miller et al., 2001), which +may imply biological plausibility of our approach. +Second, we demonstrate that models learned with CHT do not suffer from the catastrophic forgetting. We +even see cases of the positive backward transfer for smaller models, where the performance on a given task +actually improves for subsequently generated weights. +Third, while the CHT is trained to optimize for T tasks, the model can be stopped at any point t ≤ T +during the inference with weights θt that are suited for all the tasks 0 ≤ τ ≤ t. Moreover, the performance +of a given weight θt improves when the CHT is trained on more tasks T. +Finally, we designed the CHT model to be independent from a specific step and operate as a recurrent +system. It can be used to learn a larger number of tasks it was originally trained for. +2 +Related work +Few-shot learning +Many few-shot learning methods can be divided into two categories: metric-based +learning and optimization-based learning. First, metric-based methods (Vinyals et al., 2016; Snell et al., +2 + +2017; Sung et al., 2018; Oreshkin et al., 2018) train a fixed embedding network that works universally for +any task. The prediction is based on the distances between the known embeddings of the support set and +the embeddings of the query samples. These methods are not specifically tailored for the continual learning +problem, since they treat every task independently and have no memory of the past tasks. +Second, optimization-based methods (Finn et al., 2017; Nichol & Schulman, 2018; Antoniou et al., 2019; +Rusu et al., 2019) propose to learn an initial fixed embedding, which is later adapted to a specific task using +a few gradient-based steps. However, these methods are not able to learn continually, as simply adapting +the embedding for a new task will result in the catastrophic forgetting of previously learned information. +Continual learning +Most continual learning methods can be grouped into three categories based on +their approach to preventing catastrophic forgetting when learning a new task: rehearsal, regularization +and architectural (see Biesialska et al. 2020 for an overview). Rehearsal methods work by injecting some +amount of replay data from past tasks while learning the new task (Lopez-Paz & Ranzato, 2017; Riemer +et al., 2018; Rolnick et al., 2019; Gupta et al., 2020; Wang et al., 2021a) or distilling a part of a network +using task-conditioned embeddings (Mandivarapu et al., 2020; Von Oswald et al., 2019). Regularization +methods introduce an explicit regularization function when learning new tasks to ensure that old tasks +are not forgotten (Kirkpatrick et al., 2017; Zenke et al., 2017). Architectural methods modify the network +architecture with additional task-specific modules (Rusu et al., 2016), ensembles (Wen et al., 2020) or +adapters (Pfeiffer et al., 2020) that allow for separate routing of different tasks. +We believe that our approach requires the least conceptual overhead compared to the techniques above, +since it does not impose any additional explicit constraints to prevent forgetting. Instead, we reuse the same +principle that made HT work in the first place: decoupling the specialized representation model (a CNN) +from the domain-aware Transformer model. The Transformer learns how to best adapt the incoming CNN +weights in a way that the new task is learned and the old tasks are not forgotten. In this sense, the closest +analogy to our approach would be slow and fast weights (Munkhdalai & Yu, 2017), with the Transformer +weights being analogous to the slow weights that accumulate the knowledge and generate CNN weights as +fast weights. +Incremental few-shot learning +A related, but distinct area of research is incremental few-shot learning +(Gidaris & Komodakis, 2018; Ren et al., 2019; Perez-Rua et al., 2020; Chen & Lee, 2020; Tao et al., 2020; +Wang et al., 2021b; Shi et al., 2021; Zhang et al., 2021; Mazumder et al., 2021; Lee et al., 2021; Yin et al., +2022). There, the goal is to adapt a few-shot task to an existing base classifier trained on a large dataset, +without forgetting the original data. In contrast, our model learns directly from a series of few-shot tasks +presented one after the one, without relying on any prior classifier. All of our tasks are defined using only a +small number of samples. +Perhaps the closest to our setting is the paper by Antoniou et al. (2020) which focuses on the general problem +definition of the continual few-shot learning, but falls short of providing a novel method to solve it. +3 +Continual few-shot learning +We consider the problem of continual few-shot learning, where we are given a series of T tasks, where each +task t := {S(t), Q(t)} is specified via a K-way N-shot support set S(t) := (x(t) +i , y(t) +i )NK +i=0 and a query set +Q(t) := (ˆx(t) +i , ˆy(t) +i ) ˆ +NK +i=0 , where K is the number of classes in each task, N is the number of labeled examples +for each class, and ˆN (typically ˆN ≫ N) is the number of query examples to be classified. +We assume that the classes composing each individual task are drawn from the same distribution uniformly +at random without replacement. However, we consider different ways in which classes for different tasks +are chosen. +First, each task may include exactly the same set of classes. +This is similar to mini-batch +learning with T iterations, where each batch contains exactly N examples of each of K classes1. Second, +each task might include a different set of classes, but drawn from the same overall distribution of classes. +This corresponds to a lifelong learning scenario, where tasks can be thought of as observations that allow us +1This scenario does not require continual learning per se, as the classes do not change between the tasks. +3 + +Figure 2: The information flow of the HyperTransformer (HT) model (left) compared to the proposed +Continual HyperTransformer (CHT) model (right). In the original HT, the input weight embeddings +are initialized with empty placeholders. In contrast, the proposed CHT model incorporates information +from past tasks when generating weights for the current task. The weight slice information from previously +learned tasks is passed as input to the new iteration of the CHT. The CHT uses the support set for the +current task and the input weight information to generate the weights. This allows the CHT to retain +knowledge about past tasks and avoid forgetting when learning new tasks. +to learn more about the world as we encounter new classes during the inference. Finally, each task might have +its own unique semantic meaning and the classes for different tasks are drawn from different distributions. +We will evaluate all of these scenarios in our experiments. +Figure 1 illustrates the process of learning of a continual few-shot problem. For each of the tasks t ∈ 0, . . . , T, +a learner aψ (parameterized by ψ) needs to produce CNN weights θt based on the support set S(t) of task t +and previously generated weights θt−1 (except for the first task, where θt is generated only using S(0)): +θt := aψ (S(t), θt−1) , +(1) +such that θt can predict the classes from all the tasks τ ∈ 0, . . . , t. Notice that when learning from task t, +the learner does not have access to the support set of past tasks and must rely solely on the input weights +θt−1 as a source of information from previous tasks. +After the weights θt are generated, we can use the query set Q(τ) of all tasks τ ∈ 0, . . . , t to evaluate the +prediction quality of the θt and calculate the loss Lψ with respect to the learner parameters ψ. In this work, +we consider two types of predictions given the weights θt: +• Task-incremental learning, in which the goal is to identify the class attribute given the sample and +its task attribute: p(ˆy = k|ˆx, τ). +• Class-incremental learning, in which the goal is to identify both class and task attributes of the +samples: p(ˆy = k, τ|ˆx). +Finally, we can test the performance of the trained model aψ on episodes sampled from a holdout set of +classes Ctest. Notice that, in general, the total number of tasks for the test Ttest might be different from T. +4 +Continual HyperTransformer +Notice that for T = 1, the continual learning problem above reduces to a standard few-shot learning problem +defined by a single few-shot learning task t0 = {S(0), Q(0)}. One method that has been effective in solving this +type of problem is HyperTransformer (HT, Zhmoginov et al., 2022) that uses a self-attention mechanism +to generate CNN weights θ directly from the support set of the few-shot learning problem (see Figure 2, left). +These weights are constructed layer by layer using the embeddings of the support set and the activations of +4 + +the previous layer. After the weights have been generated, the cross-entropy loss Lψ (fθ(ˆx), ˆy) is calculated +by running the query set (ˆx, ˆy) through the generated CNN. +Our proposed Continual HyperTransformer (CHT) naturally extends HT to handle a continual stream +of tasks by using the generated weights from already learned tasks as input weight embeddings into the weight +generator for a new task (see Figure 2, right). In this way, the learned weights themselves act as both the +input and the output of the CHT, performing a dual function: storing information about the previous tasks +as well as serving as the weights for the CNN when evaluating on tasks that have already been seen. +For each task t, the CHT takes as input the support set of that task S(t) as well as the weights from +the previous tasks θt−1, and generates the weights using the equation (1) that are suited for all the tasks +τ ∈ 0, . . . , t. Therefore, for each step t we want to minimize the loss on the query sets of every task up to t: +Jt(ψ) = +t +� +τ=0 +Lψ +� +fθt(ˆx(τ)), ˆy(τ)� +. +(2) +The overall loss function is simply the sum of the losses for all tasks: +arg min +ψ +T +� +t=0 +Jt(ψ). +(3) +The CHT generates a sequence of weights {θτ}t +τ=0, such that each weight is suited for all tasks up to the +current task: θ0 performs well only on task 0, θ1 performs well on tasks 0 and 1, and so on. This allows +the model to effectively learn and adapt to a stream of tasks, while also maintaining good performance on +previously seen tasks. +This design allows for a “preemptive” approach to continual learning, where the CHT model can be trained +on T tasks, and run for any number of tasks τ < T, producing well-performing weights θτ for all the tasks +seen up to that point. An alternative approach would be to specify the exact number of tasks in the sequence +in advance, and only consider the performance after the final task T. This would correspond to minimizing +only the last term JT (ψ) in the equation (3). However, in our experiments, we did not observe any significant +improvement using this approach compared to the one we have described above. +Another desirable property of the proposed CHT architecture is its ability to be recurrent. The parameters +of the HT do not depend on task information, and only take the weights θ and the support set as input. +This means that it is not only possible to preempt CHT at some earlier task, but also extend the trained +model to generate weights for additional tasks beyond the ones it was trained. We will demonstrate this +ability in the experimental section. +4.1 +Prototypical loss +The last element of the algorithm that we have left to discuss is the exact form of loss function Lψ(·) in the +equation (2). The original HT used the cross-entropy loss, which is not well suited for continual learning +because the number of classes that it predicts is tied to the number of parameters in the head layer of the +weights θ. This means that as the number of tasks increases, the architecture of CNN needs to be adjusted, +which goes against our design principle of using a recurrent CHT architecture. Another option would be +to fix the head layer to the K-way classification problem across all the tasks and only predict the class +information within tasks (a problem known as domain-incremental learning; Hsu et al., 2018). However, +this would cause classes with the same label but different tasks to be minimized to the same location in +the embedding space, leading to collisions. Additionally, since class labels are assigned at random for each +training episode, the collisions would occur randomly, making it impossible for CHT learn the correct class +assignment. In the Appendix A.1, we show that the accuracy of this approach decreases dramatically as the +number of tasks increases and becomes impractical even for just two tasks. +To make the method usable, we need to decouple the class predictions of every task while keeping the +overall dimensionality of the embedding space fixed. One solution is to come up with a fixed arrangement of +5 + +Algorithm 1 Class-incremental learning using HyperTransformer with Prototypical Loss. +Input: T randomly sampled K-way N-shot episodes: {S(t); Q(t)}T +t=0. +Output: The loss value J for the generated set of tasks. +1: J ← 0 +▷ Initialize the loss. +2: θ−1 ← 0 +▷ Initialize the weights. +3: for t ← 0 to T do +4: +θt ← aψ(S(t), θt−1) +▷ Generate weight for current task. +5: +for k ← 0 to K do +▷ Compute prototypes for every class of the current task. +6: +ctk ← 1 +N +� +(x,y)∈S(t) fθt(x)1y=k +7: +end for +8: +for τ ← 0 to t do +▷ Update the loss with every seen query set using the equation (6). +9: +for k ← 0 to K do +10: +J ← J − � +(ˆx,ˆy)∈Q(τ) log p(ˆy = k, τ|ˆx)1ˆy=k +11: +end for +12: +end for +13: end for +TK points, but any kind of such arrangement is sub-optimal because it is not possible to place TK points +equidistant from each other in a fixed-dimensional space for large T. A much more elegant solution is to +learn the location of these class prototypes from the support set itself, e.g. with a prototypical loss (Snell +et al., 2017). The prototypes are computed by averaging the embeddings of support samples from a given +class k and task τ: +cτk := 1 +N +� +(x,y)∈S(τ) +fθτ (x)1y=k. +(4) +We can use the prototypes in two different continual learning scenarios. +First, for the task-incremental +learning, we are assumed to have access to the task we are solving and need to predict only the class +information. The probability of the sample belonging to a class k given the task τ is then equal to the +softmax of the ℓ2 distance between the sample and the prototype normalized over the distances to the +prototypes from all the classes from τ: +p(ˆy = k|ˆx, τ) := +exp(−∥fθt(ˆx) − cτk∥2) +� +k′ exp(−∥fθt(ˆx) − cτk′∥2). +(5) +Second, for more general class-incremental learning, we need to predict class attributes across all seen tasks. +The probability of a sample belonging to class k of task τ is equal to the softmax of the ℓ2 distance between +the sample and the prototype, normalized over the distances to the prototypes from all classes for all tasks: +p(ˆy = k, τ|ˆx) := +exp(−∥fθt(ˆx) − cτk∥2) +� +τ ′k′ exp(−∥fθt(ˆx) − cτ ′k′∥2). +(6) +The final loss function is given by minimizing the negative log probability of the chosen softmax over the +query set. The pseudo-code for the entire CHT model is described in Algorithm 1. +Empirically, we noticed that the CHT models trained with the class-incremental learning objective (6) +perform equally well in both class-incremental and task-incremental settings, while models trained with +the task-incremental objective (5) perform well only in the task-incremental setting and rarely outperform +models trained with the equation (6). Therefore, we will focus on CHT models trained with the equation +(6) and evaluate them for both task- and class-incremental learning scenarios. +Notice that the prototypes are computed using the current weights θτ in the equation (4) for task τ, but they +are used later to compare the embeddings produced by subsequent weights θt in equation (6). Ideally, once +the new weights θt are generated, the prototypes should be recomputed as well. However, in true continual +6 + +learning, we are not supposed to reuse the support samples after the task has been processed. We have +found that freezing the prototypes after they are computed provides a viable solution to this problem, and +the difference in performance compared to recomputing the prototypes every step is marginal. +Finally, we want to highlight an important use-case where recomputing the prototypes might still be possible +or even desirable. The weights θt are not affected by this issue and are computed in a continual learning +manner from the equation (1) without using information from the previous task. The support set is only +needed to update the prototypes through generated weights, which is a relatively cheap operation. This +means that it is possible to envision a privacy-preserving scenario in which the weights are updated and +passed from client to client in a continual learning manner, and the prototypes needed to “unlock” those +weights belong to the clients that hold the actual data. +5 +Connection Between Prototypical Loss and MAML +While the core idea behind the prototypical loss is very natural, this approach can also be viewed as a special +case of a simple 1-step MAML-like learning algorithm. This can be demonstrated by considering a simple +classification model q(x; φ) = s(W fθ(x) + b) with φ = (W , b, θ), where fθ(x) is the embedding and s(·) +is a softmax function. MAML algorithm identifies such initial weights φ0 that any task τ with just a few +gradient descent steps initialized at φ0 brings the model towards a task-specific local optimum of Lτ. +Notice that if any label assignment in the training tasks is equally likely, it is natural for q(x; φ0) to not prefer +any particular label over the others. Guided by this, let us choose W 0 and b0 that are label-independent. +Substituting φ = φ0 + δφ into q(x; φ), we then obtain +qℓ(x; φ) = qℓ(x; φ0) + s′ +ℓ(·) +� +δWℓfθ0(x) + δbℓ + W 0 +ℓ +∂f +∂θ (x; θ0)δθ +� ++ O(δφ2), +where ℓ is the label index and δφ = (δW , δb, δθ). The lowest-order label-dependent correction to qℓ(x; φ0) +is given simply by s′ +ℓ(·)(δWℓfθ0(x) + δbℓ). In other words, in the lowest-order, the model only adjusts the +final logits layer to adapt the pretrained embedding fθ0(x) to a new task. +For a simple softmax cross-entropy loss (between predictions q(x) and the groundtruth labels y), a single +step of the gradient descent results in the following logits weight and bias updates: +δWi,· = γ +n +� +(x,y)∈S +� +1y=k − 1 +|C| +� +fθ0(x), +δbk = γ +n +� +(x,y)∈S +� +1y=k − 1 +|C| +� +, +(7) +where the 1/|C| term results from normalization in the softmax operation. Here γ is the learning rate, n +is the total number of support-set samples, |C| is the number of classes and S is the support set. In other +words, we see that the label assignment imposed by δW and δb from the equation (7) effectively relies on +computing a dot-product of fθ0(x) with “prototypes” ck := N −1 � +(x,y)∈S fθ0(x)1y=k. +6 +Experiments +Most of our experiments were conducted using two standard benchmark problems using Omniglot and +tieredImageNet datasets. The generated weights for each task θt are composed of four convolutional +blocks and a single dense layer. Each of the convolutional blocks consist of a 3 × 3 convolutional layer, +batch norm layer, ReLU activation and a 2 × 2 max-pooling layer. For Omniglot we used 8 filters for +convolutional layers and 20-dim FC layer to demonstrate how the network works on small problems, and for +tieredImageNet we used 64 filters for convolutional and 40-dim for the FC layer2 to show that the method +2In contrast with cross-entropy, we do not need to have the head layer dimension to be equal to the number of predicted +labels when using the Prototypical Loss. +7 + +works for large problems as well. The models were trained in an episodic fashion, where the examples for each +training iteration are sampled from a given distribution of classes. The reported accuracy was calculated +from 1024 random episodic evaluations from a separate test distribution, with each episode run 16 times +with different combinations of input samples. +For the HT architecture, we tried to replicate the setup used in the original paper as closely as possible. We +used a 4-layer convolutional network as a feature extractor and a 2-layer convolutional model for computing +activation features. For Omniglot we used a 3-layer, 2-head Transformer and for tieredImageNet, we +used a simplified 1-layer Transformer with 8 heads. In all our experiments, we trained the network on a +single GPU for 4M steps with SGD with an exponential LR decay over 100 000 steps with a decay rate of +0.97. We noticed some stability issues when increasing the number of tasks and had to decrease the learning +rate to compensate: for Omniglot experiments, we used a learning rate 10−4 for up to 4 tasks and 5×10−5 +for 5 tasks. For tieredImageNet, we used the same learning rate of 5×10−6 for training with any number +of tasks T. We trained the CHT models with the class-incremental objective (6), but evaluated them for +both task-incremental and class-incremental scenarios. +6.1 +Learning from mini-batches +We first consider a case where every task includes the same set of classes. Specifically, we compared the +following three models using a set of four 5-way 1-shot support set batches S(1), . . . , S(4) that consist of the +same set of classes from tieredImageNet: +θ(a) ≡ aψ(S(1) + S(2) + S(3) + S(4), θ0), +θ(b) ≡ aψ(S(3) + S(4), aψ(S(1) + S(2), θ0)), +θ(c) ≡ aψ(S(4), aψ(S(3), aψ(S(2), aψ(S(1), θ0)))), +where + operation denotes a simple concatenation of different support set batches. For this experiment, we +used the cross-entropy loss (since the label set was the same for all S(i)) and each support set batch S(i) +contained a single example per class. We observed that the test accuracies for θ(a), θ(b) and θ(c) were equal +to 67.9%, 68.0% and 68.3% respectively, all within the statistical error range (±0.4%). At the same time, +HT trained with just S(1) or S(1) + S(2) (with 1 or 2 samples per class respectively) performed significantly +worse, reaching the test accuracies of 56.2% and 62.9% respectively. This demonstrates that the proposed +mechanism of updating generated CNN weights using information from multiple support set batches can +achieve performance comparable to processing all samples in a single pass with HT. +6.2 +Learning from tasks within a single domain +Next, we consider a scenario where the tasks consist of classes drawn from a single overall distribution. We +present the results of two models: one trained on 20-way, 1-shot tasks with classes sampled from Omniglot +dataset, and anther trained on 5-way, 5-shot tasks with classes sampled from tieredImageNet dataset. +We compare the performance of CHT to two baseline models. +The first is a Constant ProtoNet +(ConstPN), which represents a vanilla Prototypical Network, as described in Snell et al. (2017). In this +approach, a universal fixed CNN network is trained on episodes from Ctrain. This constant network can be +applied to every task separately by projecting the support set as prototypes for that task and computing +the prediction with respect to these prototypes. Strictly speaking, this is not a continual learning method, +since it treats every task independently and has no memory of previous tasks. For the best results on this +baseline, we had to increase the number of classes by a factor of 5 during training (e.g. for 20-way Omniglot +evaluation we have trained it with 100-way problems). +The second baseline we used specifically for the class-incremental learning is a Merged HyperTrans- +former (MergedHT), where we combine all the tasks and train a single original HT instance as a single +task. This method does not solve a continual learning problem, since it has the information about all the +tasks from the beginning, but it produces a solution for every class and task that we can still be compared +to the weights generated by the CHT. +8 + +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Omniglot, 8 channels +Accuracy +0 +1 +2 +3 +4 +0.84 +0.85 +0.86 +0.87 +0.88 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +tieredImageNet, 64 channels +0 +1 +2 +3 +4 +0.68 +0.70 +0.72 +0.74 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +ConstPN +Figure 3: Task-incremental learning on Omniglot and tieredImageNet. +Each column represents a +different CHT trained with a total of T = 2, 3, 4 or 5 tasks. The tasks marked with a bullet symbol (•) +correspond to the terms in the objective function (3) that are being minimized. The lines marked with the +diamond symbol (⋄) show the extrapolation of the trained CHT to a larger number of tasks. +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Omniglot, 8 channels +Accuracy +0 +0-1 +0-2 +0-3 +0-4 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +tieredImageNet, 64 channels +0 +0-1 +0-2 +0-3 +0-4 +0.60 +0.65 +0.70 +0.75 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +ConstPN +MergedHT +Figure 4: Class-incremental learning on Omniglot and tieredImageNet. +Each column represents a +different CHT trained with a total of T = 2, 3, 4 or 5 tasks. The tasks marked with a bullet symbol (•) +correspond to the terms in the objective function (3) that are being minimized. The lines marked with the +diamond symbol (⋄) show the extrapolation of the trained CHT to a larger number of tasks. +Each trained model is applied to both task-incremental (Figure 3) and class-incremental (Figure 4) settings. +To understand the effect of continual learning with multiple tasks, each column represents a separate run of +the CHT trained on T = 2, 3, 4 or 5 tasks in total (for training a higher T, see the results in the Appendix). +To demonstrate the recurrence of the method, we extended the number of tasks to 5 for the evaluation +regardless of how many tasks it was trained on. Each plot shows 5 curves corresponding to the CHT, split +into two groups: bullet marker (•) for tasks that the model was trained for and diamond marker (⋄) for +extrapolation to more tasks. +9 + +Task-incremental learning. +We start by analysing the task-incremental learning results. For the Om- +niglot dataset, we saw no signs of catastrophic forgetting for the CHT. In fact, we observed a positive +backward knowledge transfer, where the performance on past tasks improved as more weights were generated. +For example, in most cases, the performance of θ1 (green markers) was higher than θ0 (orange markers), and +θ2 was higher than both θ1 and θ0. Additionally, as the number of tasks increased, the overall performance +of the CHT also increased, with the model trained on T = 5 tasks performing better than the one trained +on T = 2 tasks. +For the tieredImageNet dataset, the results were better than the ConstPN baseline, but the positive +backward knowledge effect effect was not as pronounced as it was for the Omniglot dataset. The perfor- +mance for every training task remained roughly the same for all generated weights, indicating that the model +did not suffer from catastrophic forgetting. +Overall, the CHT consistently outperformed the ConstPN baseline, particularly when applied to the same +or lower number of tasks it was trained on. Although the accuracy of the CHT did decrease slightly when +it was applied to more tasks than it was trained on, this decrease was not significant. In fact, even when +CHT was trained on only T = 3 tasks, generating weights for one of two additional tasks still resulted in +better performance than the ConstPN baseline. +Class-incremental learning. +In the class-incremental learning setting, the task name is given by two +numbers indicating the range of tasks we used for evaluation (e.g. task name 0-3 corresponds to four tasks +from 0 to 3). The black constant dashed line is the baseline performance of the ConstPN, which uses a +fixed embedding and does not differentiate between tasks. The starred blue markers represent a separate +run of the HT for a particular configuration of merged tasks. +As one can see in the figure The accuracy of all the models decreased as more tasks were included in the +prediction. +This was expected because the size of the generated CNN did not change, but the number +of classes that needs to be predicted was increasing. For Omniglot dataset we again saw the positive +backwards transfer taking place, with CHT models trained on more tasks T performing better overall. For +a given model trained on a fixed T, the performance was comparable. This demonstrates the preemptive +property of the CHT, where models trained for a certain number of tasks can still be run for any smaller +number of tasks with similar performance. +When comparing the results to the baselines, the CHT had better results than the ConstPN up to the +number of tasks T it was trained for, and the extrapolation results improved as T increases. Interestingly, +for the case of T = 5 the CHT was able to outperform even the MergedHT baseline for the Omniglot, +even though the MergedHT had access to information about all tasks from the beginning. This suggests +that having more classes to classify makes the learning problem difficult for the original HT, as the image +embeddings may not be able to learn good embeddings. This is particularly evident in the tieredImageNet +dataset, where the performance of the MergedHT is so low that it falls below 60%, even for the 0-1 task. +6.3 +Learning from tasks across multiple domains +In the experiments described above, the support and query sets for each task were drawn from the same +general distribution, and the image domain remained consistent across all tasks. If the tasks were drawn +from different distributions and different image domains, we would expect task-agnostic ConstPN approach +to suffer in accuracy because it would need to find a universal representation that works well across all image +domains. In contrast, the CHT approach could adapt its sample representations differently for different +detected image domains, leading to improved performance. +We verify this by creating a multi-domain episode generator that includes tasks from various image datasets: +Omniglot, Caltech101, CaltechBirds2011, Cars196, OxfordFlowers102 and StanfordDogs. +We compared the accuracy of the ConstPN and CHT on this generator using episodes containing two +tasks with 5-way, 1-shot problems. The generated CNN model had 16 channels with 32 channels for the final +layer. Other parameters were the same as those used in the tieredImageNet experiments. The ConstPN +achieved the accuracy of 53% for task 0, 52.8% for task 1 and 50.8% for combined tasks. +The CHT +achieved the accuracy of 56.2% for task 0, 55.2% for task 1 and 53.8% for combined tasks. The accuracy +10 + +gap of nearly 3% between these two methods, which is larger than the gap observed in the Omniglot +and tieredImageNet experiments, suggests that the CHT is better at adapting to a multi-domain task +distribution. +7 +Conclusions +The proposed Continual HyperTransformer model has several attractive features. +As an efficient +few-shot learner, it can generate CNN weights on the fly with no training required, using only a small set +of labeled examples. As a continual learner, it is able to update the weights with information from new +tasks by iteratively passing them through HT. Empirically, we have shown that the learning occurs without +catastrophic forgetting and may even result in positive backward transfer. 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Hypertransformer: Model generation for super- +vised and semi-supervised few-shot learning. arXiv preprint arXiv:2201.04182, 2022. +13 + +0 +1 +2 +3 +4 +Task name +0.2 +0.4 +0.6 +0.8 +Accuracy +θ0 +θ1 +θ2 +θ3 +θ4 +ConstPN +Figure 5: The accuracy of the HT model trained +for T = 5 using the cross-entropy loss. +The ac- +curacy of the first weight θ0 is high and is better +than the accuracy of the ConstPN model’s em- +beddings. +However, when more tasks are added, +the accuracy drops dramatically due to collisions +between the same classes for different tasks in the +cross-entropy loss. +Accuracy +0 +1 +2 +3 +4 +0.87 +0.88 +0 +0-1 +0-2 +0-3 +0-4 +0.6 +0.7 +0.8 +0.9 +Task name +min eq. (5): +θ0 +θ1 +θ2 +θ3 +θ4 +min eq. (6): +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 6: CHT trained using task-incremental ob- +jective (5) vs. class-incremental objective (6). +A +Additional experiments and figures +A.1 +Learning with cross-entropy loss +Figure 5 shows the results of an attempt to do learn multiple tasks using a HT with a cross-entropy loss. +Since the size of the last layer’s embedding is not increased, the model can only predict the class labels +within the task and not the task themselves, which corresponds to the domain-incremental learning setup. +Additionally, the same class from different tasks are mapped to the same location in the embedding space, +leading to collisions when more tasks are added. This is why the accuracy drops significantly as the number +of tasks increases. On the other hand, ConstPN model is more flexible because the prototypes for each +task are computed from the support set of that task and do not have to be fixed to a one-hot vector as in +the cross-entropy loss. +A.2 +Training using task-incremental and class-incremental objectives +Figure 6 compares the accuracy of two different models trained with task-incremental (using equation (5)) +and class-incremental (using equation (6)) objectives. The performance of both models on task-incremental +problems are similar, while the model trained with the class-incremental objective performs better on class- +incremental problems. +A.3 +Analysis of prototypical embeddings using UMAP +To better understand the quality of the learned prototypes, we conducted an experiment on a tieredIma- +geNet 5-way, 5-shot problem. We selected a random episode of 5 tasks and ran them through the model, +producing weights θ0 to θ4 along with the prototypes for each class of every task. We then computed the +logits of the query sets, which consisted of 20 samples per class for each task. The resulting 40-dim em- +beddings for the prototypes and query sets were concatenated and projected onto a 2D space using UMAP +(Figure 7). Note that the prototypes from the earlier tasks remain unchanged for the later task, but their +2D UMAP projections are different, because UMAP is a non-parametric method and it must be re-run for +every new θk. We tried our best to align the embedding using the Procrustes alignment method. +The plot shows that the embeddings of the tasks are well separated in the logits space, which helps explain +why the model performs well for both task- and class-incremental learning. Normalizing the softmax over +the classes within the same tasks or across all tasks made little difference when the tasks are so far away +from each other. On the right of Figure 7, we show the projection of the ConstPN embedding of the same +14 + +θ0 +θ1 +θ2 +θ3 +θ4 +ConstPN +Figure 7: Left 5 plots: the UMAP projection of the CHT prototypes and the query set embeddings for +different generated weights, where the points are colored based on the task information. The query set +points are connected with their corresponding prototypes using dashed lines. Right plot: UMAP projection +of the ConstPN embedding for 25 different classes from tieredImageNet. Embeddings are aligned using +Procrustes alignment. +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 8: The UMAP projections of 20-dimensional embeddings of the prototypes and query set for different +weights obtained from incremental HT training. The query set points are connected to their corresponding +prototypes using dashed lines. In the top plot, the points are colored according to their class information, +while in the bottom plot they are colored according to their task information. Embeddings are aligned using +Procrustes alignment. +25 classes. The ConstPN model does not make a distinction between tasks and treats each class separately. +The fact that 3 clusters emerge has to do purely with the semantics of the chosen clusters and the way the +ConstPN model groups them. This also helps to explain why the CHT model performs better than the +ConstPN, as it separates the tasks before separating the classes within each task. +The UMAP embedding for the Omniglot dataset using ProtoNet (Figure 8) appears to be different from +similar embedding projection of tieredImageNet dataset. In particular, the embeddings from different +tasks seem to overlap, while in the tieredImageNet embedding they are separated. This may be due to the +fact that the classes in the Omniglot dataset are more closely connected than those in the tieredImageNet +dataset. Interestingly, despite the overlap between the classes from different tasks, the final accuracy is still +high and only slightly degrades as more tasks are added. +A.4 +Learning with more tasks +Our analysis primarily focused on the performance of the CHT on up to 5 tasks. However, as shown in +Figure 9 the CHT model is capable of handling a much larger number of tasks T. Similar to the results in +15 + +Omniglot, 8 channel +Omniglot, 32 channel +0 +0-1 +0-2 +0-3 +0-4 +0-5 +0-6 +0-7 +0-8 +0-9 +0-10 +0-11 +0-12 +0-13 +0-14 +0-15 +0-16 +0-17 +0-18 +0-19 +Task name +0.5 +0.6 +0.7 +0.8 +Accuracy +Number of tasks 10 +Number of tasks 15 +Number of tasks 20 +0 +0-1 +0-2 +0-3 +0-4 +0-5 +0-6 +0-7 +0-8 +0-9 +0-10 +Task name +0.80 +0.85 +0.90 +0.95 +Accuracy +Number of tasks 8 +Number of tasks 9 +Number of tasks 10 +Number of tasks 11 +Figure 9: Omniglot with 8 or 32 channels trained with a different number of tasks T. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 15 16 17 18 19 +Task name +0.860 +0.865 +0.870 +Accuracy +θ5 +θ19 +0-1 +0-2 +0-3 +0-4 +0-5 +0-6 +0-7 +0-8 +0-9 +0-10 +0-11 +0-12 +0-13 +0-14 +0-15 +0-16 +0-17 +0-18 +0-19 +Task name +0.5 +0.6 +0.7 +0.8 +Accuracy +θ5 +θ19 +Figure 10: Omniglot with 8 channels trained for T = 20 tasks. Here we show the final weight θ19 generated +along with an intermediate θ5 for task-incremental (left) and class-incremental (right) learning. +the main text, the nearly overlapping curves in the graph indicate that the model trained for T tasks can +maintain the same level of accuracy when applied to a larger number of tasks. +Figure 10 shows 8-channel Omniglot evaluated on task-incremental and class-incremental objectives. +A.5 +Continual HyperTransformer vs MergedHT for tieredImageNet +Figure 11 shows a zoomed out view of the results presented in Figure 4). +It illustrates the significant +difference in performance between the MergedHT and the CHT models. +A.6 +Additional figures for Omniglot task-incremental and class-incremental learning +Figures 12, 13, 14 and 15 show additional experiments with the Omniglot dataset using different number +of channels in the CNN. +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Accuracy +tieredImageNet, 64 channels +0 +0-1 +0-2 +0-3 +0-4 +0.3 +0.4 +0.5 +0.6 +0.7 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +0 +0-1 +0-2 +0-3 +0-4 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +ConstPN +MergedHT +Figure 11: Zoomed out view of Figure 4 so that the results of the MergedHT is visible. +16 + +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Task-incremental learning +Accuracy +0 +1 +2 +3 +4 +0.625 +0.650 +0.675 +0 +1 +2 +3 +4 +0.625 +0.650 +0.675 +0 +1 +2 +3 +4 +0.625 +0.650 +0.675 +0 +1 +2 +3 +4 +0.625 +0.650 +0.675 +Class-incremental learning +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 12: Task-incremental and class-incremental learning on the Omniglot dataset with 4-channels con- +volutions. +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Task-incremental learning +Accuracy +0 +1 +2 +3 +4 +0.78 +0.80 +0 +1 +2 +3 +4 +0.78 +0.80 +0 +1 +2 +3 +4 +0.78 +0.80 +0 +1 +2 +3 +4 +0.78 +0.80 +Class-incremental learning +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0.8 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0.8 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0.8 +0 +0-1 0-2 0-3 0-4 +0.4 +0.6 +0.8 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 13: Task-incremental and class-incremental learning on the Omniglot dataset with 6-channels con- +volutions. +17 + +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Task-incremental learning +Accuracy +0 +1 +2 +3 +4 +0.93 +0.94 +0.95 +0 +1 +2 +3 +4 +0.93 +0.94 +0.95 +0 +1 +2 +3 +4 +0.93 +0.94 +0.95 +0 +1 +2 +3 +4 +0.93 +0.94 +0.95 +Class-incremental learning +0 +0-1 0-2 0-3 0-4 +0.80 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.80 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.80 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.80 +0.85 +0.90 +0.95 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 14: Task-incremental and class-incremental learning on the Omniglot dataset with 16-channels +convolutions. +T = 2 tasks +T = 3 tasks +T = 4 tasks +T = 5 tasks +Task-incremental learning +Accuracy +0 +1 +2 +3 +4 +0.950 +0.955 +0.960 +0 +1 +2 +3 +4 +0.950 +0.955 +0.960 +0 +1 +2 +3 +4 +0.950 +0.955 +0.960 +0 +1 +2 +3 +4 +0.950 +0.955 +0.960 +Class-incremental learning +0 +0-1 0-2 0-3 0-4 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.85 +0.90 +0.95 +0 +0-1 0-2 0-3 0-4 +0.85 +0.90 +0.95 +Task Name +θ0 +θ1 +θ2 +θ3 +θ4 +Figure 15: Task-incremental and class-incremental learning on the Omniglot dataset with 32-channels +convolutions. +18 + diff --git a/ntE3T4oBgHgl3EQfjAo0/content/tmp_files/load_file.txt b/ntE3T4oBgHgl3EQfjAo0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3d5547cba853f6e7f293b1778e58f2250521a71 --- /dev/null +++ b/ntE3T4oBgHgl3EQfjAo0/content/tmp_files/load_file.txt @@ -0,0 +1,722 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf,len=721 +page_content='Continual Few-Shot Learning Using HyperTransformers Max Vladymyrov, Andrey Zhmoginov, Mark Sandler Google Research {mxv,azhmogin,sandler}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='com Abstract We focus on the problem of learning without forgetting from multiple tasks arriving se- quentially, where each task is defined using a few-shot episode of novel or already seen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We approach this problem using the recently published HyperTransformer (HT), a Transformer-based hypernetwork that generates a specialized task-specific CNN weights directly from the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In order to learn from a continual sequence of task, we propose to recursively re-use the generated weights as input to the HT for the next task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This way, the generated CNN weights themselves act as a representation of previ- ously learned tasks, and the HT is trained to update these weights so that the new task can be learned without forgetting past tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This approach is different from most con- tinual learning algorithms that typically rely on using replay buffers, weight regularization or task-dependent architectural changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We demonstrate that our proposed Continual HyperTransformer method equipped with a prototypical loss is capable of learning and retaining knowledge about past tasks for a variety of scenarios, including learning from mini-batches, and task-incremental and class-incremental learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 1 Introduction Continual few-shot learning involves learning from a continuous stream of tasks described by a small number of examples without forgetting previously learned information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This type of learning closely resembles how humans and other biological systems acquire new information, as we can continually learn novel concepts with a small amount of information and retain that knowledge for an extended period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Algorithms for few-shot continual learning can be useful in many real-world applications where there is a need to classify a large number of classes in a dynamic environment with limited observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Some practical applications can include enabling robots to continually adapt to changing environments based on an incoming stream of sparse demonstrations or allowing for privacy-preserving learning, where the model can be trained sequentially on private data sharing only the weights without ever exposing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' To tackle this problem, we propose using HyperTransformer (HT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Zhmoginov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 2022), a recently published few-shot learning method that utilizes a large hypernetwork (Ha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2016) to meta-learn from episodes sampled from a large set of few-shot learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The HT is trained to directly generate weights of a much smaller specialized Convolutional Neural Network (CNN) model using only few labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This works by decoupling the domain knowledge model (represented by a Transformer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 2017) from the learner itself (a CNN), generated to solve only a given specific few-shot learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We present a modification to HT method, called Continual HyperTransformer (CHT), that is aimed at exploring the capability of the HT to sequentially update the CNN weights with the information from a new task, while retaining the knowledge about the tasks that were already learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In other words, given the CNN weights θt−1 generated after seeing some previous tasks 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , t − 1 and a description of the new task t, the CHT generates the weights θt that are suited for all the tasks 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In order for the CHT to be able to absorb a continual stream of tasks, we modified the loss function from a cross-entropy that was used in the HT to a more flexible prototypical loss (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2017), that uses prototypes as a learned representation of every class from all the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' As the tasks come along, we maintain 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='04584v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='LG] 11 Jan 2023 Figure 1: In few-shot continual learning, the model learns from T tasks sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For the first task (task 0), the CNN weights θ0 are generated using only the support set S(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For each subsequent task t, the Continual HyperTransformer (CHT) uses the support set S(t) and the previously generated weights θt−1 to generate the weights θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' To update the weights ψ of the CHT, the loss is calculated by summing the individual losses computed for each generated weight θt when evaluated on the query set of all the prior tasks (Q(τ))T τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' and update a set of prototypes in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The prototypes are then used to predict the class and task attributes for a given input sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We evaluate CHT in three realistic scenarios where a continual few-shot learning model like ours might be used: the mini-batch version, where every task consists of the same classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' the lifelong learning version, where classes for all the tasks are drawn from the same overall distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' and the heterogeneous task semantic version, where every task has its own unique distribution of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We also test CHT in two different continual learning scenarios: task-incremental learning (predicting class attributes using the task information) and class-incremental learning (predicting class attributes without access to task information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' also known as lifelong learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Moreover, we show empirically that a model trained for class-incremental learning can also perform well in task-incremental learning, similar to a model specifically trained for task-incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Our approach has several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' First, as a hypernetwork, the CHT is able to generate and update the weights of the CNN on the fly with no training required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A trained Transformer holds the domain world-knowledge and can generalize from limited few-shot observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' There is also evidence to suggest that similar functions are performed by the prefrontal cortex in the human brain (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2001), which may imply biological plausibility of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Second, we demonstrate that models learned with CHT do not suffer from the catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We even see cases of the positive backward transfer for smaller models, where the performance on a given task actually improves for subsequently generated weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Third, while the CHT is trained to optimize for T tasks, the model can be stopped at any point t ≤ T during the inference with weights θt that are suited for all the tasks 0 ≤ τ ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Moreover, the performance of a given weight θt improves when the CHT is trained on more tasks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Finally, we designed the CHT model to be independent from a specific step and operate as a recurrent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' It can be used to learn a larger number of tasks it was originally trained for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 2 Related work Few-shot learning Many few-shot learning methods can be divided into two categories: metric-based learning and optimization-based learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' First, metric-based methods (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Oreshkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2018) train a fixed embedding network that works universally for any task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The prediction is based on the distances between the known embeddings of the support set and the embeddings of the query samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' These methods are not specifically tailored for the continual learning problem, since they treat every task independently and have no memory of the past tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Second, optimization-based methods (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Nichol & Schulman, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Antoniou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2019) propose to learn an initial fixed embedding, which is later adapted to a specific task using a few gradient-based steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, these methods are not able to learn continually, as simply adapting the embedding for a new task will result in the catastrophic forgetting of previously learned information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Continual learning Most continual learning methods can be grouped into three categories based on their approach to preventing catastrophic forgetting when learning a new task: rehearsal, regularization and architectural (see Biesialska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 2020 for an overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Rehearsal methods work by injecting some amount of replay data from past tasks while learning the new task (Lopez-Paz & Ranzato, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021a) or distilling a part of a network using task-conditioned embeddings (Mandivarapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Von Oswald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Regularization methods introduce an explicit regularization function when learning new tasks to ensure that old tasks are not forgotten (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Architectural methods modify the network architecture with additional task-specific modules (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2016), ensembles (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020) or adapters (Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020) that allow for separate routing of different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We believe that our approach requires the least conceptual overhead compared to the techniques above, since it does not impose any additional explicit constraints to prevent forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Instead, we reuse the same principle that made HT work in the first place: decoupling the specialized representation model (a CNN) from the domain-aware Transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The Transformer learns how to best adapt the incoming CNN weights in a way that the new task is learned and the old tasks are not forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In this sense, the closest analogy to our approach would be slow and fast weights (Munkhdalai & Yu, 2017), with the Transformer weights being analogous to the slow weights that accumulate the knowledge and generate CNN weights as fast weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Incremental few-shot learning A related, but distinct area of research is incremental few-shot learning (Gidaris & Komodakis, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Perez-Rua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Chen & Lee, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Mazumder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' There, the goal is to adapt a few-shot task to an existing base classifier trained on a large dataset, without forgetting the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In contrast, our model learns directly from a series of few-shot tasks presented one after the one, without relying on any prior classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' All of our tasks are defined using only a small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Perhaps the closest to our setting is the paper by Antoniou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (2020) which focuses on the general problem definition of the continual few-shot learning, but falls short of providing a novel method to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 3 Continual few-shot learning We consider the problem of continual few-shot learning, where we are given a series of T tasks, where each task t := {S(t), Q(t)} is specified via a K-way N-shot support set S(t) := (x(t) i , y(t) i )NK i=0 and a query set Q(t) := (ˆx(t) i , ˆy(t) i ) ˆ NK i=0 , where K is the number of classes in each task, N is the number of labeled examples for each class, and ˆN (typically ˆN ≫ N) is the number of query examples to be classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We assume that the classes composing each individual task are drawn from the same distribution uniformly at random without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, we consider different ways in which classes for different tasks are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' First, each task may include exactly the same set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This is similar to mini-batch learning with T iterations, where each batch contains exactly N examples of each of K classes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Second, each task might include a different set of classes, but drawn from the same overall distribution of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This corresponds to a lifelong learning scenario, where tasks can be thought of as observations that allow us 1This scenario does not require continual learning per se, as the classes do not change between the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 3 Figure 2: The information flow of the HyperTransformer (HT) model (left) compared to the proposed Continual HyperTransformer (CHT) model (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In the original HT, the input weight embeddings are initialized with empty placeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In contrast, the proposed CHT model incorporates information from past tasks when generating weights for the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The weight slice information from previously learned tasks is passed as input to the new iteration of the CHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The CHT uses the support set for the current task and the input weight information to generate the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This allows the CHT to retain knowledge about past tasks and avoid forgetting when learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' to learn more about the world as we encounter new classes during the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Finally, each task might have its own unique semantic meaning and the classes for different tasks are drawn from different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We will evaluate all of these scenarios in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Figure 1 illustrates the process of learning of a continual few-shot problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For each of the tasks t ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , T, a learner aψ (parameterized by ψ) needs to produce CNN weights θt based on the support set S(t) of task t and previously generated weights θt−1 (except for the first task, where θt is generated only using S(0)): θt := aψ (S(t), θt−1) , (1) such that θt can predict the classes from all the tasks τ ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Notice that when learning from task t, the learner does not have access to the support set of past tasks and must rely solely on the input weights θt−1 as a source of information from previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' After the weights θt are generated, we can use the query set Q(τ) of all tasks τ ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , t to evaluate the prediction quality of the θt and calculate the loss Lψ with respect to the learner parameters ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In this work, we consider two types of predictions given the weights θt: Task-incremental learning, in which the goal is to identify the class attribute given the sample and its task attribute: p(ˆy = k|ˆx, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Class-incremental learning, in which the goal is to identify both class and task attributes of the samples: p(ˆy = k, τ|ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Finally, we can test the performance of the trained model aψ on episodes sampled from a holdout set of classes Ctest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Notice that, in general, the total number of tasks for the test Ttest might be different from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 4 Continual HyperTransformer Notice that for T = 1, the continual learning problem above reduces to a standard few-shot learning problem defined by a single few-shot learning task t0 = {S(0), Q(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' One method that has been effective in solving this type of problem is HyperTransformer (HT, Zhmoginov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2022) that uses a self-attention mechanism to generate CNN weights θ directly from the support set of the few-shot learning problem (see Figure 2, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' These weights are constructed layer by layer using the embeddings of the support set and the activations of 4 the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' After the weights have been generated, the cross-entropy loss Lψ (fθ(ˆx), ˆy) is calculated by running the query set (ˆx, ˆy) through the generated CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Our proposed Continual HyperTransformer (CHT) naturally extends HT to handle a continual stream of tasks by using the generated weights from already learned tasks as input weight embeddings into the weight generator for a new task (see Figure 2, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In this way, the learned weights themselves act as both the input and the output of the CHT, performing a dual function: storing information about the previous tasks as well as serving as the weights for the CNN when evaluating on tasks that have already been seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For each task t, the CHT takes as input the support set of that task S(t) as well as the weights from the previous tasks θt−1, and generates the weights using the equation (1) that are suited for all the tasks τ ∈ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Therefore, for each step t we want to minimize the loss on the query sets of every task up to t: Jt(ψ) = t � τ=0 Lψ � fθt(ˆx(τ)), ˆy(τ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (2) The overall loss function is simply the sum of the losses for all tasks: arg min ψ T � t=0 Jt(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (3) The CHT generates a sequence of weights {θτ}t τ=0, such that each weight is suited for all tasks up to the current task: θ0 performs well only on task 0, θ1 performs well on tasks 0 and 1, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This allows the model to effectively learn and adapt to a stream of tasks, while also maintaining good performance on previously seen tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This design allows for a “preemptive” approach to continual learning, where the CHT model can be trained on T tasks, and run for any number of tasks τ < T, producing well-performing weights θτ for all the tasks seen up to that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' An alternative approach would be to specify the exact number of tasks in the sequence in advance, and only consider the performance after the final task T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This would correspond to minimizing only the last term JT (ψ) in the equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, in our experiments, we did not observe any significant improvement using this approach compared to the one we have described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Another desirable property of the proposed CHT architecture is its ability to be recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The parameters of the HT do not depend on task information, and only take the weights θ and the support set as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This means that it is not only possible to preempt CHT at some earlier task, but also extend the trained model to generate weights for additional tasks beyond the ones it was trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We will demonstrate this ability in the experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='1 Prototypical loss The last element of the algorithm that we have left to discuss is the exact form of loss function Lψ(·) in the equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The original HT used the cross-entropy loss, which is not well suited for continual learning because the number of classes that it predicts is tied to the number of parameters in the head layer of the weights θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This means that as the number of tasks increases, the architecture of CNN needs to be adjusted, which goes against our design principle of using a recurrent CHT architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Another option would be to fix the head layer to the K-way classification problem across all the tasks and only predict the class information within tasks (a problem known as domain-incremental learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, this would cause classes with the same label but different tasks to be minimized to the same location in the embedding space, leading to collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Additionally, since class labels are assigned at random for each training episode, the collisions would occur randomly, making it impossible for CHT learn the correct class assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='1, we show that the accuracy of this approach decreases dramatically as the number of tasks increases and becomes impractical even for just two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' To make the method usable, we need to decouple the class predictions of every task while keeping the overall dimensionality of the embedding space fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' One solution is to come up with a fixed arrangement of 5 Algorithm 1 Class-incremental learning using HyperTransformer with Prototypical Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Input: T randomly sampled K-way N-shot episodes: {S(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Q(t)}T t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Output: The loss value J for the generated set of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 1: J ← 0 ▷ Initialize the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 2: θ−1 ← 0 ▷ Initialize the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 3: for t ← 0 to T do 4: θt ← aψ(S(t), θt−1) ▷ Generate weight for current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 5: for k ← 0 to K do ▷ Compute prototypes for every class of the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 6: ctk ← 1 N � (x,y)∈S(t) fθt(x)1y=k 7: end for 8: for τ ← 0 to t do ▷ Update the loss with every seen query set using the equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 9: for k ← 0 to K do 10: J ← J − � (ˆx,ˆy)∈Q(τ) log p(ˆy = k, τ|ˆx)1ˆy=k 11: end for 12: end for 13: end for TK points, but any kind of such arrangement is sub-optimal because it is not possible to place TK points equidistant from each other in a fixed-dimensional space for large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A much more elegant solution is to learn the location of these class prototypes from the support set itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' with a prototypical loss (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The prototypes are computed by averaging the embeddings of support samples from a given class k and task τ: cτk := 1 N � (x,y)∈S(τ) fθτ (x)1y=k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (4) We can use the prototypes in two different continual learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' First, for the task-incremental learning, we are assumed to have access to the task we are solving and need to predict only the class information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The probability of the sample belonging to a class k given the task τ is then equal to the softmax of the ℓ2 distance between the sample and the prototype normalized over the distances to the prototypes from all the classes from τ: p(ˆy = k|ˆx, τ) := exp(−∥fθt(ˆx) − cτk∥2) � k′ exp(−∥fθt(ˆx) − cτk′∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (5) Second, for more general class-incremental learning, we need to predict class attributes across all seen tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The probability of a sample belonging to class k of task τ is equal to the softmax of the ℓ2 distance between the sample and the prototype, normalized over the distances to the prototypes from all classes for all tasks: p(ˆy = k, τ|ˆx) := exp(−∥fθt(ˆx) − cτk∥2) � τ ′k′ exp(−∥fθt(ˆx) − cτ ′k′∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (6) The final loss function is given by minimizing the negative log probability of the chosen softmax over the query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The pseudo-code for the entire CHT model is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Empirically, we noticed that the CHT models trained with the class-incremental learning objective (6) perform equally well in both class-incremental and task-incremental settings, while models trained with the task-incremental objective (5) perform well only in the task-incremental setting and rarely outperform models trained with the equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Therefore, we will focus on CHT models trained with the equation (6) and evaluate them for both task- and class-incremental learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Notice that the prototypes are computed using the current weights θτ in the equation (4) for task τ, but they are used later to compare the embeddings produced by subsequent weights θt in equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Ideally, once the new weights θt are generated, the prototypes should be recomputed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, in true continual 6 learning, we are not supposed to reuse the support samples after the task has been processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We have found that freezing the prototypes after they are computed provides a viable solution to this problem, and the difference in performance compared to recomputing the prototypes every step is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Finally, we want to highlight an important use-case where recomputing the prototypes might still be possible or even desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The weights θt are not affected by this issue and are computed in a continual learning manner from the equation (1) without using information from the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The support set is only needed to update the prototypes through generated weights, which is a relatively cheap operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This means that it is possible to envision a privacy-preserving scenario in which the weights are updated and passed from client to client in a continual learning manner, and the prototypes needed to “unlock” those weights belong to the clients that hold the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 5 Connection Between Prototypical Loss and MAML While the core idea behind the prototypical loss is very natural, this approach can also be viewed as a special case of a simple 1-step MAML-like learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This can be demonstrated by considering a simple classification model q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ) = s(W fθ(x) + b) with φ = (W , b, θ), where fθ(x) is the embedding and s(·) is a softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' MAML algorithm identifies such initial weights φ0 that any task τ with just a few gradient descent steps initialized at φ0 brings the model towards a task-specific local optimum of Lτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Notice that if any label assignment in the training tasks is equally likely, it is natural for q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ0) to not prefer any particular label over the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Guided by this, let us choose W 0 and b0 that are label-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Substituting φ = φ0 + δφ into q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ), we then obtain qℓ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ) = qℓ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ0) + s′ ℓ(·) � δWℓfθ0(x) + δbℓ + W 0 ℓ ∂f ∂θ (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' θ0)δθ � + O(δφ2), where ℓ is the label index and δφ = (δW , δb, δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The lowest-order label-dependent correction to qℓ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' φ0) is given simply by s′ ℓ(·)(δWℓfθ0(x) + δbℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In other words, in the lowest-order, the model only adjusts the final logits layer to adapt the pretrained embedding fθ0(x) to a new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For a simple softmax cross-entropy loss (between predictions q(x) and the groundtruth labels y), a single step of the gradient descent results in the following logits weight and bias updates: δWi,· = γ n � (x,y)∈S � 1y=k − 1 |C| � fθ0(x), δbk = γ n � (x,y)∈S � 1y=k − 1 |C| � , (7) where the 1/|C| term results from normalization in the softmax operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Here γ is the learning rate, n is the total number of support-set samples, |C| is the number of classes and S is the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In other words, we see that the label assignment imposed by δW and δb from the equation (7) effectively relies on computing a dot-product of fθ0(x) with “prototypes” ck := N −1 � (x,y)∈S fθ0(x)1y=k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 6 Experiments Most of our experiments were conducted using two standard benchmark problems using Omniglot and tieredImageNet datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The generated weights for each task θt are composed of four convolutional blocks and a single dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Each of the convolutional blocks consist of a 3 × 3 convolutional layer, batch norm layer, ReLU activation and a 2 × 2 max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For Omniglot we used 8 filters for convolutional layers and 20-dim FC layer to demonstrate how the network works on small problems, and for tieredImageNet we used 64 filters for convolutional and 40-dim for the FC layer2 to show that the method 2In contrast with cross-entropy, we do not need to have the head layer dimension to be equal to the number of predicted labels when using the Prototypical Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 7 works for large problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The models were trained in an episodic fashion, where the examples for each training iteration are sampled from a given distribution of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The reported accuracy was calculated from 1024 random episodic evaluations from a separate test distribution, with each episode run 16 times with different combinations of input samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For the HT architecture, we tried to replicate the setup used in the original paper as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We used a 4-layer convolutional network as a feature extractor and a 2-layer convolutional model for computing activation features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For Omniglot we used a 3-layer, 2-head Transformer and for tieredImageNet, we used a simplified 1-layer Transformer with 8 heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In all our experiments, we trained the network on a single GPU for 4M steps with SGD with an exponential LR decay over 100 000 steps with a decay rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We noticed some stability issues when increasing the number of tasks and had to decrease the learning rate to compensate: for Omniglot experiments, we used a learning rate 10−4 for up to 4 tasks and 5×10−5 for 5 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For tieredImageNet, we used the same learning rate of 5×10−6 for training with any number of tasks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We trained the CHT models with the class-incremental objective (6), but evaluated them for both task-incremental and class-incremental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='1 Learning from mini-batches We first consider a case where every task includes the same set of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Specifically, we compared the following three models using a set of four 5-way 1-shot support set batches S(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' , S(4) that consist of the same set of classes from tieredImageNet: θ(a) ≡ aψ(S(1) + S(2) + S(3) + S(4), θ0), θ(b) ≡ aψ(S(3) + S(4), aψ(S(1) + S(2), θ0)), θ(c) ≡ aψ(S(4), aψ(S(3), aψ(S(2), aψ(S(1), θ0)))), where + operation denotes a simple concatenation of different support set batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For this experiment, we used the cross-entropy loss (since the label set was the same for all S(i)) and each support set batch S(i) contained a single example per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We observed that the test accuracies for θ(a), θ(b) and θ(c) were equal to 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='9%, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='0% and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='3% respectively, all within the statistical error range (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' At the same time, HT trained with just S(1) or S(1) + S(2) (with 1 or 2 samples per class respectively) performed significantly worse, reaching the test accuracies of 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2% and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='9% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This demonstrates that the proposed mechanism of updating generated CNN weights using information from multiple support set batches can achieve performance comparable to processing all samples in a single pass with HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2 Learning from tasks within a single domain Next, we consider a scenario where the tasks consist of classes drawn from a single overall distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We present the results of two models: one trained on 20-way, 1-shot tasks with classes sampled from Omniglot dataset, and anther trained on 5-way, 5-shot tasks with classes sampled from tieredImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We compare the performance of CHT to two baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The first is a Constant ProtoNet (ConstPN), which represents a vanilla Prototypical Network, as described in Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In this approach, a universal fixed CNN network is trained on episodes from Ctrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This constant network can be applied to every task separately by projecting the support set as prototypes for that task and computing the prediction with respect to these prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Strictly speaking, this is not a continual learning method, since it treats every task independently and has no memory of previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For the best results on this baseline, we had to increase the number of classes by a factor of 5 during training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' for 20-way Omniglot evaluation we have trained it with 100-way problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The second baseline we used specifically for the class-incremental learning is a Merged HyperTrans- former (MergedHT), where we combine all the tasks and train a single original HT instance as a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This method does not solve a continual learning problem, since it has the information about all the tasks from the beginning, but it produces a solution for every class and task that we can still be compared to the weights generated by the CHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 8 T = 2 tasks T = 3 tasks T = 4 tasks T = 5 tasks Omniglot, 8 channels Accuracy 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='88 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 tieredImageNet, 64 channels 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='74 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Task Name θ0 θ1 θ2 θ3 θ4 ConstPN Figure 3: Task-incremental learning on Omniglot and tieredImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Each column represents a different CHT trained with a total of T = 2, 3, 4 or 5 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The tasks marked with a bullet symbol (•) correspond to the terms in the objective function (3) that are being minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The lines marked with the diamond symbol (⋄) show the extrapolation of the trained CHT to a larger number of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' T = 2 tasks T = 3 tasks T = 4 tasks T = 5 tasks Omniglot, 8 channels Accuracy 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='90 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 tieredImageNet, 64 channels 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='75 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 Task Name θ0 θ1 θ2 θ3 θ4 ConstPN MergedHT Figure 4: Class-incremental learning on Omniglot and tieredImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Each column represents a different CHT trained with a total of T = 2, 3, 4 or 5 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The tasks marked with a bullet symbol (•) correspond to the terms in the objective function (3) that are being minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The lines marked with the diamond symbol (⋄) show the extrapolation of the trained CHT to a larger number of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Each trained model is applied to both task-incremental (Figure 3) and class-incremental (Figure 4) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' To understand the effect of continual learning with multiple tasks, each column represents a separate run of the CHT trained on T = 2, 3, 4 or 5 tasks in total (for training a higher T, see the results in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' To demonstrate the recurrence of the method, we extended the number of tasks to 5 for the evaluation regardless of how many tasks it was trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Each plot shows 5 curves corresponding to the CHT, split into two groups: bullet marker (•) for tasks that the model was trained for and diamond marker (⋄) for extrapolation to more tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 9 Task-incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We start by analysing the task-incremental learning results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For the Om- niglot dataset, we saw no signs of catastrophic forgetting for the CHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In fact, we observed a positive backward knowledge transfer, where the performance on past tasks improved as more weights were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For example, in most cases, the performance of θ1 (green markers) was higher than θ0 (orange markers), and θ2 was higher than both θ1 and θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Additionally, as the number of tasks increased, the overall performance of the CHT also increased, with the model trained on T = 5 tasks performing better than the one trained on T = 2 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For the tieredImageNet dataset, the results were better than the ConstPN baseline, but the positive backward knowledge effect effect was not as pronounced as it was for the Omniglot dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The perfor- mance for every training task remained roughly the same for all generated weights, indicating that the model did not suffer from catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Overall, the CHT consistently outperformed the ConstPN baseline, particularly when applied to the same or lower number of tasks it was trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Although the accuracy of the CHT did decrease slightly when it was applied to more tasks than it was trained on, this decrease was not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In fact, even when CHT was trained on only T = 3 tasks, generating weights for one of two additional tasks still resulted in better performance than the ConstPN baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Class-incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In the class-incremental learning setting, the task name is given by two numbers indicating the range of tasks we used for evaluation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' task name 0-3 corresponds to four tasks from 0 to 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The black constant dashed line is the baseline performance of the ConstPN, which uses a fixed embedding and does not differentiate between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The starred blue markers represent a separate run of the HT for a particular configuration of merged tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' As one can see in the figure The accuracy of all the models decreased as more tasks were included in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This was expected because the size of the generated CNN did not change, but the number of classes that needs to be predicted was increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For Omniglot dataset we again saw the positive backwards transfer taking place, with CHT models trained on more tasks T performing better overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' For a given model trained on a fixed T, the performance was comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This demonstrates the preemptive property of the CHT, where models trained for a certain number of tasks can still be run for any smaller number of tasks with similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' When comparing the results to the baselines, the CHT had better results than the ConstPN up to the number of tasks T it was trained for, and the extrapolation results improved as T increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Interestingly, for the case of T = 5 the CHT was able to outperform even the MergedHT baseline for the Omniglot, even though the MergedHT had access to information about all tasks from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This suggests that having more classes to classify makes the learning problem difficult for the original HT, as the image embeddings may not be able to learn good embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This is particularly evident in the tieredImageNet dataset, where the performance of the MergedHT is so low that it falls below 60%, even for the 0-1 task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='3 Learning from tasks across multiple domains In the experiments described above, the support and query sets for each task were drawn from the same general distribution, and the image domain remained consistent across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' If the tasks were drawn from different distributions and different image domains, we would expect task-agnostic ConstPN approach to suffer in accuracy because it would need to find a universal representation that works well across all image domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In contrast, the CHT approach could adapt its sample representations differently for different detected image domains, leading to improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We verify this by creating a multi-domain episode generator that includes tasks from various image datasets: Omniglot, Caltech101, CaltechBirds2011, Cars196, OxfordFlowers102 and StanfordDogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We compared the accuracy of the ConstPN and CHT on this generator using episodes containing two tasks with 5-way, 1-shot problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The generated CNN model had 16 channels with 32 channels for the final layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Other parameters were the same as those used in the tieredImageNet experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The ConstPN achieved the accuracy of 53% for task 0, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8% for task 1 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8% for combined tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The CHT achieved the accuracy of 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2% for task 0, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2% for task 1 and 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8% for combined tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The accuracy 10 gap of nearly 3% between these two methods, which is larger than the gap observed in the Omniglot and tieredImageNet experiments, suggests that the CHT is better at adapting to a multi-domain task distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 7 Conclusions The proposed Continual HyperTransformer model has several attractive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' As an efficient few-shot learner, it can generate CNN weights on the fly with no training required, using only a small set of labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' As a continual learner, it is able to update the weights with information from new tasks by iteratively passing them through HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Empirically, we have shown that the learning occurs without catastrophic forgetting and may even result in positive backward transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' By modifying the loss function from cross-entropy to the prototype loss, we defined a learning procedure that optimizes the location of the prototypes of all the classes of every task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A single trained CHT model can be used in both task-incremental and class-incremental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 8 Acknowledgements The authors would like to thank Nolan Miller, Gus Kristiansen, Jascha Sohl-Dickstein and Johannes von Oswald for their valuable insights and feedback throughout the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' References Antreas Antoniou, Harrison Edwards, and Amos J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Storkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' How to train your MAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In 7th Interna- tional Conference on Learning Representations, 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Task name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8 Accuracy θ0 θ1 θ2 θ3 θ4 ConstPN Figure 5: The accuracy of the HT model trained for T = 5 using the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The ac- curacy of the first weight θ0 is high and is better than the accuracy of the ConstPN model’s em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, when more tasks are added, the accuracy drops dramatically due to collisions between the same classes for different tasks in the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Accuracy 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='88 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='9 Task name min eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (5): θ0 θ1 θ2 θ3 θ4 min eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' (6): θ0 θ1 θ2 θ3 θ4 Figure 6: CHT trained using task-incremental ob- jective (5) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' class-incremental objective (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A Additional experiments and figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='1 Learning with cross-entropy loss Figure 5 shows the results of an attempt to do learn multiple tasks using a HT with a cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Since the size of the last layer’s embedding is not increased, the model can only predict the class labels within the task and not the task themselves, which corresponds to the domain-incremental learning setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Additionally, the same class from different tasks are mapped to the same location in the embedding space, leading to collisions when more tasks are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This is why the accuracy drops significantly as the number of tasks increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' On the other hand, ConstPN model is more flexible because the prototypes for each task are computed from the support set of that task and do not have to be fixed to a one-hot vector as in the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='2 Training using task-incremental and class-incremental objectives Figure 6 compares the accuracy of two different models trained with task-incremental (using equation (5)) and class-incremental (using equation (6)) objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The performance of both models on task-incremental problems are similar, while the model trained with the class-incremental objective performs better on class- incremental problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='3 Analysis of prototypical embeddings using UMAP To better understand the quality of the learned prototypes, we conducted an experiment on a tieredIma- geNet 5-way, 5-shot problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We selected a random episode of 5 tasks and ran them through the model, producing weights θ0 to θ4 along with the prototypes for each class of every task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We then computed the logits of the query sets, which consisted of 20 samples per class for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The resulting 40-dim em- beddings for the prototypes and query sets were concatenated and projected onto a 2D space using UMAP (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Note that the prototypes from the earlier tasks remain unchanged for the later task, but their 2D UMAP projections are different, because UMAP is a non-parametric method and it must be re-run for every new θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' We tried our best to align the embedding using the Procrustes alignment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The plot shows that the embeddings of the tasks are well separated in the logits space, which helps explain why the model performs well for both task- and class-incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Normalizing the softmax over the classes within the same tasks or across all tasks made little difference when the tasks are so far away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' On the right of Figure 7, we show the projection of the ConstPN embedding of the same 14 θ0 θ1 θ2 θ3 θ4 ConstPN Figure 7: Left 5 plots: the UMAP projection of the CHT prototypes and the query set embeddings for different generated weights, where the points are colored based on the task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The query set points are connected with their corresponding prototypes using dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Right plot: UMAP projection of the ConstPN embedding for 25 different classes from tieredImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Embeddings are aligned using Procrustes alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' θ0 θ1 θ2 θ3 θ4 Figure 8: The UMAP projections of 20-dimensional embeddings of the prototypes and query set for different weights obtained from incremental HT training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The query set points are connected to their corresponding prototypes using dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In the top plot, the points are colored according to their class information, while in the bottom plot they are colored according to their task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Embeddings are aligned using Procrustes alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 25 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The ConstPN model does not make a distinction between tasks and treats each class separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The fact that 3 clusters emerge has to do purely with the semantics of the chosen clusters and the way the ConstPN model groups them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This also helps to explain why the CHT model performs better than the ConstPN, as it separates the tasks before separating the classes within each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' The UMAP embedding for the Omniglot dataset using ProtoNet (Figure 8) appears to be different from similar embedding projection of tieredImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' In particular, the embeddings from different tasks seem to overlap, while in the tieredImageNet embedding they are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' This may be due to the fact that the classes in the Omniglot dataset are more closely connected than those in the tieredImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Interestingly, despite the overlap between the classes from different tasks, the final accuracy is still high and only slightly degrades as more tasks are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 Learning with more tasks Our analysis primarily focused on the performance of the CHT on up to 5 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' However, as shown in Figure 9 the CHT model is capable of handling a much larger number of tasks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Similar to the results in 15 Omniglot, 8 channel Omniglot, 32 channel 0 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-10 0-11 0-12 0-13 0-14 0-15 0-16 0-17 0-18 0-19 Task name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8 Accuracy Number of tasks 10 Number of tasks 15 Number of tasks 20 0 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-10 Task name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='95 Accuracy Number of tasks 8 Number of tasks 9 Number of tasks 10 Number of tasks 11 Figure 9: Omniglot with 8 or 32 channels trained with a different number of tasks T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Task name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='870 Accuracy θ5 θ19 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0-10 0-11 0-12 0-13 0-14 0-15 0-16 0-17 0-18 0-19 Task name 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='8 Accuracy θ5 θ19 Figure 10: Omniglot with 8 channels trained for T = 20 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Here we show the final weight θ19 generated along with an intermediate θ5 for task-incremental (left) and class-incremental (right) learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' the main text, the nearly overlapping curves in the graph indicate that the model trained for T tasks can maintain the same level of accuracy when applied to a larger number of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' Figure 10 shows 8-channel Omniglot evaluated on task-incremental and class-incremental objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='5 Continual HyperTransformer vs MergedHT for tieredImageNet Figure 11 shows a zoomed out view of the results presented in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' It illustrates the significant difference in performance between the MergedHT and the CHT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 Additional figures for Omniglot task-incremental and class-incremental learning Figures 12, 13, 14 and 15 show additional experiments with the Omniglot dataset using different number of channels in the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content=' T = 2 tasks T = 3 tasks T = 4 tasks T = 5 tasks Accuracy tieredImageNet, 64 channels 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='7 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 0 0-1 0-2 0-3 0-4 Task Name θ0 θ1 θ2 θ3 θ4 ConstPN MergedHT Figure 11: Zoomed out view of Figure 4 so that the results of the MergedHT is visible.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='675 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='675 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='675 Class-incremental learning 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0 0-1 0-2 0-3 0-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQfjAo0/content/2301.04584v1.pdf'} +page_content='6 0 0-1 0-2 0-3 0-4 0.' metadata={'source': 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+1,2819 @@ +Page 1 of 37 + +Linguistic Analysis using Panini’s System of Sounds and Finite State +Machines +Shreekanth M Prabhu1 and Abhisek Midye2 +1 – Department of Computer Science and Engineering, CMR Institute of Technology, +Bengaluru +2 – Department of Information Science and Engineering, CMR Institute of Technology, +Bengaluru +Abstract +The study of spoken languages comprises phonology, morphology, and grammar. Analysis of +a language can be based on its syntax, semantics, and pragmatics. The languages can be +classified as root languages, inflectional languages, and stem languages. All these factors lead +to the formation of vocabulary which has commonality/similarity as well as distinct and subtle +differences across languages. In this paper, we make use of Panini’s system of sounds to +construct a phonetic map and then words are represented as state transitions on the phonetic +map. Each group of related words that cut across languages is represented by a m-language +(morphological language). Morphological Finite Automata (MFA) are defined that accept the +words belonging to a given m-language. This exercise can enable us to better understand the +inter-relationships between words in spoken languages in both language-agnostic and +language-cognizant manner. +Keywords: Panini’s system of sounds, State Machines, Finite Automata, Phonology, +Morphology, m-language, Comparative Linguistics +Biographical Notes +Dr. Shreekanth M Prabhu is currently working as a Professor and Head of the Department +of Computer Science and Engineering at CMR Institute of Technology, Bengaluru, India. His +research interests include Social Networks, E-Governance, and Comparative Linguistics. +Mr. Abhisek Midya is currently working as an Assistant Professor in the Department of +Information Science and Engineering. His research interest is in Theoretical Computer Science. + + + +Page 2 of 37 + +Introduction +Linguistics is a fascinating discipline going back millennia and has been a field for intense +scholarly pursuit in India. Particularly among them are contributions by Panini whose work on +the system of sounds and formal grammar has inspired significant advances worldwide. Then +there were generations of scholars enriching the field such as Kātyāyana, Patanjali, and +Bhartṛhari. In recent times pioneering work by Chomsky has been the hallmark of the +advances. According to Chomsky [1], the primary purpose of language is not communication, +rather it is cognition as language is the primary vehicle for thoughts. Chomsky [2] also +differentiated between I-language and E-language. Here I-language is a universal language that +applies to all spoken/human languages. E-language caters to specific natural languages +factoring in cultural and geographic aspects. Linguistics as a field comprises phonology which +deals with the sounds in spoken languages, morphology pertains to the construction of words, +and grammar which primarily describes the rules for the orderly usage of words to construct +sentences. Alternatively, the languages can be studied in terms of syntax which concerns +different parts of speech, semantics which deal with meaning, and pragmatics whose +preoccupation is with the usage of words that varies from milieu to milieu. +In the last few centuries, Comparative Linguistics has emerged as a fertile field for fervid +research. Here languages are compared for the similarity of words and then their structural +properties. Using that approach linguistic families are formed and even ancestral languages are +hypothesized yet times drawing far-reaching to far-fetched conclusions about the history of +populations and their movements. Not just languages but literary sources also can be +considered containers of words. +A lot of work related to comparing languages concerns itself with comparing words across +them. Comparing the words also may mean comparing root words, inflections, and derivations. +This generally calls for specialist know-how from the field of linguistics. In many cases, there +are disputes as different linguists draw different conclusions based on their own predilections. +In this paper, we take an alternate approach, where we primarily focus on morphology, and +how the words are constructed using a state machine approach. We look at the granularity of +word groups that can be related phonetically, semantically, or pragmatically. For each word +group, we propose a formal language and alphabet using Finite Automata that is useful to +decide if a given word belongs to that word group. The word groups can be extended and inter- +connected. Each m-language will have a core alphabet and an extended alphabet. We feel that +this approach can enrich the field of linguistics. We make use of Panini’s system of sounds and +construct a phonetic map that has the symbols which serve as states for representation as State +Machine. Further, we attempt to gauge the distance between words on the phonetic map and +look for insights. +The rest of the paper is as follows. Section 2, Linguistics Overview covers the literature in the +field of linguistics that is pertinent to our work. Section 3, Comparative Linguistics +Considerations explains the relevance of this paper to the field. Section 4, Analysis of words +using Panini’s using Sounds, where words are analyzed across languages and word groups are +identified. Section 5, Linguistic Analysis using Finite State Machines describes the +methodology we have proposed to arrive at unified Morphological Languages that cater to +given word groups. Section 6, Conclusions, concludes the paper. + + +Page 3 of 37 + +2. Linguistics Overview +Linguistics, by providing a structure to words and language, makes the task of understanding +language manageable. Otherwise understanding millions of words individually can prove to be +daunting and time-consuming. Without Linguistics, languages keep changing with time and +place and literature becomes incomprehensible in a matter of a century or two. +Linguistics as a field has its roots in ancient India. The Vedas are preserved for millennia by +oral transmission. To ensure accurate pronunciation, understanding, and appropriate usage of +Vedic Hymns in Yajna, the scholarly tradition mandates the study of six Vedāngas as a pre- +requisite and co-requisite for the study of Vedas. These six Vedāngas are Śiksha (phonetics, +phonology, and pronunciation), Chandas (prosody), Vyākarana (grammar and linguistic +analysis), Nirukta (etymology, explanation of words), Kalpa (ritual instructions), and Jyotish +(astronomy). Here the first four have laid the foundation for Indian Linguistics. The +expositions [3-6] give a very cogent explanation of ancient Indian Linguistics. In India +knowledge is maintained using a 4-fold mechanism that includes Sutra, Vārtika, Bhāshya, and +Kārika. Here Sutras are very compact, cryptic, and formulaic. Vārtikas are elaborations and +Bhāshyas are interpretations of Sutras. Kārika captures the essence. +There is a continuing tradition of grammarians in India and Panini’s Astādhyāyi superseded all +earlier traditions and core ideas from there spread to other languages and locales worldwide. +Astādhyāyi not only covers Vedic Sanskrit but also classical Sanskrit. Patanjali’s Bhāshya on +Panini’s grammar is the most popular. The tradition has continued for centuries with newer +Bhāshyas. Because of such rigorous discipline, the Vedas were transmitted without any +corruption for millennia. This also benefitted Classical Sanskrit as even the works such as +Ramāyana which are a few thousand years old are still intelligible to modern scholars. +Otherwise, it is common that in the case of most languages, the works done just a few centuries +ago are hard to understand for modern speakers of the language. +Generally, linguistics can be approached from the viewpoint of words (Śabda) or sentences +(Vākya). Whichever way you approach it both Śabda and Vākya are inextricably linked. The +only purposeful way of using Śabda is in the form of Vākya. The only way to decipher and +understand Vākya is by breaking it down into Śabdas. Vyakarana thus is called Shabda Śastra. +Panini’s Astādhyāyi analyses sentences, identifies words and then components, and arrives at +Dhātus (roots of words). Each word is viewed as consisting of Prakriti (the original part) and +Pratyaya (suffixes). By combining Prakriti and Pratyaya, the Padas (usable words) are formed. +With a good discipline of grammar using a single Dhātu typically 360 words can be formed. +There are at least 2000 Dhātus, resulting in lakhs of words. This framework enables Sanskrit +to be a powerful language where new words can be easily composed using components and +they become conveniently intelligible to those conversant with the language. When it comes to +the right use of words, it can be done only with meaning in mind. +Three things are critical to interpreting the meaning of individual words in a sentence in order +to arrive at the intended meaning of the sentence: Ākānkshā (expectancy), Yogyata(suitability) +and Sannidhi(proximity). According to Vedic tradition, the six objectives of precise grammar +are Rakshā (prevention from distortion), Asandeha (absence of ambiguity), Ūhā (modification +of Vedic Mantras due to the possibility of more than one interpretation, Āgama (ease of +augmentation) and Laghuh (easy means of acquiring knowledge). + +Page 4 of 37 + + +Modern linguistics like ancient linguistics comprises phonology (the science of sounds), +morphology (word formation using sounds), and grammar (deriving new words and +constructing sentences). Analysing the sentences, thus consists of syntax analysis, semantic +analysis, and pragmatics. The methodology for the analysis of natural language can be +compared with the approach taken by the compiler to analyse programming languages. A +compilation process consists of a scanning phase where a statement is broken into components +(lexemes) and then in the parsing phase, a syntax tree is constructed comprising of lexemes +and validated for grammatical correctness. Even though natural language processing is similar, +the grammar is not context-free and morphology (the constructions of words) itself makes use +of grammar in addition to the construction and analysis of sentences. However, some key +constructs such as finite automata and the concept of language from theoretical computer +science can be leveraged. That is the endeavour of this paper. +3. Comparative Linguistics Considerations +The relationship between languages did not get the attention of scholars in Europe as according +to Biblical tradition, Hebrew was considered the universal language which then broke into +other languages. In India, Sanskrit was considered the mother of all languages while scholars +were very much aware of Sanskrit words and words native to a given language. In Europe, as +acknowledged by Mallory [7], James Parsons [8] was probably one of the first to do a +systematic study of thousands of common words across European Languages. However, +according to Mallory [7], a century prior to that it was Joseph Scaliger who attempted to divide +the languages of Europe into four major groups, each labelled after their word for god. The +transparent relationship of what we today call the Romance languages was recognized in the +‘Deus’ group (for example, Latin ‘Deus’, Italian ‘Dio’, Spanish ‘Dio’, French ‘Dieu’), and +contrasted with the Germanic ‘Gott’ (English God, Dutch God, Swedish ‘Gudy’ and so on); +Greek ‘Theos’; and Slavic Bog (such as Russian ‘Bog’, Polish ‘Bog’ and Czech ‘Buh’). This +exercise of comparing languages was also undertaken by visitors to India in the 15th century. +In India, it was Filippo Sassetti and Thomas Stephens were the first two who noticed the +similarity between Indian and European Languages. Singh B [9] identifies Thomas Stephens +as the first Englishman in India. Pedro Redondo [10] explains that the motivation of Sassetti +was that of the humanist whereas that of Stephens was evangelical and theological. All these +exercises and the well-known discourse of William Jones [11] culminated in the proposal of +not only the Indo-European Family of Languages but also the acceptance of the language +family as a universal construct. +According to modern Linguistics, certain words are considered isolates i.e. they are unique to +that language or a narrow set of languages. The isoglosses cause dialectical variations. These +differences may be phonological, lexical(different words), or different linguistic features. +Cognates sound similar across languages carrying the same/related meaning. The cognates are +classified as adstrate words when these are loan words due to trade and migration. Then there +are substrate words where it is presumed that speakers of one language had dominance over +the speakers of other languages resulting in an asymmetric transfer of words. In contrast, in +Indian tradition, the words in a language are divided into three categories: Tatsama(same as +words in another language generally Sanskrit), Tadbhava(derived from word in another +language), and Deshya(native words). + +Page 5 of 37 + +Initially Sanskrit was considered the mother of the Indo-European Languages as it had cognates +across Indo-European Languages and the most complete grammar with eight cases as well as +duals in addition to singular and plurals. But then scholars who are generally known as +Indologists who call themselves mainstream changed their stance. Bryant{12] puts forward the +‘main-stream’ view that (i) There has to be a proto-language probably spoken by all speakers +before that broke into Indo-European (IE) Languages; (ii) All the IE speakers stayed in a +common homeland before they separated; (iii) The proto-language could not have been +Sanskrit; (iv) There was Proto-Indo-European(PIE) Language that broke into Celtic, Germanic, +Romance, Baltic, Slavic, Greek, and Indo-Iranian families with PIE at the root. Thus Sanskrit +was relegated as a leaf node within the Indo-Iranian family and India as yet another output of +IE speakers.. +Bryant explains how Sanskrit was dethroned using linguistic arguments. One of the reasons +given by Linguists to propose PIE is that Sanskrit has innovated a,e, and o sounds to a sound. +Greek has retained the original sounds. A typical example given is bhend in Greek becomes +bandh in Sanskrit. Another example the scholars give is Greek Deca (for number 10) is not +derivable from Sanskrit Daśa, hence there needs to be a common ancestral language to both. +The languages are further classified as Kentum and Satem languages based on the word for the +number 100 and here Kentum Languages are considered more archaic. Sanskrit is considered +Satem Language and ruled out as an archaic language. Further, since Sanskrit had retroflexes, +which many European languages did not have, some linguists say it can not be a proto- +language. To support their hypothesis scholars claimed that Sanskrit borrowed cerebralization +from Dravidian Languages and any word in Sanskrit that is not in common with European +Languages is a loan from Dravidian or Munda languages. This is in contrast to Indian tradition +where Sanskrit words appear either as Tatsama or Tadbhava forms across languages and +seldom other way around. As an example, the word for water is Neer only in Sanskrit and +Dravidian Languages but not in most Indo-Aryan Languages. So one may conclude that the +word was loan into Sanskrit. But any such conclusion may be hasty as Greek also uses neró for +water, which is likely from Sanskrit. +Bryant and Patton[13] examine the issue of Indo-Eurpean origins from multiple perspectives +in an edited volume. Among the linguists who contributed to that endeavour, Mishra[14] claims +that Sanskrit is more archaic than all others. The main features where Sanskrit is shown to +deviate from Indo-European is the merger of IE a, e, o into a in Sanskrit and the change of +palatal k etc. to palatal s etc. in Sanskrit. Mishra counters this and among many other arguments +gives the example of Gypsy language where Indo-Aryan a remains. a in Asiatic Gypsy but +becomes a, e, o in European Gypsy. This confirms that original IE a was the same as Sanskrit +a and remained a in the Indo-Iranian languages, but changed to a, e, o in their sister languages. +Then he gives the case where Sanskrit retains both Vākya and Vāchya. According to Mishra, ś +becomes k before it becomes s in Sanskrit. He maintains that ś and k are allophonic. Thus, the +k which was allophonic to ś in Sanskrit might have been generalized in the Centum languages. +He also gives examples of Lithuanian a Satem Language sporadically presenting k sound. +Witzell[15] continues to champion the mainstream view that Aryans are outsiders to India and +Vedic langauage is an import into India and he is a strong proponent of import of Munda words +into Vedic Sanskrit, whereas Kuiper[16] considered many Sanskrit words were of Dravidian +origin + +Page 6 of 37 + +The worldview of Europeans is guided by the prism of conflict, conquest, co-location, and +commerce. India was also subject to conquests from the 7th century AD onwards which +targeted Indian civilization with religious conversions and political conquests. However, the +essential characteristics of the civilization that survived have been convergence, confluence, +continuity, and contiguity aided by amalgamation, and assimilation. Thus, India has a +continuing civilization going back millennia and a sense of unity that stems from identification +with the larger sacred geography unified by common traditions, scriptures, belief systems, holy +places, and value systems. Diana Eck[17] rightly observes that India is a country united by the +footsteps of pilgrims. The migrations of people within India have been continuous and in +particular priestly classes have migrated across India and have maintained essential unity of +traditions. Many southern kings also have northern lineages. Such movements have resulted in +far greater homogenization of languages across India. The languages which were neighbours +to the Sarasvati River region such as Konkani and Punjabi are inflectional like Vedic Sanskrit. +The South Indian Languages tend to have more agglutination of consonants and less +conjunction of consonants. However, subject-object-verb order is common across all Indian +Languages. +Further, the larger geography which included Afghanistan and Central Asia was considered +contiguous to India with cultural transmission and exchange. The Central Asian Republics +continue to use Sthan as part of their names (Kazakistan, Tajikistan) showing the influence of +Sanskrit on them. Greater India thus consisted of Uttara Kuru as well as Uttara Madra regions. +Another point to be considered is the Sinhala language of Sri Lanka located to the south of +Dravida region is Indo-Aryan with commonality with Vedic Sanskrit retaining a few rather +archaic words. +Sanskrit for most of the time served as the lingua franca across India thus serving as the donor. +language of words that represented abstract concepts on one hand to mundane reality on the +other. In Sanskrit, refined and accurate pronounciation was not only important for rituals but +also considered a hallmark of the civilized. Generaly Apabramsha(mispronounced) forms of +Sanskrits word which is easier to pronounce were used by the commoners. Thus Śrāvan word +for the rainy season may change to Sāvan in Hindi. We notice that some languages(Kannada, +Konkani, Bengali) retain the original. The word for cotton Karpasa is considered to have +derived from Kāpas a Munda word. But other Indian Languages(Konkani, Marathi and +Gujarathi) use Kāpas only. Some argue that Kāpas is Apabramsha for Karpasa and not +necessarily a loan word from Munda. In India, the direction of changes is from Sanskrit to +Prākrat to vernaculars as India had tradition of Chandas(language for prosody) and +Bhasha(language for common use) concurrently evolving. This runs counter to the linguists’ +view where they expect the transformation to happen from simple/primitive to refined. +In addition, different regions of India and languages there have shown a preference for certain +sounds and a lack of preference for others. Thus retroflex sound ṇ is not in vogue in Hindi, but +very much there in Konkani, Marathi, and Punjabi. Bengali uses o instead of a and ‘b’ sound +instead of ‘v’, in certain cases. In Bihar, ‘s’ sound is used more than the ‘ś’ sound. On the other +extreme, Iranian languages have replaced ‘s’ with ‘h’. In many cases Sanskrit has more than +one sound, say for people Jana is used as well as Gaṇa is used. The same is true with Dik and +Disha both words are used for direction in Sanskrit but for different cases. Further, Sanskrit +uses a word starting with K for Kendra (center) which very few European +Languages(Greek,Armenian), use, and most use centrum which starts with the ‘s’ sound. + +Page 7 of 37 + +Thus, analysis of European Linguists using their worldview and rules may need revisiting using +a formal approach that can address voluminous vocabulary across languages. In particular, +Sanskrit commonly has more than ten words to represent the same entity or concept. European +Languages are generally compared only with Sanskrit, but not as much with other Indian +Languages. It is also worth comparing the phenomena that Indian Language words underwent +as they carried forward Sanskrit words and comparing the same with what could have happened +to Sanskrit words which are borrowed by/found in common with European Languages. Dr. +Gintaras Songaila [18] elaborates on enormous affinities which are directly there between Indo- +Aryan and Lithuanian without any connection with the Iranian language. Subhash Kak [19] +also makes a long list of common words among European languages; and Sanskrit. Both +scholars emphasize the contiguity of central Asia with India from ancient times. The borrowing +of words also spans disciplines, ‘Astipathi’ in Sanskrit becomes osteopathy and ‘Jara’ the word +for old age in Sanskrit leads to geriatrics. Same is true with common medical word sputum +which has natural association with Sphut, Sanskrit word than spit, an English/Latin verb.The +word pa(a)th is due to path in Sanskrit(as used in RajPath i.e. King’s Road) leading to words +such as allopathy and homeopathy. Hence the transmission of words has continued for +centuries and millennia. +Also, there are few studies that compare Dravidian Languages with other Indian languages. A +study by Swaminath Aiyar[20] is a rare exception. Aiyar after a very unique and highly detailed +comparative study of langauges says “My views differ from those of all previous scholars +because they contended themselves with comparing Dravidian Languages with Classical +Sanskrit and naturally saw no deep-seated affinities. When one language is extensively affected +by another, we need to look for the source of influence not in the artificial language of high +literature but in the spoken idioms of common people. It is necessary to compare Dravidian +idioms with the Vedic Dialects and the Prākrats of pre-Christian Centuries, before we can +decide the question of Aryo-Dravidian affinities”. It was Bishop Caldwell who compared +Classical Sanskrit and Dravidian Languages and pronounced the differences. At the same time +there were other scholars such as Pope, who also was a missionary did not agree. He felt the +decision to consider Dravidian Languages as disjoint from Aryan Languages was rather abrupt. +He expressed the opinion “(i) that between the languages of Southern India and those of the +Aryan family there are many deeply seated and radical affinities and (ii) that the differences +between the Dravidian Tongues and Aryan are not so great as between the Celtic (for instance) +languages and the Sanskrit; and (iii) that by consequence the doctrine that the place of +Dravidian dialects is rather with the Aryan than with Turanian families is still capable of +defence”. He illustrated these positions by means of copious illustrations and pointed out that +“the resemblances appeared in the most uncultivated Dravidian dialects’ and that “the identity +was most striking in the names of instruments, places, and acts connected with a simple life”. +He promised to follow on with a paper that looked at derivative words and show that the +prefixes and affixes were Aryan. The work of Aiyar thus fills that gap. +In summary, dethroning of Sanskrit as a proto-language needs to be revisited. In the least, +confining Sanskrit as a daughter language under the Indo-Iranian branch is a travesty. Further, +the inter-relationship between Dravidian Languages and Indo-Aryan Languages needs many +more studies. + +Page 8 of 37 + +4. Analysis of words using Panini’s System of Sounds +In this section, we introduce the concept of m-alphabet which is the set of phonemes used to +construct a word. The core m-alphabet is the set of sounds that pertain to the original part +(Prakriti) of the word, that too where the chosen sounds are common cutting across languages +or that pertain to the suspected original word. The m-languages consist of words belonging to +a word group there are related phonetically, semantically, grammatically, and ontologically. +The word groups across different languages are compared and analyzed using these +morphology- based constructs. We make use of Panini’s System of Sounds which represents +natural language sounds comprehensively in a scientific manner. +4.1 Panini’s System of Sounds +Panini developed the system of human/natural language sounds after a careful study of how +they are generated by the vocal box. Panini’s Śiksha (phonology) explains the form of each +Varṇa ((letter/sound) is determined by Svara (intonation), Kāla (time taken to pronounce it), +Sthana (place of articulation), and Karaṇa. Abhyantara Prayatna (effort within the oral cavity) +and Bāhya Prayatna (effort outside the oral cavity) are two additional factors. Figure 1, +illustrates Panini’s System of Sounds. + + + +Figure 1: Panini’s System of Sounds +Sounds that do not face any obstruction when we speak are termed vowels. These may vary +depending on whether they are short, long or very long. In his scheme there are 13 vowels and +two additional vowels which can be used only in conjunction with other sounds namely am and +ah. The sounds that face obstruction are termed consonants. He classifies them based on place +of articulation. The guttural/velar/Kanṭavya sounds are produced in the throat. Next, palatal/ +Tālavya Sounds are generated by touching one’s tongue to the pallet. Next set of sounds are +Cerebral/Murdhya sounds. They are also called hard palatal sounds or retroflex sounds as it +requires one to reverse the direction of the tongue while generating them. The fourth set of +consonants are dental/Dantavya. They are generated by touching the tongue to the teeth. Fifth +set of consonants are labial/Austa. Here the lips are involved in generating the sounds. Each + +Vouh +h (anundn) b(turg) +Coasonaths +Gutturaler +kha +Te +h +Palatalsr: +a +tp +Certbeils +Denals: +Ttha +Labialar +Hm +7 +Ocmnd +Senivovcker +Shlnte +Apinate +OuaPage 9 of 37 + +of these group of 5 consonants can be further classified – (i) unvoiced and unaspirated/tenuis +ii) aspirated, (iii) voiced (iv) voiced and aspirated and (v) nasal. +Then there are other consonants which are called semivowels, sibilants and aspirates. Figure 1 +below illustrates Panini’s System of Sounds. Rajesh Kumar [21] and Anuradha Chaudhari [22] +explain Panini’s system of sounds covering modern linguistics and traditional Indian +vocabulary. +Whereas Panini’s System of Sounds is very comprehensive and representative, there are sounds +that are not represented specifically. Vedic Sanskrit and many Indian Languages have a +cerebral ḷ sound which is at times used in lieu of the ḍ sound as in Iḍa, and Iḷa. This is not +represented above. +Alveolar sounds are intermediate sounds typically used when English say “Tea”, “Table” or +“Tennis”. They are not fully dental. A person who is a native speaker of a language that has +retroflex sounds; may treat them as such. Then there are additional alveolar sounds in Tamil +which are not there in North Indian Languages. In fact, Tamil and probably other Dravidian +Languages early on had far too limited an alphabet or far fewer phonemes. Tamil continues to +have a limited alphabet consisting of vowels: a, ā, i, ī , u, ū., e, ai, o, ō, au, with the omission +of r, rr, lr. The consonants are k, nasal (k), c, nasal(c), t, n, ṭ, ṇ p, m, y, r, l,v, l,l,r,n. The last +four are alveolar sounds and unknown to Sanskrit Alphabet. In each class of consonants, +instead of 5 members, only tenuis (the first), and nasal (the last) sounds are there. +Generally, European Languages do not use cerebral/retroflex sounds, except in a few North +European Languages such as Swedish. Some languages such as French use only dental sounds. +The Tamil Language also has far fewer sounds and the script uses the same symbol for four +consonants of the same category. +Further, there are a total of nine fricative consonants in English: /f, θ, s, ∫, v, ð, z, З, h/, and +eight of them (all except for/h/) are produced by partially obstructing the airflow through the +oral cavity. These are: /f/: far, /v/: save, of, /θ/: think, /ð/: those, /s/: sir, race, /z/: zoo, rise, +/ʃ/: sharp, chef, pressure, sugar, motion, /h/: ahead. +4.2 Analyzing Words using Sounds +In this section, we build a word bank cutting across languages. Table 1 indicates the encoding +we have used for the languages. +Table 1: Encoding to indicate the language of the word +European Languages +Indian Languages +English(En), German(Ge), Russin(Ru), Greek(Gr), +Romanian(Ro), Latin(La), Latvian(Latv), French(Fr), +Lithuanian(Li), Italian(It), Welsh(We), Danish(Da), +Dutch(Du), Spanish(Sp), Polish(Po), +Portuguese(Por), Bulgarian(Bu), Corsican(Co), +Croatian(Cr), Ukraian(Uk), Scot Galeic(SG), +Irish(Ir), Slovak(Sl) +Sanskrit(Sa), Prakrat(Pr), Hindi(Hi), Marathi(Ma), +Punjabi(Pu), Konkani(Ko), Bengali(Be), +Gujarati,Kannada(Ka), Tamil(Ta), Telugu(Te), +Malayalam(Mal), Sinhala(Si) + + +Page 10 of 37 + +Subhash Kak did a study of words derived from Sanskrit in European Languages. Table 2 +below lists Sanskrit words, and corresponding cognates in European Languages. We have also +added a word in Kannada and Konkani for water. +Here we also list basic sounds used in Sanskrit word which we call as m-alphabet +(Morphological Alphabet). This is followed by an extended alphabet to represent all words, +sounds gained, replaced, and lost. Also listed are related words. All words in a given row can +be considered to constitute an m-language (Morphological Language). + + +Table 2: Sanskrit Words and Cognates in European Languages +Sanskrit +Word +m-alphabet +(Sanskrit) +Word +(Language) +m-alphabet +(Extended) +Sounds +Gained +Sounds +Changed +Sounds +Lost +Related +words +āvāsa +a,ā,v,s +house(En) +haus(Ge) +a,ā,v,s,h,o,u +h, o, u +v to u + + +dam +d,a,m +dom(Ru) +damus(La) +d,a,m,o,u,s,h +o, u, s ,h +a to o + +domicile, +domestic +grha +g,r,h,a +casa(La) +cass(Sp) +g,r,h,a,k,s +k, s +g to k, s to +h +r + +vāri +v,r,ā,i +water(Du) +v,r,ā,i,t +t + +i + +udaka +u,d,k,a +uda(Ko) +voda(Ru) +u,d,k,a,v +v +u to v + + +āp +a,p +apa(Ro) +a,p, + + + + +nīr +n,r, ī +nero(Gr) +dur(We) +neeru(Ka) +n,r, ī, d, u +d, u, o +n to d + + +dhara +dh,r,ā +terra(It) +dal(We) +dh,r,ā,d,l,t,e +d, l, t, e +dh to t, dh +to d + + +nabha +n,a,bh +nebo(Ru,Cr) +nebe(Cz) +n,a,bh,b,e,o +b, e, o +bh,a + + +Varun +v, r, ṇ, u,a +ouranos(Gr) +v, r, ṇ,u,a,o +o +v + + +yuva +y,u,v,a +youth(En) +jeunesse(Fr) +y, u, v ,a, +t, h + + +Juvenile +Thus, the formation of cognate words may involve sound shifts, and closely related sounds +(voiced versus voiceless, aspirated versus unaspirated, changes of vowels) as well as changes +to grammar (gender-related or other changes) or due to any other peculiarities of receiving +languages. Thus, we can define a grammar which can cater to such scenarios which can +determine if a word belongs to a word group or not. Additionally, we may be able to generate +candidate words which can prospectively belong to the same word group. +The words from Vedic Sanskrit have gone through variety of transformations in Indian +Languages. This is accepted by all. Now our hypothesis is that the transformation of those + +Page 11 of 37 + +words in European Languages can also be considered the manifestations of the same +phenomena that happened as the words got carried over to European Languages. For example, +Graha in Sanskrit becomes Kar in spoken Punjabi but in Hindi it remains as Ghar. The same +word becomes Casa in Latin, presenting k sound as initial sound. +Tables 3 to 7 illustrate the concept of m-alphabet and m-language with additional examples +which we have collected. Note that this is based on Google Translate output and our own +knowledge which may have missed certain synonyms that are cognate. Annexure 1[23] has a +bank of Indian and European words, where we have enumerated. nearly two hundred groups +of words for which m-languages can be defined. +Table 3: m-language for word group “Being in the middle” +Theme +Being in the middle, in between +m-language +madhya (Sa), mādhyam(Sa), middle, medium, mediate, media , midten(Da), +midden(Du), madhala(Ma), madhyama(Ka), milieu(Fr), mezzo(It), mitte(Ge), +meio(Po), mijloc(Ro), maeda(Si), meadhan(SG), mesaio(Gr)} +Non-members +natuttara(Ta), lar(Ir), vidu(latv), vidurio(Li),sredina(Ru) +m-alphabet(core) +{m, d, y, a,i} +m-alphabet(Extended) +{m, d,y, a, I, t, n, l, c, z} +Remarks +Sanskrit, Indic, Germanic, Greek and Romance language and Scot Gaelic, use the +above m-alphabet. +Extended Vocabulary +mezzanine floor, meso (between micro and marco) +Table 4: m-language for word group “Face, Mouth” + Theme + Face, Mouth +m-language +mukh(Sat), moga (Ka)}, muh(Hi)}, mouth, mukhya(Sa:Main), mund(Da), +mond(Du), mute(Latv), tond(Ko) +Non-members +Face, Chehera(Hindi), beul(Irish), Bayi(Kannada) Usta(Slovenian) +m-alphabet(core) +{m, u, kh,o,g, t, n, h,d} +m-alphabet +(Extended) + {m u, k, kh, h, o, g, y, d, n, t} +Remarks +Face and mouth words get overlapped. Tond may belong to another m-language with +Sanskrit Connection, Tunda – trunk. Germanic and Sanskrit languages have +commonality. + + + + + + +Page 12 of 37 + +Table 5: m-language for word group “Long. Tall” +Theme +Long, Tall +m-language +long, lamba(Hi), lāmb(Ma), labi(Gu), long(Fr), lang(Sw) +Non-members +dugo – Baltic and Slavic languages use words cognate with deergha. fada(Irish), +makrys(Greek) +m-alphabet(core) +l, n, m, b, g, a, o, i +m-alphabet +(Extended) + NA +Remarks +Here Indian Languages have direct cognates with European Languages. Sanskrit +tends to use Deergh. However Sanskrit word vilamb(delay) indicates Sanskrit origin +of the above words. +Table 6: m-language for word group “High” +Theme +High +m-language +unc(Hi), ucca(Sa), ucca(Be) hoch(Ge), hoog(Du) hog(Sw), Haut(Fr) +Non-members + Uyar(Ta) +m-alphabet(core) +{u, c } +m-alphabet +(Extended) + {u, n, c, t, g, a, u, e} +Table 7: m-language for word group “Below, Low, Lowly” +Theme +Lowly/below +m-language +Lowly:nīc(Sa), Below: nīce(Hi), nizhe(Ru) nizsie(Sl) +Non-members +Many +m-alphabet(core) +n, c +m-alphabet +(Extended) + n, c, ī, e, zh, s +Next, we analyze the Dravidian Language words using sounds. In Table 8 below, we analyse +how the words for numbers are constructed in Dravidian Languages. There are sound shifts +from pa to ha (Pattu and Hattu) in Kannada. The ‘b’, ‘p’ and ‘v’ sounds also seem to be used +interchangeably. Malayalam and in some cases, Tamil manage without a suffix ‘u’, whereas +others customarily use it. + + + + +Page 13 of 37 + +Table 8: Words for numbers in Dravidian Languages +Number +Kannada +Tulu +Telugu +Tamil +Malayal +am +m-alphabet +(Extended) +m- +alphabet +(core) +One +ondu +onji +okati +onru +onn +o,n,d,u,j,I,k,a,t,r +o,n +Two +eraḍu +radd +ranḍu +irand +rand +e,r,a,d,u,n,i +r,a,d +Three +mooru +mooji +muḍu +munr +munn +m, ū,r,u,j,I,d,r +m, ū +Four +nālku +nāl +nālugu +nānku +nal +n,ā,l,k,u,g,n +n,ā,l +Five +aidu +ain +aidu +aintu +anj +ai,d,u,n,t,a,j +ai,n +Six +āru +āji +aru +āru +ār +ā,r,u,j,i +ā,r +Seven +elu +el +edu +elu +el +e,l,u,d +e,l +Eight +entu +edma +enimidi +ettu +ett +e,n,t,u,d,m,ā,I,d +e,t +Nine +ombattu +ormbā +tommidi +onpatu +ompat +o,m,b,a,t,u,r,ā,d, +n,p +o,m,t +ten +hattu +patt +padi +pattu +patt +h,a,t,u,p,d +p,a,t +twenty +ippattu +irva +irvai +irupat +irupat +I,p,a,t,u,r,v,i +I,r,v,p,a,t +thirty +muvattu +muppa +muppai +muppat +u +muppat +m,u,v,a,t,p +m,,u,p,a,t +fourty +naluvattu +nālpa +nalabhai +narpatu +nalpat +n,ā,l,u,v,a,t,u,p,b +h,r +n,ā,l,p +fifty +aivattu +aiva +yabhai +aimpat +u +ampat +ai,v,a,t,u +ai, v, p, + Phonemic +Affinity +u, v, d +j,ā +d,bh +n, r + +a, n, m + + + +Excluded +Phonemes + + + +v +v + + +Next, we look at the study of inter-relationships between Indo-Aryan and Dravidian Languages +done by Swaminath Aiyar [20]. The Drāvidian Languages were historically divided into +Andhra Group with Telugu and a set of languages and the Drāvida group consisting of Tamil, +Kannada, Malayaḷam and Tuḷu . Andhra Group is independently influenced by neighbouring +Prākrats as well as greater propensity to use Sanskrit words. Aiyar’s main conclusion is that in +addition to a large number of clearly Sanskrit (Tatsama) words in the Drāvidian Languages, +there are a significant number of Tadbhava words that are derived from Sanskrit. He claims +that when Caldwell came up with the hypothesis that Dravidian Languages have a low affinity + +Page 14 of 37 + +for other Indian Languages, he compared words from Classical Sanskrit which indeed were +different for the sample he had chosen. Aiyar invalidates Caldwell’s conclusions by comparing +South Indian Language words with other Sanskrit words which are closer to Vedic Sanskrit, +Prākrats and other Indian Languages. Table 9 contrasts Caldwell’s approach with that of +Aiyar’s. + +Table 9 Comparison of Sanskrit and Tamil Words +Sr,No. +English Word +Sanskrit Word +(Caldwell) +Tamil, Telugu, +Kannada, +Malayalam +Proposed +Word (Aiyar) +Remarks +1 +hair +kesha +Mayir(Ta) +Śmashru(Sa) + +2 +mouth +mukha +Vay(Ta) +Vac(Sa) +Vac is alternate word +from Vedic Sanskrit +2(a) +nose + +Mūkku(Ta), +Mūgu(K), +Mukku(Te) + +Words derived from +Mukha are used for face +and mouth. Here it is +proposed to be used for +nose as well +3 +ear +karna +Shevi(Ta) +Śrava(Sa), +shravika(Sa) + +4 +hear +sru +Kel(Ta) +Karna(Sa) + +5 +eat +bhaks +Tin(Ta) +Trṇu(Sa), +Tr(Sa), + +6 +walk +car, cel +Egu(Ta) +Ya(Sa), i(Sa) + +7 +night +nak +Ira, Iravu +Rātri(Sa) + +8 +mother +matr +Āyi(Ta) +Yāyi(Paisc.) + +9 +tiger +vyaghra +Puli(Ta) +Vengai(Tamil) + +10 +deer, beast +mrga +Marai, Man, +Ma(Ta) +Mrga(S), +Maga(Pr_ + +11 +Fire +Agni +Ti(ta) +Tejas(Sa), +Tij(Sa) + +12 +Snake +Sarpa +Pāmbu.(Ta), +Aravu (Ta), +Arava(Ma) +Prasarpa, +Sarpa, Sarpaks + + +13 +Village +grama +Ūr(Ta), Ūru(Ka) +Pura(Sa) + +14 +buffalo +mahiSa +Erumai(Ta), +Emme(Ka) + +Heramba(ka) +Associated words are +swapped +14(a) + + +M āDu(Ta) +MahiSa(Sa) +15 +horse +ashva +Kuthirai(Ta) +Ashvatara(ka) + +16 +hill +parvata +Malai(Ta) +Paruppu(Tam) +Matching Associations +found +According to Swaminath Aiyar, a large number of Dravidian words, in particular in Tamil that +appears to have no affinity with Sanskrit, in fact, are Tadbhava words from Sanskrit. As Tamil +has a highly constrained Alphabet, they went through a lot more transformation and corruption +compared to North Indian Vernaculars and appear unrelated. To get the whole picture one +needs to look at a plurality of Sanskrit words and Prākrat words and inter-relationships between +Dravidian Languages, as the closest word could belong to Telugu or Tamil in most cases and +then further transformed in modern Kannada and Malayalam. Table 10, contains a sample of +words analyzed by Aiyar and inferred as Sanskrit words. Aiyar derives Dravidian words from + +Page 15 of 37 + +Sanskrit/Prākrat words with a variety of rules such as sound elision, sound substitution and +suffix additions. +Table 10: Tadbhava Dravidian Words which are derived from Sanskrit +Sr. No +Sanskrit +Word +Meaning +Tamil/Dravidian Word/Other +Indian Language +Meaning +1 +Paksha +Wing, Side +Pakka(Ta) +Side +2 +See +Pashya +Paar(Ta), Paḷe(Ko) +See +3 +Dakshina +South +Tenkaṇa(Ta) +South +4 +Bhru +Brow +Pubbu(Ta), Hubbu(Ka) +Eyebrow +5 +Satya +Truth +Sari(Ka), Sahi(Hi) +Correct +6 +Vayalah +Bangle +Baḷe(Ka), Vaḷai(Ta) +Bangle +7 +Lokah +People, Word +Olaku(Ta) +People, World +8 +Mridu +Soft +Mella(Ka) +Slowly, Gently +9 +Mrda +Mud +Maṇṇu(Ka),Maṇṇ (Ta) +Soil, Earth +10 +Dhvani +Voice, Sound +Toni(Ta) +Sound +11 +Vandyah +Barren Woman +Banje(Ka), Vandi(Ta) +Barren woman +12 +Shabdah +Word +Sadd(Pu), Saddu(Ka) +Sound +13 +kāṣṭakah +Wood +Koṭṭai(Ta), Kaṭṭige(Ka) +Wood (Collected from Forest) +14 +Mrtya +Perishable +(Body) +Mai(Ka) +Body +15 +Svithra +Silver/White +Velli(Ta), Belli(Ka), +Belagu(Ka). Belaku +Silver, White,Light +16 +Sreṇi +Line +Eṇi(Ka) +Ladder +17 +Chayah +Hand +Kai(Ka, Ta) +Hand +18 +Śirah +Head +Sir(Hi), Tale(Ka), Tare(Tu) +Head +19 +Kārṣapaṇa +Coin or weight +Kāṇam(Ta) Kāhavaṇo(Pr) +Kāhāṇ(Or) + +20 +Meṣa +Sheep/Goat +Meḍam(Ta), Meke(Ka) + +Goat +According to Aiyar, the original Dravidian Languages were under the influence of Aryan +Languages from the early days. He claimed after omitting clear Sanskrit words, there may be +1000 root words in Dravidian Languages. The tense and mood signs are highly influenced by +Indo-Aryan Languages. In conclusion, he says the basic portion of Dravidian vocabulary +consists largely of words of Indo-European origin. But owing to the extremely limited character +of Tamil and Dravidian Alphabet (sounds), these words have been greatly corrupted and are +very difficult to recognize as similar. In addition, he identifies around a hundred suffixes in +Dravidian languages used for indicating tenses and modes of verb forms as of Aryan origin. +He disputes the contention of other scholars that Dravidian Languages have influenced Vedic +Sanskrit. He claims cerebralization of sounds in Sanskrit is internal development. Dravidian +Languages all along have retained a few alveolar forms from historic times and two still retain +them. They have no particular preference for cerebral sounds via-s-vis alveolar sounds or +dental sounds. In fact, Languages like Telugu do not tolerate cerebral sounds ṣ and ṇ. Other +changes in Indian Languages are due to the transition from the synthetic stage to the analytical +stage. In summary, he says Dravidian scholars have mistaken the reflection for the original and +the original for reflection. +Annexure 2[24] has a list of Dravidian words which are Tadbhava words, derived from Sanskrit +words that appear very distinct to lay persons. As against commonly accepted view that mainly +the abstract forms in Dravidian languages are from Sanskrit, Aiyar demonstrates that even day- + +Page 16 of 37 + +to-day and common words are Tadbhava forms from Sanskrit that too in large numbers. One +only needs to trace the transformation journey. +5. Linguistic Analysis using Finite State Machines +Panini’s method to understand the language consists of +• Breaking the sentence into words +• Words into Prakriti (original part) and Pratyaya (suffix). +• Further break Prakriti into components if possible and needed. +• These components are repeatedly seen in multiple words +• Map these repeating components with repeating meanings +• Assigning meanings to these components +• Also observe how these meanings in a sentence are connected +Panini’s method of analyzing words consists of +• Observing the repeated occurrences of letters or groups of letters in different words +• Observe the repetition of the same meaning in different words +• Map repeating sounds with repeating meanings. +• Assigning meaning to the component of a word. +This process results in deriving common Dhātus (root words) out of the Prakriti component +and identification of Pratyayas/common suffixes) that get attached to multiple words depending +on the meaning to be conveyed. Panini ordains a step-by-step process for joining the Prakriti +and Pratyaya. Phonetic and intonation changes when words come together (Sandhi and +Samāsa) also need to be considered. +The proposed methodology builds on these foundational concepts. +5.1 Proposed Methodology +In this paper, we propose the following methodology. +• We construct a phonetic map using Panini’s System of sounds. +• We represent sounds and words including parts of words under construction as states +and represent each word as state-transition diagram. +• Construct a unified state transition diagram for words belonging to a word-group with +associated m-language and m-alphabet. Here a completed word is represented as an +accepting state. +• Compute distances on the phonetic map, each word traverses as it gets constructed. +Compute inter-word distances for word group. This can be useful to identify central +words or original words that have led to other words. +• Associate a grammar (NT,T,P,S) where NT is set of non-terminals, T is set of Terminal +Symbols, S is the starting Symbol and P is set of production rules, with each m- +language. +• Derive a Finite Automaton that accepts words that belong to given m-language. +• The m-languages can be expanded to include groups based on ontological +considerations when words express related concepts and grammatical considerations +when words are used to convey related constructs. + +Page 17 of 37 + +• The Finite Automata can be extended to accommodate suffixes which also have +commonality across languages as well as undergo transformation within languages. +Once we have a repository of m-languages we can derive additional words and in some cases +discover linkages between words that were not widely known. The overall idea is to analyze +words beyond the confines of individual languages and improve their intelligibility without +necessarily requiring one to know the corresponding language in entirety The proposed +approach can enable us to appreciate how the words change over temporal, geospatial, cultural, +religious, professional locales, landscapes and milieu. +Here we have used Google Translate (translate.google.com) extensively. We also have used +dictionaries (learn.sanskrit.com) and our own knowledge of languages as native speakers. +5.2 Proposed Phonetic Map of Sounds +First, we lay out a geometric space of sounds as per Panini’s System of Sounds. This is used to +create the phonetic map. In this map, each word is a path traversed. Comparing two words is a +matter of comparing two paths. Words with common roots may get naturally represented as +they share the first part of the word. Words that have sound shifts may show divergence only +at those points where the shift has happened. Figure 2, illustrates the proposed Phonetic Map. +The topology of the map, we have constructed using the following thought process. Origin is +when no sound is produced and no effort is exercised. On Y axis, lower coordinates are given +for vowels and higher Coordinates are given for consonants. The semi-vowels are +accommodated next to vowels. Sibilants and aspirate are accommodated just before +consonants. On X axis, the velar sounds have low coordinates and labial sounds have higher +coordinates. Thus, we have depicted voice box on the left bottom extreme and mouth at the +right bottom extreme. Then among consonants, we have given lower X coordinate for an +unaspirated sound and higher coordinate for aspirated. The voiced sounds are placed higher +compared to unvoiced sounds. +Certain vowels are considered as combination of basic vowels. For example, we consider sound +ai gets constructed due to the quick succession of sounds a and i. Then we consider sound e is +composed due to the combination of sounds ‘a’ and ‘i’. Similar considerations apply to au and +o sounds which make use of ‘a’ and ‘u’ sounds. +Alternative topologies also may be considered where labials get low X-coordinates and velars +get high X coordinates. In such as case, the distance from origin may be a better indicator of +the effort required to generate a sound. However, the present layout, we feel is acceptable and +easier to relate to. + + +Page 18 of 37 + + + + + + + + + +17 +ङ +ञ +ण(ṇ) +न्(n) +म्(m) +Nasal +16 + +घ्(gh) + +झ्(jh) + +ढ(ḍh) + +ध(dh) + +भ(bh) +Voiced- +Aspirated +15 +ग्(g) + +ज्(j) + +ड +ळ(ḷ) +द(d) + +ब(b) + +Voiced +14 + +ख्(kh) + +छ् (ch) + +ठ(ṭh) + +त्(t) + +फ(ph) +Aspirated +13 +क् (k) + +च्(c) + +ट(ṭ) + +त्(t) + +प्(p) + +Tenue +12 +ह्(h) aspirate +श्(ś) +ष्(ṣ) +स्(s) + + +Sibilant + +Kantavya +Talavya +Murdhva +Datavya +Austa + + +Guttaral +Palatal +Cerebral +Dental +Labial + +11 +व्(v) +Semi- +vowels +10 +ल्(l) + + +9 +र्(r) + + + + +8 +य्(y) + + + + + + +7 +अ(a) +आ(ā) +इ(i) +ई(ī) +ऋ +ॠ +ऌ +ॡ +उ(u) +ऊ(ū) +Vowels +6 +ऐ(ai) + + + + + + +5 +ए(e) + + + + + + +4 +औ(au) +3 +ओ(o) +2 +ं : (am) +. 1 +ं (ah) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 + + + + + + + + + + +Figure 2: Phonetic Map of Indic Sounds (Devanagari) + +Nose +Mouth +Vocal Box + +Page 19 of 37 + +Next we tabulate the coordinates of sounds on the phonetic map in tables 11-13. Table 14 +contains examples of words. +Table 11: Vowel Sounds +Sound +Coordinate +Sound +Coordinate +Sound +Coordinate +अ +(7,1) +आ +(7,2) +इ +(7,3) +ई +(7,4) +ऋ +(7,5) +ॠ +(7,6) +ऌ +(7,7) +ॡ +(7,8) +उ +(7,9) +ऊ +(7,10) +ऐ +(6,2) +ए +(5,2) +औ +(4,5) +ओ +(3,5) +ं +(2,5) +ं +(1,1) + + + + + + + + +Table 12: Consonant Sounds +Sound +Coordinate +Sound +Coordinate +Sound +Coordinate +क् +(13, 1) +ख् +(14, 2) +ग् +(15,1)) +घ् +(16,2) +ङ +(17, 1.5) +च् +(13,3) +छ् +(14,4) +ज् +(15,3) +झ् +(16,4) +ञ + +(17, 3.5) +ट +(13,5) +ठ +(14,6) +ड +(15,5) +ळ +(15,6) +ढ +(16,6) +ण +(17,5.5) +त् +(13,7) +थ +(14,8) +द +(15,7) +ध +(16,8) +न् +(17, 7.5) +प् +(13,9) +फ +(14,10) +ब +(15, 9) +भ +(16, 10) +म् +(17, 9.5) + + + + + + +Page 20 of 37 + +Table 13: Sibilants and Semivowels +Sound +Coordinate +Sound +Coordinate +Sound +Coordinate +श् +(12, 3.5) +ष् +(12,5.5) +स् +(12, 7.5) +ह् +(12, 1.5) +य् +(8, 2.5) +र् +((9, 3.5) +ल् +(10,4.5) +व् +(11, 5.5) + + +Table 14: Word Examples +Word +Path +Word +Path +kapi +(13,1) (7,1) (13,9) (7,4) +hrudaya +(12,1.5) (9, 3.5) (7,1) (15,7) (7,1) +(8,2.5), (7,1) +ape /eip/ +(5,2) (13,9) +heart /ha:t/ +(12,1.5) (7.2) (9,3.5) (14,8) +go +(15,1) (3,5) +mana +(17,9.5) (7,1) (17,7.5) (7,1) +cow/kau/ +(13,1) (4,5) +mind mʌɪnd/ +(17,9,5) (6,2) (17,7.5)(15,7) (7,1) +bo +(15,9) (3,5) +mental /ˈmɛnt(ə)l +(17, 9.5) (5,2) (17,7.5) (13,7) (7,1) +(10,4.5) +In the above table, it can be argued that the English word mental is closer to the Sanskrit word +mana rather than ‘mind’. In the case of hrudaya, ‘hrut’ is the root word that is close to the heart +as well. The Irish word ‘bo’ is the word for cow. This may be unrelated but it ends with the +same vowel sound as go, the Sanskrit word for cow. The old English word for cow is coo. +English uses the word bovine as a generic term to mean “affecting cattle”. The German word +for cow is kuh. Persian has retained go. Latvian also has retained govs. Otherwise, most +European Languages use words starting from k for the cow. In contrast, when it comes to +interrogatives, Sanskrit and Indian Languages as well as a majority of European Languages, +use words starting with the ”k’ sound whereas Germanic languages use words such as who and +hvem. . Thus, which word is original can become a matter of debate and controversy. +The sounds which are not included in Panini’s System of Sounds such as Alveolar or fricative +sounds can be given intermediate coordinates on the phonetic map. +5.3 Finite State Machine Preliminaries +A state machine consists of states and transitions. There may be one or more initial states and +one or more terminal states. From the terminal States, no further transitions happen. There +can be transitions back to the same state as well. Figure 3 below illustrates a state machine. +Here S1, S2, S3, and S4 are states represented by circles, and T1, T2, T3, and T4 are transitions +depicted using arrows. S1 is the start state. S4 the terminal state is represented using a donut- +shaped circle. The transitions happen from state to state depending on the input given to the +system in a particular state. + +Page 21 of 37 + + + +Figure 3: Finite State Machine +A Finite Automata is a State Machine that takes string of symbols as input and changes the +state accordingly. For a given input, the automaton can move to another state or remain in the +same state. After processing a symbol string if the Automaton reaches an accepting state, then +it has accepted that string as a valid string. One can also configure bad states, where from a +given state when a particular input symbol is encountered it will reach the bad state, when the +string is rejected. There are two kinds of Finite Automata: Deterministic and Non- +deterministic. Here a string w=a1a2…an, where a1,a2, … are input symbols. +A deterministic finite automata (M) is a Quintuple +M= (Q, ∑, δ, qo, F) +Q: finite set of states +q0: Start State, where q0 € Q. +∑: finite set of input symbols +F: final states where F ⊆Q +δ : Transition function where δ: Q x ∑ -> Q +The language accepted by DFA M is +L(M) = {w | δ^(q0,w) € F} +If for a given input, more than one kind of transition happens such an automata is non- +deterministic. If for a given input if there is no clarity on what happens such automata are non- +deterministic. Finite automata with multiple start states are non-deterministic. Thus, only that +automata which has a single start state and has a uniquely defined transition for every input is +considered Deterministic. +The most basic and foundational construct for processing symbols is the Atomic Proposition. +Here AP is a set of Atomic Propositions and AP-INF is a set of infinite words over Power Set +(AP). A set of words is termed as language. To form words, one needs an alphabet. For +example, let us say (a, b) is alphabet. Then, a formal/rule-based language can be one that +accepts only a’s, only b’s or a’s and b’s alternating. In the case of a language that takes only + +T1 +S2 +T3 +S1 +S4 +T2 +T4 +S3Page 22 of 37 + +a’s as input, when we model it as a finite automaton, the initial and end-states are the same. In +this case, since there is no transition defined when the input is b, it is considered a Non- +deterministic Finite Automaton. Figure 4 below shows an automaton that accepts only ‘a’ as +the input. Here ‘a’, ‘aa’, and ‘aaa’ are the words of the language. + + + + +Figure 4: Finite Automaton which accepts only “a” + +Thus, we have: +Alphabet {a,b} +Language a* = { ȅ, a, aa, aaa, aaaa, a5 , …} +Another example of Language using same alphabet is +L1 = { ȅ, ab, abab, ababab, … } +Here ȅ is an empty symbol and a word of length 0. The language accepts alternating ‘a’s and +‘b’s or empty symbols. +The following finite automaton illustrates a language where the initial symbol is a, and one or +more b’s. Figure 5, illustrates the same. The language +L2= {a, ab, ab2, ab3,… } + + +Figure 5: Finite Automaton that accepts a and then one or more b’s + +Page 23 of 37 + +For example, if ∑.is alphabet, ∑* is the set of all words over ∑, a word starting with ‘a’ and +ending with ‘a’ can be represented as a∑*a. +The languages that are accepted by finite automata are called regular languages and for every +regular language there is a DFA that accepts it. Every NFA (Non-deterministic Finite +Automaton) can be converted to an equivalent DFA (Deterministic Finite Automaton). +5.4 Application of Proposed Methodology +We take a group of words that relate to each other phonetically, semantically, grammatically, +and/or ontologically. This we call m-language and give it a unique identifier. The sounds that +are used in constructing the words of the m-language constitute m-alphabet. This analysis and +construction of m-language requires reasonable knowledge about the words and languages +involved. At the same time, the process of analysis itself can be educative. We can extend the +m-language and cover related concepts. In certain languages, by adding specific sounds we end +up with an antonym. +Next, we look at representative cases. In the following m-language, we address the poetry +theme. Here starting phoneme is common. The Figure 6, illustrates the state transition diagram +where each phoneme as well as word under construction are states. The completed word is +accepting state. + + +Figure 6: State Transition Diagram for words related to Poetry Theme +Here we have represented Kavi(poet), Kavita(poem), Kavana(poem), Kāvya(Epic in poetic +form), and Kavana(poem). The last word is found only in Kannada. Other words are common +across Indic languages. With each m-alphabet, we associate the coordinates on the phonetic +map covered in the last section. Thus, corresponding +m-language = { kavi, kavita, kāvya, kavana} + +V +( +kaa +kaav +kaavy +(11,7.5) +(8,2.5) +kaavya +(7,1) +aa +(7,2) +a +(G +kav +kava +(7,1) +K +(11,7.5) +(7,1) +n +(13,1) +(7,3) +(17,7.5) +kavan +(13,7) +kavi +(7,1) +(7,2) +a +kavit +t +aa +kavana +kavitaPage 24 of 37 + +m-alphabet = { k,v,t,y,n,a,ā,i} = {(13,1), (11,5.5), (13,7), (8,2.5), (17,7.5), (7,1), (7,2) (7,3)} +Here k and v are basic alphabets that are extended to make new words. Here basic sounds +remain the same and new word forms are due to grammar. The way sounds were associated +with coordinates on phoentic map, the combination of souds words can be associated with +phonetic distances that traverse. Table 15 illustrates the method used to compute distances for +states. We express distance as X and Y components. +Table 15: Words with Poetry theme +Input and Coordinates +State and Manhattan +Distance +Input and Coordinates +State and Manhattan +Distance +Null +0 +0 +Null +0 +0 +Null +0 +0 +Null +0 +0 +k +13 +1 +k +13 +1 +k +13 +2 +k +13 +1 +a +7 +1 +ka +19 +1 +a +7 +1 +ka +19 +1 +v +11 +5.5 +kav +23 +5.5 +v +11 +5.5 +kav +23 +5.5 +i +7 +3 +kavi +27 +8 +a +7 +1 +kava +27 +10 +t +13 +6 +kavit +33 +12 +n +17 +7.5 +kavan +37 +16.5 +ā +7 +2 +kavita +39 +17 +a +7 +1 +kavana +47 +23 +Null +0 +0 +Null +0 +0 +k +13 +1 +k +13 +1 +ā +7 +2 +kā + +2 +v +11 +5.5 +kāv +23 +5.5 +y +8 +2.5 +kāvy +26 +8.5 +a +7 +1 +kāvya +27 +10 +Next we can tabulate inter-word distances. See Table 16 below. + Table 16: Inter-word distances Poetry Theme + +Kavi +Kavita +Kāvya +Kavana +Row Sum +Kavi +0,0 +12,9 +0,2 +20,15 +32,26 +Kavita +12,9 +0,0 +12,7 +8, 6 +32, 15 +Kāvya +0,2 +12,7 +0,0 +20,13 +32, 22 +Kavana +20,15 +8,6 +20,13 +0,0 +48,34 +The above analysis alludes to the possibility that Kavita and Kāvya are central words. Kavi +here is the most basic word. We can repeat the same analysis by excluding Kavana. Here +Kāvya is more central than Kavita. + + + +Page 25 of 37 + + + +Table 17: Inter-word distances Poetry Theme excluding Kavana + +Kavi +Kavita +Kāvya +Row Sum +Kavi +0,0 +12,9 +0,2 +12, 11 +Kavita +12,9 +0,0 +12,7 +24, 16 +Kāvya +0,2 +12,7 +0,0 +12,9 +For the above case, Figure 7 below illustrates the Deterministic Finite Automata, which we +term as Morphological Finite Automata(MFA). Here Q0 is the Starting Symbol, Q5,Q7,Q11 +and Q4 are accepting states. We have made use of null symbols to end with an accepting +state and continue to form more words in parallel. Along with word, in the paranthesis the +language is indicated. + +Figure 7: MFA for Kavita and related words +Corresponding the above MFA, the production rules for the grammar can be written as +follows. +Q0 ->kQ1; Q1->aQ2; Q2->vQ3;Q3->i|iQ4; Q4-> tQ6; Q6 ->ā + Q0->kQ1;Q1->āQ8; Q8->vQ9;Q9->yQ10; Q10->a + +Kavana (Ka) +Q14 +Q13 +Q12 +Q0 +Q8 +Q5 +Kavita(Sa,Ka) +Kavi(Sa.Ka) +Q9 +Q6 +Q10 +Q11 +Kavya(Sa,Ka)Page 26 of 37 + +Here tā and ya are standard and commonly used suffixees in Indian Languages. The +production rules can be rewrriten as follows by accommodating the suffixes as terminal +symbols in their own right. Similar words are Savita, Kartaya etc. +Q0 ->kQ1; Q1->aQ2; Q2->vQ3;Q3->iQ4->tā + Q0->kQ1;Q1->āQ8; Q8->vQ9;Q9->ya + m-language(L) = {S->* W, W is related to Poetry Theme} +Below we look at words that mean “the well’, cutting across languages. Sanskrit uses Koopa +for deep well and Vapi for a broad well. The figure 8 below depicts the corresponding MFA. + + + +Figure 8: MFA for words meaning “the well”. +The production rules can be arrived at similarly as in the previous case. Here the m-alphabet +corresponding to Koopa is {k,p,v} and vowels. By adding b to the same alphabet, we can +accommodate second set of words i.e. Vāpi and Bāvi. +Next, we look at an example that also starts with a common phoneme but cuts across languages. +We take up the word for God in Indo-Europeam Languages, which starts with the sound ‘d’ in +a majority of the languages except Germanic and Russian which uses Bhag derivative. See +Figure 9. +Corresponding m-language = {deva, devs, dio, dia, theos, dieu, devaru, devudu} +m-alphabet = {d, th, a, i, u,o, s, d, r} +Greek is using “th’ sound with coordinate (14,8) instead of ‘d’ sound with coordinate (15,7). +Both sounds are dental. Other than that sounds used are nearly the same. The ‘s’ sound is used +for plurals in Vedic Sanskrit and in Indo-European Language. In Kannada and Telugu, the +word for God is in the plural form and they use the ‘r’ and retroflex ‘D’ sounds respectively + + +h +Q24 +Q23 +p +a +Q2 +Q3 +Q4 +Koopa(Sa) +u +khuha(Pu) +Kunva(Hi) +u +A +a +Q1 +Q12 +Q13 +Q14 +K +kh +a +Q15 +Q17 +Q18 +Kuval(Ta) +Q0 +A +0 +b +Q6 +Q19 +Kuvo(Gu) +ICU +Q9 +p +vapi(Sa) +Q10 +Q11 +n +bavi(Ka, Te) +Q20 +024 +Q22 +banyi(Ko)Page 27 of 37 + + +Figure 9 State Transition Digram for words cognate with Deva +The state computation digram for the MFA in Figure 7 is given in Table 8 below. +Table 18: Distances on Phonetcic Map for Words with Sanskrit Deva +deva +deu +dio +dia +devs +theos +divine(davain) +35, 4.5 +23,6 +27,2 +23,2 +35,3.5 +29,4 +43, 12/5 +The corresponding inter-word distances are given in Table 19 below. + + +Table 19: Inter-word Distances words cognate with Deva + +deva +deu +dio +dia +devs +theos +Row Sum +deva +0,0 +12,1.5 8,2.5 +12,2.5 0,1 +6,0.5 +38,8 +deu +12,1.5 +0,0 +4,4 +0,4 +12,2.5 6,2 +34,14 +dio +8,2.5 +4,4 +0,0 +4,0 +8,1.5 +2,2 +26,10 +dia +12,2.5 +0,4 +4,0 +0,0 +12,1.5 6,2 +34,10 +devs +0,1 +6,2.5 +8,1.5 +12,1.5 0,0 +6,0.5 +32,7 +theos +6,0.5 +6,2 +2.2 +6,2 +6,0.5 +0,0 +26,7 +Here ‘theos’ seems to be the basic form whereas ‘deva’ and ‘deu’ seem to be more refined +forms. However, if you compare the distance between ‘divine’ and words for God, the + +可 +Sanskrit +dev +(7,1) +deva +Konkani +de +(11,5.5) +h +2 +(5,2) +(12,7.5) +(7,9) +(9,3.5) +(15,7) +devu +devar +(7,3) +dia +deus +(7,9) +(15,5) +(14,8) +(7,9) +Latin +(3,5) +Irish +devud +dieu +devaru +(7,9) +3 +(7,3) +dio +French +Kannada +thi +Italian +devudu +(3,5) +(12,7.5) +theo +theos +Telugu +GreekPage 28 of 37 + +following picture emerges. Phonetically the word ‘divine’ is rendered as ‘davain’. Table 20 +below gives the distance of ‘divine’ between different words for God. +Table 20 Distance between divine and cognate words for God + +deva +deu +dio +dia +devs +theos +divine +8,8 +20, 6.5 +16,10.5 20,10.5 +8,9 +14, 4.5 + +The MFA for the above set of words is depicted in a compact manner below. + + + +Figure 10 MFA for words cognate with Deva +The production rules in the corresponding grammar are as follows: + +Q0->dQ1|thQ1; Q1->eQ2; Q2->vQ3; Q3->aQ4; Q4->Q5|rQ7; Q7->uQ8 + + Q1->iQ12; Q12->{a,u,o}Q13->Q14. + +Q0->thQ1; Q1->iQ12; Q12->oQ13; Q13->sQ15; +Overall, our claim is that Vedic Sanskrit in prosodic form has retained the most accurate form +of a word with a high degree of fidelity, while Indian and European Languages have tended to +retain simpler and at times mispronounced forms in colloquial and then written forms. When + +Deva(Sa) +Q1 +Q2 +Q3 +p +Q4 +th +S +Q0 +Q9 +Q12 +Q6 +Devs(La) +p +Q8 +Devaru(Ka) +[a, u, o] +Q10 +Q11 +Q13 +dio(It) +dia(Ir) +Q15 +Q14 +dieu(Fr) +theos(Gr)Page 29 of 37 + +you analyse a group of words(cognates and related words), the root word across languages is +likely to be from Sanskrit. In India, Chandas(prosodic form) used by scholars and +Bhasha(colloquial forms) used by commoners have been concurrent traditions. +Next we look at kinship words that end with “ta” sound. These incude Pita, Mata, Bhrāta, +Duhita, Tata in Sanskrit. In Figure 11, we cover these and cognate words in other languages +and illustrate the State Transition Diagram.. + +Figure 11: Kinship words ending with “ta” + + + + + +Figure 12 MFA for Kinship words ending with Ta + +pi +pit +(7,2) +pita +p +(7,3) +(13,7) +(13,9) +(17,9.5) +ma +mat +mata +m +(7,2) +(13,7) +(7,2) +(16,10) +15,7 +bh +(9,3.5) +bhr +p +(7,9) +bhra +bhrat +bhrata +(13,7) +(7,2) +(7,2) +du +(12,1.5) +(7,3) +duhit +duh +duhi +duhita +(13,7) +(7,2)Q12 +Q2 +m +Q11 +Q0 +Q3 +Q4 +bh +Q13 +Q5 +Q6 +Pita +Mata +Q8 +Bhrata +u +h +Q9 +Q10 +DuhitaPage 30 of 37 + +The corresponding MFA is illustrated in Figure 9. Here we have represented common +endings by using null transitions in between. +Corresponding to the above kinship words m-language={pita, mata,bhrata, duhita} and m- +alphabet = {p,m,bh,r,d,t,h,a,i,u} The state computation table for the MFA in Figure 8 is given +in Table 21. +Table 21: Kinship words +Null +0 +0 Null +0 +0 +Null +0 +0 +Null +0 +0 +p +13 +9 p +13 +9 +d +15 +7 +d +15 +7 + + + +i +7 +3 pi +19 +15 +u +7 +9 +du +23 +9 + + + +t +13 +7 pit +25 +19 +h +12 +1.5 +duh +28 16.5 + + + +ā +7 +2 pitā +31 +24 +i +7 +3 +duhi +33 +18 + + + +m +17 9.5 m +17 +9.5 +t +17 +9.5 +duhit +43 24.5 + + + +ā +7 +2 ā +27 +17 +ā +7 +2 duhitā +53 +32 + + + +t +13 +7 māt +33 +22 +t +17 +9.5 +t +17 +9.5 + + + +ā +7 +2 mātā +39 +27 +ā +7 +2 +ta +27 +17 + + + +bh +16 +10 bh +16 +10 +t +17 +9.5 +tāt +37 24.5 + + + +r +9 3.5 bhr +23 16.5 +ā +7 +2 +tātā +47 +32 + + + +ā +7 +2 bhrā +25 +18 + + + + + + +t +17 9.5 bhrāt +35 25.5 + + + + + + +ā +7 +2 bhrātā +45 +33 + + + + + + +Using the same alphabet we can derive Pitr, Matr, Bhratr, and Duhitar which correspond to +father, mother, brother, and daughter as well as Pateras. Mitera in Greek and by adding ‘k’ +sound, Dukra in Lithuanian. Other cognate words for daughter are Dushterya(Bulgarian), +Doch(Russian), Dcera(Slovak). Among Indian languages only Duva(Konkani), Dhi(Punjabi), +Dikari(Gujarati) and Diyania(Sinhala) well as have retained the word. In Gujarati, Dikara(son) +is related to the word for daughter Dikari. Incidentally, Dikari(Gujarati) and Dukra +(Lithuanian) sound similar. Nepali uses Chori (word for a girl used for daughter) sounds akin +to Corka(Polish). Many Indian Languages use Chokri. Here Romance Languages do not seem +to take part in the cognate word group related to daughter. +The word for sister is Bhagini in Sanskrit which goes with Bhrāta and thus Indian Languages +use words such as Behen (Hindi), Bahini(Konkani), Bona(Bengali). Then Sanskrit uses Svasa +for sister with cognates Seusa (Lithuanian), Soror(French), and Sistra(Russian). Even Finnish +has Sisko. Only exceptions are Celtic Languages and Greek which seem to use very different +words. +Next, we look at words for son and daughter-in-law across languages. + +Page 31 of 37 + + +Figure 13: Words for son and daughter-in-law +Here Sanskrit word ‘sunu’ has a cognate word in Germanic as well as Baltic languages but not +so much in Romance languages. The concept of Daughter-in-law when interpreted as a son’s +wife is ‘snusha’ in Sanskrit. Similar constructs are Snuka(Bulgarian) and Soon/Suna(Konkani) +Words Nuha(Punjabi), Nos(Kashmiri), Nuos(Ancient Greek) and Nora(Portuguese) seem to +have commonality with the same word group Incidentally the word in Kananda for Daughter- +in-law is Sose. The state computation table for the above MFA is given in Table 22 below. +Only a subset of words is represented. + + +Table 22: Words for son and daughter-in-law and distances +san +sunu +sunus +son +nora +soon +snusha +snuka +nuha +sose +27,20.5 +37,12 +42,12.5 +35,12 +39,12 +27,12.5 +37,16 +39.18 +37,17 +37,18 + + +nora +Daughter-in-law (Portuguese) +nor +no +nuh +nuha +daughter-in-law (Punjabi) +nu +sa +san +son (English) +(17,7.5) +(7,1) +(13,7) +(7,1) +(7,9) +su +sut +suta +son (Sanskrit) +3 +(17,7.5) +sunus +son (Lithuanian) +Swedish +(7,9) +son +(12,7,5) +sun +sunu +上 +son (Sanskrit) +(3,5) +S +(7,1) +(7,3) +(12,7,5) +suna +Daughter-in-law +(17,7.5) +(Konkani) +SOS +(7,1) +S +(5,2) +Daughter-in-law +(17,7.5) +(13,1) +(Kannada) +sna +snak +(7,2) +sin +son (bulgarian) +(7,9) +3 +snaka +(12, 5.5) +Daughter-in-law +snu +(Bulgarian) +h +snuSh +(7,2) +(3,5) +可 +syno +snuShaa +Daughter-in-law(Sanskrit) +synov +(11,5.5) +synova +daughter-in-law (Polish) +(7,2)Page 32 of 37 + +The MFA for words meaning the daughter-in-law is shown in Figure 14 below. + + + + +Figure 14 MFA for words meaning Daughter-in-law +Corresponding to the above MFA, basic m-alphabet ={s,n,u,a,o} Here we can consider +derivations such as Snusha and Snuka as language specific. Thus a minor extension of m- +alphabet as m-alphabet = {s, sh, h, u, a, k, o, r} can enable the generation of all the above words. +In summary, Sanskrit words in the kinship category have cognates cutting across the Indo- +European Languages. The kinship word group in Sanskrit as a whole is coherent and self- +contained/derived. +Next, we look at the Apabramsha phenomenon using the word for long. It is in Sanskrit and +the corresponding word is Dīg in Konkani. Other Indian Languages either use Dīrgh as is or +use some other word. Cognates are available also in Croatian, Czech, Bosnian, Macedonian, +Bulgarian, Polish, Serbian, Slovak and Russian. The m-language = {Dīrgha, Deeg, Dugo, +Dluho, Dulgi, Duohi, Dlugi, Dlinyy}. Here two words have same sounds but with a swap of +neighbouring sounds. Thus, languages either drop r or replace r with l and arrive at the +Apabramsha form. Thus, core m-alphabet for this word = {d, g}. Sinhala old and isolated +Indo-European Language has retained Digu. The words and distances on the phonetic map are +given in Table 23 and the corresponding MFA is depicted in Figure 15. + + + +n +Q2 +Q1 +[a,u,o) +0 +Q3 +(s,k,h,r) +Q0 +06 +Q8 +Q4 +n +60 +Q7 +Snusa(Sa) +nuha(Pu) +Snaka(Bu) +nora(Por) +Q5 +sun(Ko) +Q10 +Sose(Ka)Page 33 of 37 + + +Table 23: Words cognate with Dīrgha and Distances +dīrgha +dīg +dugo +dulgi +dlugi +digu +41,13 +31,13 +43,21 +39,19 +39,24 +39,21 + + + + + + +Figure 15: MFA for words cognate with dīrgha(“long”) +Most Indian languages use the words lamba or lambi which is closer to long in English. Both +Germanic and Romance languages also use similar forms. Konkani uses lāmb to mean hang +from a height (or become longer). Sanskrit uses lamb as verb to hang/linger, with vilamba used +for delay, but the direct word for long continues to be Dīrgha. We can make a point that inter- +relationships between individual Indian Languages and European Languages should also be +studied. Some Wiktionaries attempt to derive long from ’dlogos’. +The word for a boy is ‘Chello’ in Konkani and ‘Chele’ in Bengali. The word for girl is ‘Chelli’ +in Konkani, but Bengali uses ‘Meye’ for the girl. Some connection may be there with the +English word boy and, the Sanskrit word ‘Bālaka’, Lativian ‘Puika’, and Lithuanian +‘Berniukas’. +Finally, we take up Sanskrit forms and Dravidian Forms which were worked on by Aiyar. +Figure 16 illustrates the MFAs’s for Sanskrit words and their Tadbhava forms in Drāvidian +Languages. + + + +(i,u,,1) +(r,,I,u) +Q3 +Q2 +Q1 +d +(g.gh) +Q0 +Q4 +(a,u.o,i) +05 +dirgha(Sa) +dig(Ko) +digu(Si) +dulgi(Bu) +dlugi(Po)Page 34 of 37 + + + +Figure 16 MFA for Sanskrit words and their Tadbhava Forms + +Serpent(En) +Aravu(Ta) +Q4 +Q3 +shravika(Sa) +Q4 +cevi(Te) +Kivi(Ka) +pashya(sa) +par(Ta) +Irulu(Ka) +Q3 +ratri(Sa) +medam(Ta) +elam +sid(Hi) +=Straight +sidi(Hi) +trnu(Sa) +tin(Ka, Ta) +Lokanam(Sa) +Nokali(Ta) +Nodali(Ta)Page 35 of 37 + +In the first example, from Sarpa Sanskrit word first syllable is elided and sound shift between +pa and va sound results in Aravu, Tamil form which includes the suffix. The second example +alludes to common origin for the word for ear in Sanskrit and Dravidian Languages. In the third +case, Pashya word for seeing, is close to the Tamil form. In similar vein, common words for +night, sheep, night and perceiving also seem to have commonalities. In summary Finite State +Machines serve as useful mechanism for linguistic analysis across languages and can throw up +not so obvious inter-relationships. +6. Conclusions +In this paper, we have analysed languages with a focus on words. The words are divided into +word groups where a set of these words form m-language (morphological language). With a +given m-language, we associate an m-alphabet. The m-alphabet may have a basic version with +common sounds and an extended version with all sounds. Corresponding to these morphology- +based constructs we construct state transition diagrams, here every phoneme is a state and so +are sequence of phonemes. A valid word, a member of m-language is an accepting state. A +suitable grammar can thus determine whether a word belongs to the word group or not. To +enable that we construct a unified Morpohological Finite Automata which is expressed in a +compact manner and accepts all words belonging the m-language, that cuts across multiple +natural languages.. Secondly this exercise can enable us to infer new words which may belong +to the same word group and give insights on hitherto unknown associations between two words +either belonging to the same or different languages. +We have used Panini’s System of Sounds to represent sounds and words. In addition, we have +defined a phonetic map that manifests these sounds in a geometric fashion on a 2-dimensional +plane. Thus, each phoneme has a coordinate on the phonetic map. Each word has an associated +distance measure that gives an indication of the quantum of traversal required on the phonetic +map. This measure we have used to analyse differences between words. Thus, based on the +distance we can term some words as basic words, some as refined words and some others as +central words. These ideas we believe are useful in comparative linguistics. +The phonetic-map distance measure we believe is an improvement on the current mechanism +to compare words in natural languages. One approach is to use Levenshtein Distance, where +natural language words need to be transliterated first in English. Here the number of +substitutions/modifications required to get two words to match is used as distance. This misses +the phonetic dimension. The second well-known measure in Soundex works well for European +Languages, in particular for de-duplication of names. Here each word is associated with a code +such +as +M460. +Soundex +uses +the +following +codes: +1=B,P,F,V; +2=C,S,G,J,K,Q,X,Z;3=D,T;4=L;5=M,N;6 = R The letters A, E, I, O, U, Y, H, and W are not +coded. Compared to these measures the scheme we have proposed is more elaborate and +promising. In our earlier paper [25], we had used Soundex based measures for language +classification. +Based on our analysis in this paper, we surmise the following: Vedic Sanskrit as part of +Chandas (prosody) has retained the most refined forms from which simpler forms can be +derived. Thus, in certain cases, a word in Sanskrit may result in a high distance measure on the +phonetic map. Also, the Sanskrit word in many cases is a central word that has cognates cutting +across languages, and language groups. If we were to use a genetic or clustering viewpoint, + +Page 36 of 37 + +Sanskrit words have some relationship or other in some manner/context or other with all other +languages among the Indo-European Languages. At times it may appear that Greek/some other +language has a more basic or original word compared to Sanskrit, but when you do the same +analysis at the word group level that includes derived and related words, Sanskrit words are +indeed central. Secondly, Sanskrit is the donor language when it comes to the Dravidian +Languages, even for day-to-day words. Hence, based on morphological analysis, a more +accurate representation for the comparative linguistics field may be Sanskrit occupying the hub +from which words have been transmitted to all other languages and groups of languages that +underwent transformations in transit. The process of transformation of Sanskrit words in Indian +Languages and European Languages are similar. This process has very likely happened over +millennia due to well-acknowledged migrations within India and less understood outward +transmissions to Europe. +References +1. Noam Chomsky, Understanding Linguistics, Talks at Google, 2014, +https://www.youtube.com/watch?v=Y3PwG4UoJ0Y +2. Noam Chomsky. 1995. Language Arts & Disciplines. MIT Press +3. Nagendra Pavana, Śikśa – The Art and Science of Vedic Chanting, Open Learning for +All, Video 27, Chinmaya Vishwa Vidyapeetha, November 2020. +https://www.youtube.com/watch?v=WUDgKX_CbnM&list=PLbQHD8oHpmE15FcG2rP +WejiQnGL0TC_m2&index=28 +4. Shreehari Gokranakar, Chandas – The Vedic Meters, Open Learning for All, Video 28, +Chinmaya Vishwa Vidyapeetha, November 2021, +https://www.youtube.com/watch?v=7BNkWHUXcds&list=PLbQHD8oHpmE15FcG2rP +WejiQnGL0TC_m2&index=29 +5. Gauri Mahulkar, Nirukta - The Etymological Studies in the Veda, Open Learning for All, +Video +31, +Chinmaya +Vishwa +Vidyapeetha, +November +2021, +https://www.youtube.com/watch?v=q_AYmuaXA- +8&list=PLbQHD8oHpmE15FcG2rPWejiQnGL0TC_m2&index=32 +6. Nagendra Pavana, Vyakarana – Linguistics from Vedas, Open Learning for All, Video +32, Chinmaya Vishwa Vidyapeetha, November 2021. +https://www.youtube.com/watch?v=Hi1yItWodw0&list=PLbQHD8oHpmE15FcG2rPWe +jiQnGL0TC_m2&index=33 +7. J.P. Mallory, In Search of the Indo-Europeans, Language, Archaeology and Myth, +Thames and Hudson, 1991, ISBN-13 : 978-0500276167 +8. James Parsons, Remains of Japhet: : being historical enquiries into the affinity and origin +of the European languages.1705-1770, +https://archive.org/details/remainsofjaphetb00pars/page/n19/mode/2up +9. Singh B (1995) The first Englishman in India Thomas Stephens (1547–1619). J South +Asian Literature 30(1/2):146–161 +10. Pedro Redondo, Filippo Sassetti and Thomas Stephens in the beginnings of Indo- +European linguistics, Academia Letters, 10.20935/AL2158. +11. Jones SW. Discourses delivered before the Asiatic society; and miscellaneous papers, on +the religion, poetry, literature, etc., of the nations of India. C. S. Arnold, Michigan, 1824 +12. Edwyn Bryant, The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration +Debate Paperback – Illustrated, 11 March 2004, OUP USA, ISBN-13 : 978-0195169478 + +Page 37 of 37 + +13. Edwin Bryant and Laurie Patton,The Indo-Aryan Controversy +Evidence and Inference in Indian History, Edited By Edwin Bryant, Laurie Patton, 2005, +ISBN 9780203641880, Published August 2, 2004 by Routledge +14. Satya Swaroop Mishra, The date of the Rigveda and the Indian Migration, Fresh +Linguistic Evidence, ibid +15. Michael Witzel, Indocentricism, Autochthonous Visions of Ancient India, ibid +16. F.B.J. Kuiper, Selected Writings on Indian Linguistics and Philology, Leiden Studies in +Indo-European, Volume: 8, 1997, ISBN: 978-90-420-0235-7 +17. Diana L Eck, India A Sacred Geography, Harmony; Reprint edition (26 March 2013); +ISBN-13 : 978-0385531924 +18. Dr. Gintaras Songaila Affinities between Vedic and Baltic Cultures | | Sangam Talks, Aug +22, 2020. https://www.youtube.com/watch?v=-OlsA9KMf-0 +19. Subhash Kak, Sanskrit and Ancient Migrations, 2021, Itihas Darpan, vol. 26, pp. 12-18 +20. Swaminatha Aiyar, Dravidian Theories, Motilal Banarsidass Publishers (1 January 1987), +ISBN-13 : 978-8120803312 +21. Rajesh Kumar, Basics of Language Science, NPTEL Swayam, April 2021, +https://onlinecourses.nptel.ac.in/noc21_hs12/preview +22. Anuradha Chaudhary, Lecture 02: Sounds of Spoken Sanskrit: Its Alphabet, IIT +Kharagpur, October 2018, +https://www.youtube.com/watch?v=UgVwzueOKRU&list=PLbRMhDVUMngfYG2GVf +2bQnIgsI0Y923g3 +23. Shreekanth Prabhu, Annexure 1: Word Groups for Indian and European Languages, +ResearchGate, January 2023, +https://www.researchgate.net/publication/367361269_Annexure_1_Word_Groups_for_In +dian_and_European_Languages +24. Shreekanth Prabhu, Annexure 2: Dravidian Theories, ResearchGate, January 2023, +https://www.researchgate.net/publication/367411879_Annexure_2_Dravidian_Theories +25. Girdhar, R., Nayak, P.S., Prabhu, S.M. (2022). Linguistic Classification Using Instance- +Based Learning. In: Saraswat, M., Sharma, H., Balachandran, K., Kim, J.H., Bansal, J.C. +(eds) Congress on Intelligent Systems. Lecture Notes on Data Engineering and +Communications Technologies, vol 111. Springer, Singapore. +https://doi.org/10.1007/978-981-16-9113-3_63 + + + + + + + + + + + + diff --git a/otFMT4oBgHgl3EQf7TEy/content/tmp_files/load_file.txt b/otFMT4oBgHgl3EQf7TEy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ff503cff58da452b783e2eb4da623fc421549c3 --- /dev/null +++ b/otFMT4oBgHgl3EQf7TEy/content/tmp_files/load_file.txt @@ -0,0 +1,1238 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf,len=1237 +page_content='Page 1 of 37 Linguistic Analysis using Panini’s System of Sounds and Finite State Machines Shreekanth M Prabhu1 and Abhisek Midye2 1 – Department of Computer Science and Engineering, CMR Institute of Technology, Bengaluru 2 – Department of Information Science and Engineering, CMR Institute of Technology, Bengaluru Abstract The study of spoken languages comprises phonology, morphology, and grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Analysis of a language can be based on its syntax, semantics, and pragmatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The languages can be classified as root languages, inflectional languages, and stem languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' All these factors lead to the formation of vocabulary which has commonality/similarity as well as distinct and subtle differences across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In this paper, we make use of Panini’s system of sounds to construct a phonetic map and then words are represented as state transitions on the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Each group of related words that cut across languages is represented by a m-language (morphological language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Morphological Finite Automata (MFA) are defined that accept the words belonging to a given m-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This exercise can enable us to better understand the inter-relationships between words in spoken languages in both language-agnostic and language-cognizant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Keywords: Panini’s system of sounds, State Machines, Finite Automata, Phonology, Morphology, m-language, Comparative Linguistics Biographical Notes Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Shreekanth M Prabhu is currently working as a Professor and Head of the Department of Computer Science and Engineering at CMR Institute of Technology, Bengaluru, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' His research interests include Social Networks, E-Governance, and Comparative Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Abhisek Midya is currently working as an Assistant Professor in the Department of Information Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' His research interest is in Theoretical Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 2 of 37 Introduction Linguistics is a fascinating discipline going back millennia and has been a field for intense scholarly pursuit in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Particularly among them are contributions by Panini whose work on the system of sounds and formal grammar has inspired significant advances worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then there were generations of scholars enriching the field such as Kātyāyana, Patanjali, and Bhartṛhari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In recent times pioneering work by Chomsky has been the hallmark of the advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' According to Chomsky [1], the primary purpose of language is not communication, rather it is cognition as language is the primary vehicle for thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Chomsky [2] also differentiated between I-language and E-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here I-language is a universal language that applies to all spoken/human languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' E-language caters to specific natural languages factoring in cultural and geographic aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Linguistics as a field comprises phonology which deals with the sounds in spoken languages, morphology pertains to the construction of words, and grammar which primarily describes the rules for the orderly usage of words to construct sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Alternatively, the languages can be studied in terms of syntax which concerns different parts of speech, semantics which deal with meaning, and pragmatics whose preoccupation is with the usage of words that varies from milieu to milieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the last few centuries, Comparative Linguistics has emerged as a fertile field for fervid research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here languages are compared for the similarity of words and then their structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Using that approach linguistic families are formed and even ancestral languages are hypothesized yet times drawing far-reaching to far-fetched conclusions about the history of populations and their movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Not just languages but literary sources also can be considered containers of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A lot of work related to comparing languages concerns itself with comparing words across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Comparing the words also may mean comparing root words, inflections, and derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This generally calls for specialist know-how from the field of linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In many cases, there are disputes as different linguists draw different conclusions based on their own predilections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In this paper, we take an alternate approach, where we primarily focus on morphology, and how the words are constructed using a state machine approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We look at the granularity of word groups that can be related phonetically, semantically, or pragmatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' For each word group, we propose a formal language and alphabet using Finite Automata that is useful to decide if a given word belongs to that word group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word groups can be extended and inter- connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Each m-language will have a core alphabet and an extended alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We feel that this approach can enrich the field of linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We make use of Panini’s system of sounds and construct a phonetic map that has the symbols which serve as states for representation as State Machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, we attempt to gauge the distance between words on the phonetic map and look for insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The rest of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Section 2, Linguistics Overview covers the literature in the field of linguistics that is pertinent to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Section 3, Comparative Linguistics Considerations explains the relevance of this paper to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Section 4, Analysis of words using Panini’s using Sounds, where words are analyzed across languages and word groups are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Section 5, Linguistic Analysis using Finite State Machines describes the methodology we have proposed to arrive at unified Morphological Languages that cater to given word groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Section 6, Conclusions, concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 3 of 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Linguistics Overview Linguistics, by providing a structure to words and language, makes the task of understanding language manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Otherwise understanding millions of words individually can prove to be daunting and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Without Linguistics, languages keep changing with time and place and literature becomes incomprehensible in a matter of a century or two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Linguistics as a field has its roots in ancient India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Vedas are preserved for millennia by oral transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' To ensure accurate pronunciation, understanding, and appropriate usage of Vedic Hymns in Yajna, the scholarly tradition mandates the study of six Vedāngas as a pre- requisite and co-requisite for the study of Vedas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These six Vedāngas are Śiksha (phonetics, phonology, and pronunciation), Chandas (prosody), Vyākarana (grammar and linguistic analysis), Nirukta (etymology, explanation of words), Kalpa (ritual instructions), and Jyotish (astronomy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here the first four have laid the foundation for Indian Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The expositions [3-6] give a very cogent explanation of ancient Indian Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In India knowledge is maintained using a 4-fold mechanism that includes Sutra, Vārtika, Bhāshya, and Kārika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here Sutras are very compact, cryptic, and formulaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vārtikas are elaborations and Bhāshyas are interpretations of Sutras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Kārika captures the essence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There is a continuing tradition of grammarians in India and Panini’s Astādhyāyi superseded all earlier traditions and core ideas from there spread to other languages and locales worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Astādhyāyi not only covers Vedic Sanskrit but also classical Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Patanjali’s Bhāshya on Panini’s grammar is the most popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The tradition has continued for centuries with newer Bhāshyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Because of such rigorous discipline, the Vedas were transmitted without any corruption for millennia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This also benefitted Classical Sanskrit as even the works such as Ramāyana which are a few thousand years old are still intelligible to modern scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Otherwise, it is common that in the case of most languages, the works done just a few centuries ago are hard to understand for modern speakers of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Generally, linguistics can be approached from the viewpoint of words (Śabda) or sentences (Vākya).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Whichever way you approach it both Śabda and Vākya are inextricably linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The only purposeful way of using Śabda is in the form of Vākya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The only way to decipher and understand Vākya is by breaking it down into Śabdas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vyakarana thus is called Shabda Śastra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Panini’s Astādhyāyi analyses sentences, identifies words and then components, and arrives at Dhātus (roots of words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Each word is viewed as consisting of Prakriti (the original part) and Pratyaya (suffixes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' By combining Prakriti and Pratyaya, the Padas (usable words) are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' With a good discipline of grammar using a single Dhātu typically 360 words can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There are at least 2000 Dhātus, resulting in lakhs of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This framework enables Sanskrit to be a powerful language where new words can be easily composed using components and they become conveniently intelligible to those conversant with the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' When it comes to the right use of words, it can be done only with meaning in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Three things are critical to interpreting the meaning of individual words in a sentence in order to arrive at the intended meaning of the sentence: Ākānkshā (expectancy), Yogyata(suitability) and Sannidhi(proximity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' According to Vedic tradition, the six objectives of precise grammar are Rakshā (prevention from distortion), Asandeha (absence of ambiguity), Ūhā (modification of Vedic Mantras due to the possibility of more than one interpretation, Āgama (ease of augmentation) and Laghuh (easy means of acquiring knowledge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 4 of 37 Modern linguistics like ancient linguistics comprises phonology (the science of sounds), morphology (word formation using sounds), and grammar (deriving new words and constructing sentences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Analysing the sentences, thus consists of syntax analysis, semantic analysis, and pragmatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The methodology for the analysis of natural language can be compared with the approach taken by the compiler to analyse programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A compilation process consists of a scanning phase where a statement is broken into components (lexemes) and then in the parsing phase, a syntax tree is constructed comprising of lexemes and validated for grammatical correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Even though natural language processing is similar, the grammar is not context-free and morphology (the constructions of words) itself makes use of grammar in addition to the construction and analysis of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, some key constructs such as finite automata and the concept of language from theoretical computer science can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' That is the endeavour of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Comparative Linguistics Considerations The relationship between languages did not get the attention of scholars in Europe as according to Biblical tradition, Hebrew was considered the universal language which then broke into other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In India, Sanskrit was considered the mother of all languages while scholars were very much aware of Sanskrit words and words native to a given language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Europe, as acknowledged by Mallory [7], James Parsons [8] was probably one of the first to do a systematic study of thousands of common words across European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, according to Mallory [7], a century prior to that it was Joseph Scaliger who attempted to divide the languages of Europe into four major groups, each labelled after their word for god.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The transparent relationship of what we today call the Romance languages was recognized in the ‘Deus’ group (for example, Latin ‘Deus’, Italian ‘Dio’, Spanish ‘Dio’, French ‘Dieu’), and contrasted with the Germanic ‘Gott’ (English God, Dutch God, Swedish ‘Gudy’ and so on);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Greek ‘Theos’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' and Slavic Bog (such as Russian ‘Bog’, Polish ‘Bog’ and Czech ‘Buh’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This exercise of comparing languages was also undertaken by visitors to India in the 15th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In India, it was Filippo Sassetti and Thomas Stephens were the first two who noticed the similarity between Indian and European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Singh B [9] identifies Thomas Stephens as the first Englishman in India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Pedro Redondo [10] explains that the motivation of Sassetti was that of the humanist whereas that of Stephens was evangelical and theological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' All these exercises and the well-known discourse of William Jones [11] culminated in the proposal of not only the Indo-European Family of Languages but also the acceptance of the language family as a universal construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' According to modern Linguistics, certain words are considered isolates i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' they are unique to that language or a narrow set of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The isoglosses cause dialectical variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These differences may be phonological, lexical(different words), or different linguistic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Cognates sound similar across languages carrying the same/related meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The cognates are classified as adstrate words when these are loan words due to trade and migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then there are substrate words where it is presumed that speakers of one language had dominance over the speakers of other languages resulting in an asymmetric transfer of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In contrast, in Indian tradition, the words in a language are divided into three categories: Tatsama(same as words in another language generally Sanskrit), Tadbhava(derived from word in another language), and Deshya(native words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 5 of 37 Initially Sanskrit was considered the mother of the Indo-European Languages as it had cognates across Indo-European Languages and the most complete grammar with eight cases as well as duals in addition to singular and plurals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' But then scholars who are generally known as Indologists who call themselves mainstream changed their stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Bryant{12] puts forward the ‘main-stream’ view that (i) There has to be a proto-language probably spoken by all speakers before that broke into Indo-European (IE) Languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (ii) All the IE speakers stayed in a common homeland before they separated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (iii) The proto-language could not have been Sanskrit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (iv) There was Proto-Indo-European(PIE) Language that broke into Celtic, Germanic, Romance, Baltic, Slavic, Greek, and Indo-Iranian families with PIE at the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus Sanskrit was relegated as a leaf node within the Indo-Iranian family and India as yet another output of IE speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='. Bryant explains how Sanskrit was dethroned using linguistic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' One of the reasons given by Linguists to propose PIE is that Sanskrit has innovated a,e, and o sounds to a sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Greek has retained the original sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A typical example given is bhend in Greek becomes bandh in Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Another example the scholars give is Greek Deca (for number 10) is not derivable from Sanskrit Daśa, hence there needs to be a common ancestral language to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The languages are further classified as Kentum and Satem languages based on the word for the number 100 and here Kentum Languages are considered more archaic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sanskrit is considered Satem Language and ruled out as an archaic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, since Sanskrit had retroflexes, which many European languages did not have, some linguists say it can not be a proto- language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' To support their hypothesis scholars claimed that Sanskrit borrowed cerebralization from Dravidian Languages and any word in Sanskrit that is not in common with European Languages is a loan from Dravidian or Munda languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This is in contrast to Indian tradition where Sanskrit words appear either as Tatsama or Tadbhava forms across languages and seldom other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' As an example, the word for water is Neer only in Sanskrit and Dravidian Languages but not in most Indo-Aryan Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' So one may conclude that the word was loan into Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' But any such conclusion may be hasty as Greek also uses neró for water, which is likely from Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Bryant and Patton[13] examine the issue of Indo-Eurpean origins from multiple perspectives in an edited volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Among the linguists who contributed to that endeavour, Mishra[14] claims that Sanskrit is more archaic than all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The main features where Sanskrit is shown to deviate from Indo-European is the merger of IE a, e, o into a in Sanskrit and the change of palatal k etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' to palatal s etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' in Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Mishra counters this and among many other arguments gives the example of Gypsy language where Indo-Aryan a remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' a in Asiatic Gypsy but becomes a, e, o in European Gypsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This confirms that original IE a was the same as Sanskrit a and remained a in the Indo-Iranian languages, but changed to a, e, o in their sister languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then he gives the case where Sanskrit retains both Vākya and Vāchya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' According to Mishra, ś becomes k before it becomes s in Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He maintains that ś and k are allophonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, the k which was allophonic to ś in Sanskrit might have been generalized in the Centum languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He also gives examples of Lithuanian a Satem Language sporadically presenting k sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Witzell[15] continues to champion the mainstream view that Aryans are outsiders to India and Vedic langauage is an import into India and he is a strong proponent of import of Munda words into Vedic Sanskrit, whereas Kuiper[16] considered many Sanskrit words were of Dravidian origin Page 6 of 37 The worldview of Europeans is guided by the prism of conflict, conquest, co-location, and commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' India was also subject to conquests from the 7th century AD onwards which targeted Indian civilization with religious conversions and political conquests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, the essential characteristics of the civilization that survived have been convergence, confluence, continuity, and contiguity aided by amalgamation, and assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, India has a continuing civilization going back millennia and a sense of unity that stems from identification with the larger sacred geography unified by common traditions, scriptures, belief systems, holy places, and value systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Diana Eck[17] rightly observes that India is a country united by the footsteps of pilgrims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The migrations of people within India have been continuous and in particular priestly classes have migrated across India and have maintained essential unity of traditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Many southern kings also have northern lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Such movements have resulted in far greater homogenization of languages across India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The languages which were neighbours to the Sarasvati River region such as Konkani and Punjabi are inflectional like Vedic Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The South Indian Languages tend to have more agglutination of consonants and less conjunction of consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, subject-object-verb order is common across all Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, the larger geography which included Afghanistan and Central Asia was considered contiguous to India with cultural transmission and exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Central Asian Republics continue to use Sthan as part of their names (Kazakistan, Tajikistan) showing the influence of Sanskrit on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Greater India thus consisted of Uttara Kuru as well as Uttara Madra regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Another point to be considered is the Sinhala language of Sri Lanka located to the south of Dravida region is Indo-Aryan with commonality with Vedic Sanskrit retaining a few rather archaic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sanskrit for most of the time served as the lingua franca across India thus serving as the donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' language of words that represented abstract concepts on one hand to mundane reality on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Sanskrit, refined and accurate pronounciation was not only important for rituals but also considered a hallmark of the civilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Generaly Apabramsha(mispronounced) forms of Sanskrits word which is easier to pronounce were used by the commoners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus Śrāvan word for the rainy season may change to Sāvan in Hindi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We notice that some languages(Kannada, Konkani, Bengali) retain the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word for cotton Karpasa is considered to have derived from Kāpas a Munda word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' But other Indian Languages(Konkani, Marathi and Gujarathi) use Kāpas only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Some argue that Kāpas is Apabramsha for Karpasa and not necessarily a loan word from Munda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In India, the direction of changes is from Sanskrit to Prākrat to vernaculars as India had tradition of Chandas(language for prosody) and Bhasha(language for common use) concurrently evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This runs counter to the linguists’ view where they expect the transformation to happen from simple/primitive to refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In addition, different regions of India and languages there have shown a preference for certain sounds and a lack of preference for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus retroflex sound ṇ is not in vogue in Hindi, but very much there in Konkani, Marathi, and Punjabi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Bengali uses o instead of a and ‘b’ sound instead of ‘v’, in certain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Bihar, ‘s’ sound is used more than the ‘ś’ sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' On the other extreme, Iranian languages have replaced ‘s’ with ‘h’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In many cases Sanskrit has more than one sound, say for people Jana is used as well as Gaṇa is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The same is true with Dik and Disha both words are used for direction in Sanskrit but for different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, Sanskrit uses a word starting with K for Kendra (center) which very few European Languages(Greek,Armenian), use, and most use centrum which starts with the ‘s’ sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 7 of 37 Thus, analysis of European Linguists using their worldview and rules may need revisiting using a formal approach that can address voluminous vocabulary across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In particular, Sanskrit commonly has more than ten words to represent the same entity or concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' European Languages are generally compared only with Sanskrit, but not as much with other Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' It is also worth comparing the phenomena that Indian Language words underwent as they carried forward Sanskrit words and comparing the same with what could have happened to Sanskrit words which are borrowed by/found in common with European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Gintaras Songaila [18] elaborates on enormous affinities which are directly there between Indo- Aryan and Lithuanian without any connection with the Iranian language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Subhash Kak [19] also makes a long list of common words among European languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' and Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Both scholars emphasize the contiguity of central Asia with India from ancient times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The borrowing of words also spans disciplines, ‘Astipathi’ in Sanskrit becomes osteopathy and ‘Jara’ the word for old age in Sanskrit leads to geriatrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Same is true with common medical word sputum which has natural association with Sphut, Sanskrit word than spit, an English/Latin verb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='The word pa(a)th is due to path in Sanskrit(as used in RajPath i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' King’s Road) leading to words such as allopathy and homeopathy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Hence the transmission of words has continued for centuries and millennia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Also, there are few studies that compare Dravidian Languages with other Indian languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A study by Swaminath Aiyar[20] is a rare exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Aiyar after a very unique and highly detailed comparative study of langauges says “My views differ from those of all previous scholars because they contended themselves with comparing Dravidian Languages with Classical Sanskrit and naturally saw no deep-seated affinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' When one language is extensively affected by another, we need to look for the source of influence not in the artificial language of high literature but in the spoken idioms of common people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' It is necessary to compare Dravidian idioms with the Vedic Dialects and the Prākrats of pre-Christian Centuries, before we can decide the question of Aryo-Dravidian affinities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' It was Bishop Caldwell who compared Classical Sanskrit and Dravidian Languages and pronounced the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' At the same time there were other scholars such as Pope, who also was a missionary did not agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He felt the decision to consider Dravidian Languages as disjoint from Aryan Languages was rather abrupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He expressed the opinion “(i) that between the languages of Southern India and those of the Aryan family there are many deeply seated and radical affinities and (ii) that the differences between the Dravidian Tongues and Aryan are not so great as between the Celtic (for instance) languages and the Sanskrit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' and (iii) that by consequence the doctrine that the place of Dravidian dialects is rather with the Aryan than with Turanian families is still capable of defence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He illustrated these positions by means of copious illustrations and pointed out that “the resemblances appeared in the most uncultivated Dravidian dialects’ and that “the identity was most striking in the names of instruments, places, and acts connected with a simple life”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He promised to follow on with a paper that looked at derivative words and show that the prefixes and affixes were Aryan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The work of Aiyar thus fills that gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In summary, dethroning of Sanskrit as a proto-language needs to be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the least, confining Sanskrit as a daughter language under the Indo-Iranian branch is a travesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, the inter-relationship between Dravidian Languages and Indo-Aryan Languages needs many more studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 8 of 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Analysis of words using Panini’s System of Sounds In this section, we introduce the concept of m-alphabet which is the set of phonemes used to construct a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The core m-alphabet is the set of sounds that pertain to the original part (Prakriti) of the word, that too where the chosen sounds are common cutting across languages or that pertain to the suspected original word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The m-languages consist of words belonging to a word group there are related phonetically, semantically, grammatically, and ontologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word groups across different languages are compared and analyzed using these morphology- based constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We make use of Panini’s System of Sounds which represents natural language sounds comprehensively in a scientific manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='1 Panini’s System of Sounds Panini developed the system of human/natural language sounds after a careful study of how they are generated by the vocal box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Panini’s Śiksha (phonology) explains the form of each Varṇa ((letter/sound) is determined by Svara (intonation), Kāla (time taken to pronounce it), Sthana (place of articulation), and Karaṇa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Abhyantara Prayatna (effort within the oral cavity) and Bāhya Prayatna (effort outside the oral cavity) are two additional factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 1, illustrates Panini’s System of Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 1: Panini’s System of Sounds Sounds that do not face any obstruction when we speak are termed vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These may vary depending on whether they are short, long or very long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In his scheme there are 13 vowels and two additional vowels which can be used only in conjunction with other sounds namely am and ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The sounds that face obstruction are termed consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He classifies them based on place of articulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The guttural/velar/Kanṭavya sounds are produced in the throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next, palatal/ Tālavya Sounds are generated by touching one’s tongue to the pallet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next set of sounds are Cerebral/Murdhya sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' They are also called hard palatal sounds or retroflex sounds as it requires one to reverse the direction of the tongue while generating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The fourth set of consonants are dental/Dantavya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' They are generated by touching the tongue to the teeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Fifth set of consonants are labial/Austa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here the lips are involved in generating the sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Each Vouh h (anundn) b(turg) Coasonaths Gutturaler kha Te h Palatalsr: a tp Certbeils Denals: Ttha Labialar Hm 7 Ocmnd Senivovcker Shlnte Apinate OuaPage 9 of 37 of these group of 5 consonants can be further classified – (i) unvoiced and unaspirated/tenuis ii) aspirated, (iii) voiced (iv) voiced and aspirated and (v) nasal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then there are other consonants which are called semivowels, sibilants and aspirates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 1 below illustrates Panini’s System of Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Rajesh Kumar [21] and Anuradha Chaudhari [22] explain Panini’s system of sounds covering modern linguistics and traditional Indian vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Whereas Panini’s System of Sounds is very comprehensive and representative, there are sounds that are not represented specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vedic Sanskrit and many Indian Languages have a cerebral ḷ sound which is at times used in lieu of the ḍ sound as in Iḍa, and Iḷa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This is not represented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Alveolar sounds are intermediate sounds typically used when English say “Tea”, “Table” or “Tennis”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' They are not fully dental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A person who is a native speaker of a language that has retroflex sounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' may treat them as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then there are additional alveolar sounds in Tamil which are not there in North Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In fact, Tamil and probably other Dravidian Languages early on had far too limited an alphabet or far fewer phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tamil continues to have a limited alphabet consisting of vowels: a, ā, i, ī , u, ū.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', e, ai, o, ō, au, with the omission of r, rr, lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The consonants are k, nasal (k), c, nasal(c), t, n, ṭ, ṇ p, m, y, r, l,v, l,l,r,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The last four are alveolar sounds and unknown to Sanskrit Alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In each class of consonants, instead of 5 members, only tenuis (the first), and nasal (the last) sounds are there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Generally, European Languages do not use cerebral/retroflex sounds, except in a few North European Languages such as Swedish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Some languages such as French use only dental sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Tamil Language also has far fewer sounds and the script uses the same symbol for four consonants of the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Further, there are a total of nine fricative consonants in English: /f, θ, s, ∫, v, ð, z, З, h/, and eight of them (all except for/h/) are produced by partially obstructing the airflow through the oral cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These are: /f/: far, /v/: save, of, /θ/: think, /ð/: those, /s/: sir, race, /z/: zoo, rise, /ʃ/: sharp, chef, pressure, sugar, motion, /h/: ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='2 Analyzing Words using Sounds In this section, we build a word bank cutting across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 1 indicates the encoding we have used for the languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 1: Encoding to indicate the language of the word European Languages Indian Languages English(En),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' German(Ge),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Russin(Ru),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Greek(Gr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Romanian(Ro),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Latin(La),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Latvian(Latv),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' French(Fr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Lithuanian(Li),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Italian(It),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Welsh(We),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Danish(Da),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Dutch(Du),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Spanish(Sp),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Polish(Po),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Portuguese(Por),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Bulgarian(Bu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Corsican(Co),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Croatian(Cr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Ukraian(Uk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Scot Galeic(SG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Irish(Ir),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Slovak(Sl) Sanskrit(Sa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Prakrat(Pr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Hindi(Hi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Marathi(Ma),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Punjabi(Pu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Konkani(Ko),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Bengali(Be),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Gujarati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Kannada(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tamil(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Telugu(Te),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Malayalam(Mal),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sinhala(Si) Page 10 of 37 Subhash Kak did a study of words derived from Sanskrit in European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 2 below lists Sanskrit words, and corresponding cognates in European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We have also added a word in Kannada and Konkani for water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here we also list basic sounds used in Sanskrit word which we call as m-alphabet (Morphological Alphabet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This is followed by an extended alphabet to represent all words, sounds gained, replaced, and lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Also listed are related words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' All words in a given row can be considered to constitute an m-language (Morphological Language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 2: Sanskrit Words and Cognates in European Languages Sanskrit Word m-alphabet (Sanskrit) Word (Language) m-alphabet (Extended) Sounds Gained Sounds Changed Sounds Lost Related words āvāsa a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='s house(En) haus(Ge) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u v to u dam d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='m dom(Ru) damus(La) d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='h o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='h a to o domicile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' domestic grha g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a casa(La) cass(Sp) g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='s k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' s g to k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' s to h r vāri v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i water(Du) v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t t i udaka u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a uda(Ko) voda(Ru) u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v v u to v āp a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p apa(Ro) a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' nīr n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ī nero(Gr) dur(We) neeru(Ka) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ī,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' o n to d dhara dh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā terra(It) dal(We) dh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' e dh to t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' dh to d nabha n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='bh nebo(Ru,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Cr) nebe(Cz) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='bh,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ṇ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a ouranos(Gr) v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ṇ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='o o v yuva y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a youth(En) jeunesse(Fr) y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' v ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' h Juvenile Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' the formation of cognate words may involve sound shifts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' and closely related sounds (voiced versus voiceless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' aspirated versus unaspirated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' changes of vowels) as well as changes to grammar (gender-related or other changes) or due to any other peculiarities of receiving languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, we can define a grammar which can cater to such scenarios which can determine if a word belongs to a word group or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Additionally, we may be able to generate candidate words which can prospectively belong to the same word group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The words from Vedic Sanskrit have gone through variety of transformations in Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This is accepted by all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Now our hypothesis is that the transformation of those Page 11 of 37 words in European Languages can also be considered the manifestations of the same phenomena that happened as the words got carried over to European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' For example, Graha in Sanskrit becomes Kar in spoken Punjabi but in Hindi it remains as Ghar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The same word becomes Casa in Latin, presenting k sound as initial sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tables 3 to 7 illustrate the concept of m-alphabet and m-language with additional examples which we have collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Note that this is based on Google Translate output and our own knowledge which may have missed certain synonyms that are cognate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Annexure 1[23] has a bank of Indian and European words, where we have enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' nearly two hundred groups of words for which m-languages can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 3: m-language for word group “Being in the middle” Theme Being in the middle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' in between m-language madhya (Sa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mādhyam(Sa),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' middle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mediate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' media ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' midten(Da),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' midden(Du),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' madhala(Ma),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' madhyama(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' milieu(Fr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mezzo(It),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mitte(Ge),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' meio(Po),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mijloc(Ro),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' maeda(Si),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' meadhan(SG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' mesaio(Gr)} Non-members natuttara(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' lar(Ir),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' vidu(latv),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' vidurio(Li),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='sredina(Ru) m-alphabet(core) {m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i} m-alphabet(Extended) {m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' z} Remarks Sanskrit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Indic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Germanic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Greek and Romance language and Scot Gaelic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' use the above m-alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Extended Vocabulary mezzanine floor, meso (between micro and marco) Table 4: m-language for word group “Face, Mouth” Theme Face, Mouth m-language mukh(Sat), moga (Ka)}, muh(Hi)}, mouth, mukhya(Sa:Main), mund(Da), mond(Du), mute(Latv), tond(Ko) Non-members Face, Chehera(Hindi), beul(Irish), Bayi(Kannada) Usta(Slovenian) m-alphabet(core) {m, u, kh,o,g, t, n, h,d} m-alphabet (Extended) {m u, k, kh, h, o, g, y, d, n, t} Remarks Face and mouth words get overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tond may belong to another m-language with Sanskrit Connection, Tunda – trunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Germanic and Sanskrit languages have commonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 12 of 37 Table 5: m-language for word group “Long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tall” Theme Long, Tall m-language long, lamba(Hi), lāmb(Ma), labi(Gu), long(Fr), lang(Sw) Non-members dugo – Baltic and Slavic languages use words cognate with deergha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' fada(Irish), makrys(Greek) m-alphabet(core) l, n, m, b, g, a, o, i m-alphabet (Extended) NA Remarks Here Indian Languages have direct cognates with European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sanskrit tends to use Deergh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However Sanskrit word vilamb(delay) indicates Sanskrit origin of the above words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 6: m-language for word group “High” Theme High m-language unc(Hi), ucca(Sa), ucca(Be) hoch(Ge), hoog(Du) hog(Sw), Haut(Fr) Non-members Uyar(Ta) m-alphabet(core) {u, c } m-alphabet (Extended) {u, n, c, t, g, a, u, e} Table 7: m-language for word group “Below, Low, Lowly” Theme Lowly/below m-language Lowly:nīc(Sa), Below: nīce(Hi), nizhe(Ru) nizsie(Sl) Non-members Many m-alphabet(core) n, c m-alphabet (Extended) n, c, ī, e, zh, s Next, we analyze the Dravidian Language words using sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Table 8 below, we analyse how the words for numbers are constructed in Dravidian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There are sound shifts from pa to ha (Pattu and Hattu) in Kannada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The ‘b’, ‘p’ and ‘v’ sounds also seem to be used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Malayalam and in some cases, Tamil manage without a suffix ‘u’, whereas others customarily use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 13 of 37 Table 8: Words for numbers in Dravidian Languages Number Kannada Tulu Telugu Tamil Malayal am m-alphabet (Extended) m- alphabet (core) One ondu onji okati onru onn o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n Two eraḍu radd ranḍu irand rand e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d Three mooru mooji muḍu munr munn m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ū,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ū Four nālku nāl nālugu nānku nal n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l Five aidu ain aidu aintu anj ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='j ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n Six āru āji aru āru ār ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r Seven elu el edu elu el e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l Eight entu edma enimidi ettu ett e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t Nine ombattu ormbā tommidi onpatu ompat o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t ten hattu patt padi pattu patt h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='d p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t twenty ippattu irva irvai irupat irupat I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t thirty muvattu muppa muppai muppat u muppat m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p m,,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t fourty naluvattu nālpa nalabhai narpatu nalpat n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='b h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='r n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='p fifty aivattu aiva yabhai aimpat u ampat ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='u ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Phonemic Affinity u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' d j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ā d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='bh n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' r a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' m Excluded Phonemes v v Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' we look at the study of inter-relationships between Indo-Aryan and Dravidian Languages done by Swaminath Aiyar [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Drāvidian Languages were historically divided into Andhra Group with Telugu and a set of languages and the Drāvida group consisting of Tamil, Kannada, Malayaḷam and Tuḷu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Andhra Group is independently influenced by neighbouring Prākrats as well as greater propensity to use Sanskrit words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Aiyar’s main conclusion is that in addition to a large number of clearly Sanskrit (Tatsama) words in the Drāvidian Languages, there are a significant number of Tadbhava words that are derived from Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He claims that when Caldwell came up with the hypothesis that Dravidian Languages have a low affinity Page 14 of 37 for other Indian Languages, he compared words from Classical Sanskrit which indeed were different for the sample he had chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Aiyar invalidates Caldwell’s conclusions by comparing South Indian Language words with other Sanskrit words which are closer to Vedic Sanskrit, Prākrats and other Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 9 contrasts Caldwell’s approach with that of Aiyar’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 9 Comparison of Sanskrit and Tamil Words Sr,No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' English Word Sanskrit Word (Caldwell) Tamil, Telugu, Kannada, Malayalam Proposed Word (Aiyar) Remarks 1 hair kesha Mayir(Ta) Śmashru(Sa) 2 mouth mukha Vay(Ta) Vac(Sa) Vac is alternate word from Vedic Sanskrit 2(a) nose Mūkku(Ta), Mūgu(K), Mukku(Te) Words derived from Mukha are used for face and mouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here it is proposed to be used for nose as well 3 ear karna Shevi(Ta) Śrava(Sa), shravika(Sa) 4 hear sru Kel(Ta) Karna(Sa) 5 eat bhaks Tin(Ta) Trṇu(Sa), Tr(Sa), 6 walk car, cel Egu(Ta) Ya(Sa), i(Sa) 7 night nak Ira, Iravu Rātri(Sa) 8 mother matr Āyi(Ta) Yāyi(Paisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=') 9 tiger vyaghra Puli(Ta) Vengai(Tamil) 10 deer, beast mrga Marai, Man, Ma(Ta) Mrga(S), Maga(Pr_ 11 Fire Agni Ti(ta) Tejas(Sa), Tij(Sa) 12 Snake Sarpa Pāmbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Aravu (Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Arava(Ma) Prasarpa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sarpa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sarpaks 13 Village grama Ūr(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Ūru(Ka) Pura(Sa) 14 buffalo mahiSa Erumai(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Emme(Ka) Heramba(ka) Associated words are swapped 14(a) M āDu(Ta) MahiSa(Sa) 15 horse ashva Kuthirai(Ta) Ashvatara(ka) 16 hill parvata Malai(Ta) Paruppu(Tam) Matching Associations found According to Swaminath Aiyar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' a large number of Dravidian words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' in particular in Tamil that appears to have no affinity with Sanskrit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' in fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' are Tadbhava words from Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' As Tamil has a highly constrained Alphabet, they went through a lot more transformation and corruption compared to North Indian Vernaculars and appear unrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' To get the whole picture one needs to look at a plurality of Sanskrit words and Prākrat words and inter-relationships between Dravidian Languages, as the closest word could belong to Telugu or Tamil in most cases and then further transformed in modern Kannada and Malayalam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 10, contains a sample of words analyzed by Aiyar and inferred as Sanskrit words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Aiyar derives Dravidian words from Page 15 of 37 Sanskrit/Prākrat words with a variety of rules such as sound elision, sound substitution and suffix additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 10: Tadbhava Dravidian Words which are derived from Sanskrit Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' No Sanskrit Word Meaning Tamil/Dravidian Word/Other Indian Language Meaning 1 Paksha Wing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Side Pakka(Ta) Side 2 See Pashya Paar(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Paḷe(Ko) See 3 Dakshina South Tenkaṇa(Ta) South 4 Bhru Brow Pubbu(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Hubbu(Ka) Eyebrow 5 Satya Truth Sari(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sahi(Hi) Correct 6 Vayalah Bangle Baḷe(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vaḷai(Ta) Bangle 7 Lokah People,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Word Olaku(Ta) People,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' World 8 Mridu Soft Mella(Ka) Slowly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Gently 9 Mrda Mud Maṇṇu(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Maṇṇ (Ta) Soil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Earth 10 Dhvani Voice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sound Toni(Ta) Sound 11 Vandyah Barren Woman Banje(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vandi(Ta) Barren woman 12 Shabdah Word Sadd(Pu),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Saddu(Ka) Sound 13 kāṣṭakah Wood Koṭṭai(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Kaṭṭige(Ka) Wood (Collected from Forest) 14 Mrtya Perishable (Body) Mai(Ka) Body 15 Svithra Silver/White Velli(Ta),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Belli(Ka),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Belagu(Ka).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Belaku Silver, White,Light 16 Sreṇi Line Eṇi(Ka) Ladder 17 Chayah Hand Kai(Ka, Ta) Hand 18 Śirah Head Sir(Hi), Tale(Ka), Tare(Tu) Head 19 Kārṣapaṇa Coin or weight Kāṇam(Ta) Kāhavaṇo(Pr) Kāhāṇ(Or) 20 Meṣa Sheep/Goat Meḍam(Ta), Meke(Ka) Goat According to Aiyar, the original Dravidian Languages were under the influence of Aryan Languages from the early days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He claimed after omitting clear Sanskrit words, there may be 1000 root words in Dravidian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The tense and mood signs are highly influenced by Indo-Aryan Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In conclusion, he says the basic portion of Dravidian vocabulary consists largely of words of Indo-European origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' But owing to the extremely limited character of Tamil and Dravidian Alphabet (sounds), these words have been greatly corrupted and are very difficult to recognize as similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In addition, he identifies around a hundred suffixes in Dravidian languages used for indicating tenses and modes of verb forms as of Aryan origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He disputes the contention of other scholars that Dravidian Languages have influenced Vedic Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' He claims cerebralization of sounds in Sanskrit is internal development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Dravidian Languages all along have retained a few alveolar forms from historic times and two still retain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' They have no particular preference for cerebral sounds via-s-vis alveolar sounds or dental sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In fact, Languages like Telugu do not tolerate cerebral sounds ṣ and ṇ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Other changes in Indian Languages are due to the transition from the synthetic stage to the analytical stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In summary, he says Dravidian scholars have mistaken the reflection for the original and the original for reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Annexure 2[24] has a list of Dravidian words which are Tadbhava words, derived from Sanskrit words that appear very distinct to lay persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' As against commonly accepted view that mainly the abstract forms in Dravidian languages are from Sanskrit, Aiyar demonstrates that even day- Page 16 of 37 to-day and common words are Tadbhava forms from Sanskrit that too in large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' One only needs to trace the transformation journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Linguistic Analysis using Finite State Machines Panini’s method to understand the language consists of • Breaking the sentence into words • Words into Prakriti (original part) and Pratyaya (suffix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Further break Prakriti into components if possible and needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • These components are repeatedly seen in multiple words • Map these repeating components with repeating meanings • Assigning meanings to these components • Also observe how these meanings in a sentence are connected Panini’s method of analyzing words consists of • Observing the repeated occurrences of letters or groups of letters in different words • Observe the repetition of the same meaning in different words • Map repeating sounds with repeating meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Assigning meaning to the component of a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This process results in deriving common Dhātus (root words) out of the Prakriti component and identification of Pratyayas/common suffixes) that get attached to multiple words depending on the meaning to be conveyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Panini ordains a step-by-step process for joining the Prakriti and Pratyaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Phonetic and intonation changes when words come together (Sandhi and Samāsa) also need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The proposed methodology builds on these foundational concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='1 Proposed Methodology In this paper, we propose the following methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We construct a phonetic map using Panini’s System of sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • We represent sounds and words including parts of words under construction as states and represent each word as state-transition diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Construct a unified state transition diagram for words belonging to a word-group with associated m-language and m-alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here a completed word is represented as an accepting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Compute distances on the phonetic map, each word traverses as it gets constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Compute inter-word distances for word group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This can be useful to identify central words or original words that have led to other words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Associate a grammar (NT,T,P,S) where NT is set of non-terminals, T is set of Terminal Symbols, S is the starting Symbol and P is set of production rules, with each m- language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • Derive a Finite Automaton that accepts words that belong to given m-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' • The m-languages can be expanded to include groups based on ontological considerations when words express related concepts and grammatical considerations when words are used to convey related constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 17 of 37 The Finite Automata can be extended to accommodate suffixes which also have commonality across languages as well as undergo transformation within languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Once we have a repository of m languages we can derive additional words and in some cases discover linkages between words that were not widely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The overall idea is to analyze words beyond the confines of individual languages and improve their intelligibility without necessarily requiring one to know the corresponding language in entirety The proposed approach can enable us to appreciate how the words change over temporal, geospatial, cultural, religious, professional locales, landscapes and milieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here we have used Google Translate (translate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com) extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We also have used dictionaries (learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com) and our own knowledge of languages as native speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='2 Proposed Phonetic Map of Sounds First, we lay out a geometric space of sounds as per Panini’s System of Sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This is used to create the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In this map, each word is a path traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Comparing two words is a matter of comparing two paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Words with common roots may get naturally represented as they share the first part of the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Words that have sound shifts may show divergence only at those points where the shift has happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 2, illustrates the proposed Phonetic Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The topology of the map, we have constructed using the following thought process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Origin is when no sound is produced and no effort is exercised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' On Y axis, lower coordinates are given for vowels and higher Coordinates are given for consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The semi vowels are accommodated next to vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sibilants and aspirate are accommodated just before consonants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' On X axis, the velar sounds have low coordinates and labial sounds have higher coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, we have depicted voice box on the left bottom extreme and mouth at the right bottom extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then among consonants, we have given lower X coordinate for an unaspirated sound and higher coordinate for aspirated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The voiced sounds are placed higher compared to unvoiced sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Certain vowels are considered as combination of basic vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' For example, we consider sound ai gets constructed due to the quick succession of sounds a and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then we consider sound e is composed due to the combination of sounds ‘a’ and ‘i’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Similar considerations apply to au and o sounds which make use of ‘a’ and ‘u’ sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Alternative topologies also may be considered where labials get low X coordinates and velars get high X coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In such as case, the distance from origin may be a better indicator of the effort required to generate a sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, the present layout, we feel is acceptable and easier to relate to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Page 18 of 37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ङ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ञ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ण(ṇ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='न्(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='म्(m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Nasal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='घ्(gh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='झ्(jh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ढ(ḍh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ध(dh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='भ(bh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Voiced Aspirated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ग्(g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ज्(j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ड ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ळ(ḷ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='द(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ब(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Voiced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ख्(kh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='छ् (ch) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ठ(ṭh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='त्(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='फ(ph) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Aspirated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='क् (k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='च्(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ट(ṭ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='त्(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='प्(p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Tenue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ह्(h) aspirate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='श्(ś) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ष्(ṣ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='स्(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Sibilant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Kantavya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Talavya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Murdhva ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Datavya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Austa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Guttaral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Palatal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Cerebral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Dental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Labial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='व्(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Semi vowels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ल्(l) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='र्(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='य्(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='अ(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='आ(ā) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='इ(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ई(ī) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ऋ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ॠ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ऌ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ॡ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='उ(u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ऊ(ū) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Vowels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ऐ(ai) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ए(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='औ(au) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ओ(o) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ं : (am) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 1 ं (ah) 0 1 2 3 4 5 6 7 8 9 10 Figure 2: Phonetic Map of Indic Sounds (Devanagari) Nose Mouth Vocal Box Page 19 of 37 Next we tabulate the coordinates of sounds on the phonetic map in tables 11-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 14 contains examples of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 11: Vowel Sounds Sound Coordinate Sound Coordinate Sound Coordinate अ (7,1) आ (7,2) इ (7,3) ई (7,4) ऋ (7,5) ॠ (7,6) ऌ (7,7) ॡ (7,8) उ (7,9) ऊ (7,10) ऐ (6,2) ए (5,2) औ (4,5) ओ (3,5) ं (2,5) ं (1,1) Table 12: Consonant Sounds Sound Coordinate Sound Coordinate Sound Coordinate क् (13, 1) ख् (14, 2) ग् (15,1)) घ् (16,2) ङ (17, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) च् (13,3) छ् (14,4) ज् (15,3) झ् (16,4) ञ (17, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) ट (13,5) ठ (14,6) ड (15,5) ळ (15,6) ढ (16,6) ण (17,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) त् (13,7) थ (14,8) द (15,7) ध (16,8) न् (17, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) प् (13,9) फ (14,10) ब (15, 9) भ (16, 10) म् (17, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) Page 20 of 37 Table 13: Sibilants and Semivowels Sound Coordinate Sound Coordinate Sound Coordinate श् (12, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) ष् (12,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) स् (12, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) ह् (12, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) य् (8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) र् ((9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) ल् (10,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) व् (11, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) Table 14: Word Examples Word Path Word Path kapi (13,1) (7,1) (13,9) (7,4) hrudaya (12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,1) (15,7) (7,1) (8,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5), (7,1) ape /eip/ (5,2) (13,9) heart /ha:t/ (12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='2) (9,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (14,8) go (15,1) (3,5) mana (17,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,1) (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,1) cow/kau/ (13,1) (4,5) mind mʌɪnd/ (17,9,5) (6,2) (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5)(15,7) (7,1) bo (15,9) (3,5) mental /ˈmɛnt(ə)l (17, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (5,2) (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (13,7) (7,1) (10,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) In the above table, it can be argued that the English word mental is closer to the Sanskrit word mana rather than ‘mind’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the case of hrudaya, ‘hrut’ is the root word that is close to the heart as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Irish word ‘bo’ is the word for cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This may be unrelated but it ends with the same vowel sound as go, the Sanskrit word for cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The old English word for cow is coo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' English uses the word bovine as a generic term to mean “affecting cattle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The German word for cow is kuh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Persian has retained go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Latvian also has retained govs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Otherwise, most European Languages use words starting from k for the cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In contrast, when it comes to interrogatives, Sanskrit and Indian Languages as well as a majority of European Languages, use words starting with the ”k’ sound whereas Germanic languages use words such as who and hvem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, which word is original can become a matter of debate and controversy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The sounds which are not included in Panini’s System of Sounds such as Alveolar or fricative sounds can be given intermediate coordinates on the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='3 Finite State Machine Preliminaries A state machine consists of states and transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There may be one or more initial states and one or more terminal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' From the terminal States, no further transitions happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There can be transitions back to the same state as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 3 below illustrates a state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here S1, S2, S3, and S4 are states represented by circles, and T1, T2, T3, and T4 are transitions depicted using arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' S1 is the start state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' S4 the terminal state is represented using a donut- shaped circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The transitions happen from state to state depending on the input given to the system in a particular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 21 of 37 Figure 3: Finite State Machine A Finite Automata is a State Machine that takes string of symbols as input and changes the state accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' For a given input, the automaton can move to another state or remain in the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' After processing a symbol string if the Automaton reaches an accepting state, then it has accepted that string as a valid string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' One can also configure bad states, where from a given state when a particular input symbol is encountered it will reach the bad state, when the string is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' There are two kinds of Finite Automata: Deterministic and Non- deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here a string w=a1a2…an, where a1,a2, … are input symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A deterministic finite automata (M) is a Quintuple M= (Q, ∑, δ, qo, F) Q: finite set of states q0: Start State, where q0 € Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ∑: finite set of input symbols F: final states where F ⊆Q δ : Transition function where δ: Q x ∑ -> Q The language accepted by DFA M is L(M) = {w | δ^(q0,w) € F} If for a given input, more than one kind of transition happens such an automata is non- deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' If for a given input if there is no clarity on what happens such automata are non- deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Finite automata with multiple start states are non-deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, only that automata which has a single start state and has a uniquely defined transition for every input is considered Deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The most basic and foundational construct for processing symbols is the Atomic Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here AP is a set of Atomic Propositions and AP-INF is a set of infinite words over Power Set (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A set of words is termed as language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' To form words, one needs an alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' For example, let us say (a, b) is alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then, a formal/rule-based language can be one that accepts only a’s, only b’s or a’s and b’s alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the case of a language that takes only T1 S2 T3 S1 S4 T2 T4 S3Page 22 of 37 a’s as input, when we model it as a finite automaton, the initial and end-states are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In this case, since there is no transition defined when the input is b, it is considered a Non- deterministic Finite Automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 4 below shows an automaton that accepts only ‘a’ as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here ‘a’, ‘aa’, and ‘aaa’ are the words of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 4: Finite Automaton which accepts only “a” Thus, we have: Alphabet {a,b} Language a* = { ȅ, a, aa, aaa, aaaa, a5 , …} Another example of Language using same alphabet is L1 = { ȅ, ab, abab, ababab, … } Here ȅ is an empty symbol and a word of length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The language accepts alternating ‘a’s and ‘b’s or empty symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The following finite automaton illustrates a language where the initial symbol is a, and one or more b’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 5, illustrates the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The language L2= {a, ab, ab2, ab3,… } Figure 5: Finite Automaton that accepts a and then one or more b’s Page 23 of 37 For example, if ∑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='is alphabet, ∑* is the set of all words over ∑, a word starting with ‘a’ and ending with ‘a’ can be represented as a∑*a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The languages that are accepted by finite automata are called regular languages and for every regular language there is a DFA that accepts it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Every NFA (Non-deterministic Finite Automaton) can be converted to an equivalent DFA (Deterministic Finite Automaton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='4 Application of Proposed Methodology We take a group of words that relate to each other phonetically, semantically, grammatically, and/or ontologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This we call m-language and give it a unique identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The sounds that are used in constructing the words of the m-language constitute m-alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This analysis and construction of m-language requires reasonable knowledge about the words and languages involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' At the same time, the process of analysis itself can be educative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We can extend the m-language and cover related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In certain languages, by adding specific sounds we end up with an antonym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next, we look at representative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the following m-language, we address the poetry theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here starting phoneme is common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The Figure 6, illustrates the state transition diagram where each phoneme as well as word under construction are states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The completed word is accepting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 6: State Transition Diagram for words related to Poetry Theme Here we have represented Kavi(poet), Kavita(poem), Kavana(poem), Kāvya(Epic in poetic form), and Kavana(poem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The last word is found only in Kannada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Other words are common across Indic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' With each m-alphabet, we associate the coordinates on the phonetic map covered in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, corresponding m-language = { kavi, kavita, kāvya, kavana} V ( kaa kaav kaavy (11,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (8,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) kaavya (7,1) aa (7,2) a (G kav kava (7,1) K (11,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,1) n (13,1) (7,3) (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) kavan (13,7) kavi (7,1) (7,2) a kavit t aa kavana kavitaPage 24 of 37 m-alphabet = { k,v,t,y,n,a,ā,i} = {(13,1), (11,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5), (13,7), (8,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5), (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5), (7,1), (7,2) (7,3)} Here k and v are basic alphabets that are extended to make new words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here basic sounds remain the same and new word forms are due to grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The way sounds were associated with coordinates on phoentic map, the combination of souds words can be associated with phonetic distances that traverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 15 illustrates the method used to compute distances for states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We express distance as X and Y components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 15: Words with Poetry theme Input and Coordinates State and Manhattan Distance Input and Coordinates State and Manhattan Distance Null 0 0 Null 0 0 Null 0 0 Null 0 0 k 13 1 k 13 1 k 13 2 k 13 1 a 7 1 ka 19 1 a 7 1 ka 19 1 v 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 kav 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 v 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 kav 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 i 7 3 kavi 27 8 a 7 1 kava 27 10 t 13 6 kavit 33 12 n 17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 kavan 37 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 kavita 39 17 a 7 1 kavana 47 23 Null 0 0 Null 0 0 k 13 1 k 13 1 ā 7 2 kā 2 v 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 kāv 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 y 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 kāvy 26 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 a 7 1 kāvya 27 10 Next we can tabulate inter-word distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' See Table 16 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 16: Inter-word distances Poetry Theme Kavi Kavita Kāvya Kavana Row Sum Kavi 0,0 12,9 0,2 20,15 32,26 Kavita 12,9 0,0 12,7 8, 6 32, 15 Kāvya 0,2 12,7 0,0 20,13 32, 22 Kavana 20,15 8,6 20,13 0,0 48,34 The above analysis alludes to the possibility that Kavita and Kāvya are central words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Kavi here is the most basic word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We can repeat the same analysis by excluding Kavana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here Kāvya is more central than Kavita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 25 of 37 Table 17: Inter-word distances Poetry Theme excluding Kavana Kavi Kavita Kāvya Row Sum Kavi 0,0 12,9 0,2 12, 11 Kavita 12,9 0,0 12,7 24, 16 Kāvya 0,2 12,7 0,0 12,9 For the above case, Figure 7 below illustrates the Deterministic Finite Automata, which we term as Morphological Finite Automata(MFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here Q0 is the Starting Symbol, Q5,Q7,Q11 and Q4 are accepting states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We have made use of null symbols to end with an accepting state and continue to form more words in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Along with word, in the paranthesis the language is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 7: MFA for Kavita and related words Corresponding the above MFA, the production rules for the grammar can be written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q0 ->kQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q1->aQ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q2->vQ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q3->i|iQ4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q4-> tQ6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q6 ->ā Q0->kQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q1->āQ8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q8->vQ9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q9->yQ10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q10->a Kavana (Ka) Q14 Q13 Q12 Q0 Q8 Q5 Kavita(Sa,Ka) Kavi(Sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Ka) Q9 Q6 Q10 Q11 Kavya(Sa,Ka)Page 26 of 37 Here tā and ya are standard and commonly used suffixees in Indian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The production rules can be rewrriten as follows by accommodating the suffixes as terminal symbols in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Similar words are Savita, Kartaya etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q0 ->kQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q1->aQ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q2->vQ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q3->iQ4->tā Q0->kQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q1->āQ8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q8->vQ9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='Q9->ya m-language(L) = {S->* W, W is related to Poetry Theme} Below we look at words that mean “the well’, cutting across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sanskrit uses Koopa for deep well and Vapi for a broad well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The figure 8 below depicts the corresponding MFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 8: MFA for words meaning “the well”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The production rules can be arrived at similarly as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here the m-alphabet corresponding to Koopa is {k,p,v} and vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' By adding b to the same alphabet, we can accommodate second set of words i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Vāpi and Bāvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next, we look at an example that also starts with a common phoneme but cuts across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We take up the word for God in Indo-Europeam Languages, which starts with the sound ‘d’ in a majority of the languages except Germanic and Russian which uses Bhag derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' See Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Corresponding m-language = {deva, devs, dio, dia, theos, dieu, devaru, devudu} m-alphabet = {d, th, a, i, u,o, s, d, r} Greek is using “th’ sound with coordinate (14,8) instead of ‘d’ sound with coordinate (15,7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Both sounds are dental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Other than that sounds used are nearly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The ‘s’ sound is used for plurals in Vedic Sanskrit and in Indo-European Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Kannada and Telugu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' the word for God is in the plural form and they use the ‘r’ and retroflex ‘D’ sounds respectively h Q24 Q23 p a Q2 Q3 Q4 Koopa(Sa) u khuha(Pu) Kunva(Hi) u A a Q1 Q12 Q13 Q14 K kh a Q15 Q17 Q18 Kuval(Ta) Q0 A 0 b Q6 Q19 Kuvo(Gu) ICU Q9 p vapi(Sa) Q10 Q11 n bavi(Ka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Te) Q20 024 Q22 banyi(Ko)Page 27 of 37 Figure 9 State Transition Digram for words cognate with Deva The state computation digram for the MFA in Figure 7 is given in Table 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 18: Distances on Phonetcic Map for Words with Sanskrit Deva deva deu dio dia devs theos divine(davain) 35, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 23,6 27,2 23,2 35,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 29,4 43, 12/5 The corresponding inter-word distances are given in Table 19 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 19: Inter-word Distances words cognate with Deva deva deu dio dia devs theos Row Sum deva 0,0 12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 8,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 12,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 0,1 6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 38,8 deu 12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 0,0 4,4 0,4 12,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 6,2 34,14 dio 8,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 4,4 0,0 4,0 8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 2,2 26,10 dia 12,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 0,4 4,0 0,0 12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 6,2 34,10 devs 0,1 6,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 8,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 0,0 6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 32,7 theos 6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 6,2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='2 6,2 6,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 0,0 26,7 Here ‘theos’ seems to be the basic form whereas ‘deva’ and ‘deu’ seem to be more refined forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' However, if you compare the distance between ‘divine’ and words for God, the 可 Sanskrit dev (7,1) deva Konkani de (11,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) h 2 (5,2) (12,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,9) (9,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (15,7) devu devar (7,3) dia deus (7,9) (15,5) (14,8) (7,9) Latin (3,5) Irish devud dieu devaru (7,9) 3 (7,3) dio French Kannada thi Italian devudu (3,5) (12,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) theo theos Telugu GreekPage 28 of 37 following picture emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Phonetically the word ‘divine’ is rendered as ‘davain’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 20 below gives the distance of ‘divine’ between different words for God.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 20 Distance between divine and cognate words for God deva deu dio dia devs theos divine 8,8 20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 16,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 20,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 8,9 14, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 The MFA for the above set of words is depicted in a compact manner below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 10 MFA for words cognate with Deva The production rules in the corresponding grammar are as follows: Q0 >dQ1|thQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q1 >eQ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q2 >vQ3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q3 >aQ4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q4 >Q5|rQ7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q7 >uQ8 Q1 >iQ12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q12 >{a,u,o}Q13 >Q14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q0->thQ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q1->iQ12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q12->oQ13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Q13->sQ15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Overall, our claim is that Vedic Sanskrit in prosodic form has retained the most accurate form of a word with a high degree of fidelity, while Indian and European Languages have tended to retain simpler and at times mispronounced forms in colloquial and then written forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' When Deva(Sa) Q1 Q2 Q3 p Q4 th S Q0 Q9 Q12 Q6 Devs(La) p Q8 Devaru(Ka) [a, u, o] Q10 Q11 Q13 dio(It) dia(Ir) Q15 Q14 dieu(Fr) theos(Gr)Page 29 of 37 you analyse a group of words(cognates and related words), the root word across languages is likely to be from Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In India, Chandas(prosodic form) used by scholars and Bhasha(colloquial forms) used by commoners have been concurrent traditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next we look at kinship words that end with “ta” sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These incude Pita, Mata, Bhrāta, Duhita, Tata in Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Figure 11, we cover these and cognate words in other languages and illustrate the State Transition Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='. Figure 11: Kinship words ending with “ta” Figure 12 MFA for Kinship words ending with Ta pi pit (7,2) pita p (7,3) (13,7) (13,9) (17,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) ma mat mata m (7,2) (13,7) (7,2) (16,10) 15,7 bh (9,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) bhr p (7,9) bhra bhrat bhrata (13,7) (7,2) (7,2) du (12,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,3) duhit duh duhi duhita (13,7) (7,2)Q12 Q2 m Q11 Q0 Q3 Q4 bh Q13 Q5 Q6 Pita Mata Q8 Bhrata u h Q9 Q10 DuhitaPage 30 of 37 The corresponding MFA is illustrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here we have represented common endings by using null transitions in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Corresponding to the above kinship words m-language={pita, mata,bhrata, duhita} and m- alphabet = {p,m,bh,r,d,t,h,a,i,u} The state computation table for the MFA in Figure 8 is given in Table 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 21: Kinship words Null 0 0 Null 0 0 Null 0 0 Null 0 0 p 13 9 p 13 9 d 15 7 d 15 7 i 7 3 pi 19 15 u 7 9 du 23 9 t 13 7 pit 25 19 h 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 duh 28 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 pitā 31 24 i 7 3 duhi 33 18 m 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 m 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 t 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 duhit 43 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 ā 27 17 ā 7 2 duhitā 53 32 t 13 7 māt 33 22 t 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 t 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 mātā 39 27 ā 7 2 ta 27 17 bh 16 10 bh 16 10 t 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 tāt 37 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 r 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 bhr 23 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 tātā 47 32 ā 7 2 bhrā 25 18 t 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 bhrāt 35 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 ā 7 2 bhrātā 45 33 Using the same alphabet we can derive Pitr, Matr, Bhratr, and Duhitar which correspond to father, mother, brother, and daughter as well as Pateras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Mitera in Greek and by adding ‘k’ sound, Dukra in Lithuanian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Other cognate words for daughter are Dushterya(Bulgarian), Doch(Russian), Dcera(Slovak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Among Indian languages only Duva(Konkani), Dhi(Punjabi), Dikari(Gujarati) and Diyania(Sinhala) well as have retained the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In Gujarati, Dikara(son) is related to the word for daughter Dikari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Incidentally, Dikari(Gujarati) and Dukra (Lithuanian) sound similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Nepali uses Chori (word for a girl used for daughter) sounds akin to Corka(Polish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Many Indian Languages use Chokri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here Romance Languages do not seem to take part in the cognate word group related to daughter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word for sister is Bhagini in Sanskrit which goes with Bhrāta and thus Indian Languages use words such as Behen (Hindi), Bahini(Konkani), Bona(Bengali).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Then Sanskrit uses Svasa for sister with cognates Seusa (Lithuanian), Soror(French), and Sistra(Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Even Finnish has Sisko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Only exceptions are Celtic Languages and Greek which seem to use very different words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next, we look at words for son and daughter-in-law across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Page 31 of 37 Figure 13: Words for son and daughter-in-law Here Sanskrit word ‘sunu’ has a cognate word in Germanic as well as Baltic languages but not so much in Romance languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The concept of Daughter-in-law when interpreted as a son’s wife is ‘snusha’ in Sanskrit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Similar constructs are Snuka(Bulgarian) and Soon/Suna(Konkani) Words Nuha(Punjabi), Nos(Kashmiri), Nuos(Ancient Greek) and Nora(Portuguese) seem to have commonality with the same word group Incidentally the word in Kananda for Daughter- in-law is Sose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The state computation table for the above MFA is given in Table 22 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Only a subset of words is represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Table 22: Words for son and daughter-in-law and distances san sunu sunus son nora soon snusha snuka nuha sose 27,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 37,12 42,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 35,12 39,12 27,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5 37,16 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='18 37,17 37,18 nora Daughter in law (Portuguese) nor no nuh nuha daughter in law (Punjabi) nu sa san son (English) (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (7,1) (13,7) (7,1) (7,9) su sut suta son (Sanskrit) 3 (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) sunus son (Lithuanian) Swedish (7,9) son (12,7,5) sun sunu 上 son (Sanskrit) (3,5) S (7,1) (7,3) (12,7,5) suna Daughter in law (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (Konkani) SOS (7,1) S (5,2) Daughter in law (17,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) (13,1) (Kannada) sna snak (7,2) sin son (bulgarian) (7,9) 3 snaka (12, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) Daughter in law snu (Bulgarian) h snuSh (7,2) (3,5) 可 syno snuShaa Daughter in law(Sanskrit) synov (11,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5) synova daughter in law (Polish) (7,2)Page 32 of 37 The MFA for words meaning the daughter-in-law is shown in Figure 14 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 14 MFA for words meaning Daughter-in-law Corresponding to the above MFA, basic m-alphabet ={s,n,u,a,o} Here we can consider derivations such as Snusha and Snuka as language specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus a minor extension of m- alphabet as m-alphabet = {s, sh, h, u, a, k, o, r} can enable the generation of all the above words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In summary, Sanskrit words in the kinship category have cognates cutting across the Indo- European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The kinship word group in Sanskrit as a whole is coherent and self- contained/derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Next, we look at the Apabramsha phenomenon using the word for long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' It is in Sanskrit and the corresponding word is Dīg in Konkani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Other Indian Languages either use Dīrgh as is or use some other word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Cognates are available also in Croatian, Czech, Bosnian, Macedonian, Bulgarian, Polish, Serbian, Slovak and Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The m-language = {Dīrgha, Deeg, Dugo, Dluho, Dulgi, Duohi, Dlugi, Dlinyy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here two words have same sounds but with a swap of neighbouring sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, languages either drop r or replace r with l and arrive at the Apabramsha form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, core m-alphabet for this word = {d, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sinhala old and isolated Indo-European Language has retained Digu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The words and distances on the phonetic map are given in Table 23 and the corresponding MFA is depicted in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' n Q2 Q1 [a,u,o) 0 Q3 (s,k,h,r) Q0 06 Q8 Q4 n 60 Q7 Snusa(Sa) nuha(Pu) Snaka(Bu) nora(Por) Q5 sun(Ko) Q10 Sose(Ka)Page 33 of 37 Table 23: Words cognate with Dīrgha and Distances dīrgha dīg dugo dulgi dlugi digu 41,13 31,13 43,21 39,19 39,24 39,21 Figure 15: MFA for words cognate with dīrgha(“long”) Most Indian languages use the words lamba or lambi which is closer to long in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Both Germanic and Romance languages also use similar forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Konkani uses lāmb to mean hang from a height (or become longer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Sanskrit uses lamb as verb to hang/linger, with vilamba used for delay, but the direct word for long continues to be Dīrgha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We can make a point that inter- relationships between individual Indian Languages and European Languages should also be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Some Wiktionaries attempt to derive long from ’dlogos’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word for a boy is ‘Chello’ in Konkani and ‘Chele’ in Bengali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The word for girl is ‘Chelli’ in Konkani, but Bengali uses ‘Meye’ for the girl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Some connection may be there with the English word boy and, the Sanskrit word ‘Bālaka’, Lativian ‘Puika’, and Lithuanian ‘Berniukas’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Finally, we take up Sanskrit forms and Dravidian Forms which were worked on by Aiyar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Figure 16 illustrates the MFAs’s for Sanskrit words and their Tadbhava forms in Drāvidian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (i,u,,1) (r,,I,u) Q3 Q2 Q1 d (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='gh) Q0 Q4 (a,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='i) 05 dirgha(Sa) dig(Ko) digu(Si) dulgi(Bu) dlugi(Po)Page 34 of 37 Figure 16 MFA for Sanskrit words and their Tadbhava Forms Serpent(En) Aravu(Ta) Q4 Q3 shravika(Sa) Q4 cevi(Te) Kivi(Ka) pashya(sa) par(Ta) Irulu(Ka) Q3 ratri(Sa) medam(Ta) elam sid(Hi) =Straight sidi(Hi) trnu(Sa) tin(Ka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Ta) Lokanam(Sa) Nokali(Ta) Nodali(Ta)Page 35 of 37 In the first example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' from Sarpa Sanskrit word first syllable is elided and sound shift between pa and va sound results in Aravu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Tamil form which includes the suffix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The second example alludes to common origin for the word for ear in Sanskrit and Dravidian Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In the third case, Pashya word for seeing, is close to the Tamil form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In similar vein, common words for night, sheep, night and perceiving also seem to have commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In summary Finite State Machines serve as useful mechanism for linguistic analysis across languages and can throw up not so obvious inter-relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Conclusions In this paper, we have analysed languages with a focus on words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The words are divided into word groups where a set of these words form m-language (morphological language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' With a given m-language, we associate an m-alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The m-alphabet may have a basic version with common sounds and an extended version with all sounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Corresponding to these morphology- based constructs we construct state transition diagrams, here every phoneme is a state and so are sequence of phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A valid word, a member of m-language is an accepting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' A suitable grammar can thus determine whether a word belongs to the word group or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' To enable that we construct a unified Morpohological Finite Automata which is expressed in a compact manner and accepts all words belonging the m-language, that cuts across multiple natural languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='. Secondly this exercise can enable us to infer new words which may belong to the same word group and give insights on hitherto unknown associations between two words either belonging to the same or different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' We have used Panini’s System of Sounds to represent sounds and words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In addition, we have defined a phonetic map that manifests these sounds in a geometric fashion on a 2-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, each phoneme has a coordinate on the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Each word has an associated distance measure that gives an indication of the quantum of traversal required on the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This measure we have used to analyse differences between words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, based on the distance we can term some words as basic words, some as refined words and some others as central words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' These ideas we believe are useful in comparative linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The phonetic-map distance measure we believe is an improvement on the current mechanism to compare words in natural languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' One approach is to use Levenshtein Distance, where natural language words need to be transliterated first in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here the number of substitutions/modifications required to get two words to match is used as distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This misses the phonetic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The second well-known measure in Soundex works well for European Languages, in particular for de-duplication of names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Here each word is associated with a code such as M460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Soundex uses the following codes: 1=B,P,F,V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 2=C,S,G,J,K,Q,X,Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='3=D,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='4=L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='5=M,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='6 = R The letters A, E, I, O, U, Y, H, and W are not coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Compared to these measures the scheme we have proposed is more elaborate and promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In our earlier paper [25], we had used Soundex based measures for language classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Based on our analysis in this paper, we surmise the following: Vedic Sanskrit as part of Chandas (prosody) has retained the most refined forms from which simpler forms can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Thus, in certain cases, a word in Sanskrit may result in a high distance measure on the phonetic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Also, the Sanskrit word in many cases is a central word that has cognates cutting across languages, and language groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' If we were to use a genetic or clustering viewpoint, Page 36 of 37 Sanskrit words have some relationship or other in some manner/context or other with all other languages among the Indo-European Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' At times it may appear that Greek/some other language has a more basic or original word compared to Sanskrit, but when you do the same analysis at the word group level that includes derived and related words, Sanskrit words are indeed central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Secondly, Sanskrit is the donor language when it comes to the Dravidian Languages, even for day-to-day words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Hence, based on morphological analysis, a more accurate representation for the comparative linguistics field may be Sanskrit occupying the hub from which words have been transmitted to all other languages and groups of languages that underwent transformations in transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' The process of transformation of Sanskrit words in Indian Languages and European Languages are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' This process has very likely happened over millennia due to well-acknowledged migrations within India and less understood outward transmissions to Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Noam Chomsky, Understanding Linguistics, Talks at Google, 2014, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=Y3PwG4UoJ0Y 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Language Arts & Disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' MIT Press 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Nagendra Pavana, Śikśa – The Art and Science of Vedic Chanting, Open Learning for All, Video 27, Chinmaya Vishwa Vidyapeetha, November 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=WUDgKX_CbnM&list=PLbQHD8oHpmE15FcG2rP WejiQnGL0TC_m2&index=28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Shreehari Gokranakar, Chandas – The Vedic Meters, Open Learning for All, Video 28, Chinmaya Vishwa Vidyapeetha, November 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=7BNkWHUXcds&list=PLbQHD8oHpmE15FcG2rP WejiQnGL0TC_m2&index=29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Gauri Mahulkar, Nirukta - The Etymological Studies in the Veda, Open Learning for All, Video 31, Chinmaya Vishwa Vidyapeetha, November 2021, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=q_AYmuaXA- 8&list=PLbQHD8oHpmE15FcG2rPWejiQnGL0TC_m2&index=32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Nagendra Pavana, Vyakarana – Linguistics from Vedas, Open Learning for All, Video 32, Chinmaya Vishwa Vidyapeetha, November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=Hi1yItWodw0&list=PLbQHD8oHpmE15FcG2rPWe jiQnGL0TC_m2&index=33 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Mallory, In Search of the Indo-Europeans, Language, Archaeology and Myth, Thames and Hudson, 1991, ISBN-13 : 978-0500276167 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' James Parsons, Remains of Japhet: : being historical enquiries into the affinity and origin of the European languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='1705-1770, https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='org/details/remainsofjaphetb00pars/page/n19/mode/2up 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Singh B (1995) The first Englishman in India Thomas Stephens (1547–1619).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' J South Asian Literature 30(1/2):146–161 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Pedro Redondo, Filippo Sassetti and Thomas Stephens in the beginnings of Indo- European linguistics, Academia Letters, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='20935/AL2158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Jones SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Discourses delivered before the Asiatic society;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' and miscellaneous papers, on the religion, poetry, literature, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', of the nations of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Arnold, Michigan, 1824 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Edwyn Bryant, The Quest for the Origins of Vedic Culture: The Indo-Aryan Migration Debate Paperback – Illustrated, 11 March 2004, OUP USA, ISBN-13 : 978-0195169478 Page 37 of 37 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Edwin Bryant and Laurie Patton,The Indo-Aryan Controversy Evidence and Inference in Indian History, Edited By Edwin Bryant, Laurie Patton, 2005, ISBN 9780203641880, Published August 2, 2004 by Routledge 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Satya Swaroop Mishra, The date of the Rigveda and the Indian Migration, Fresh Linguistic Evidence, ibid 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Michael Witzel, Indocentricism, Autochthonous Visions of Ancient India, ibid 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Kuiper, Selected Writings on Indian Linguistics and Philology, Leiden Studies in Indo-European, Volume: 8, 1997, ISBN: 978-90-420-0235-7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Diana L Eck, India A Sacred Geography, Harmony;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Reprint edition (26 March 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' ISBN-13 : 978-0385531924 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Gintaras Songaila Affinities between Vedic and Baltic Cultures | | Sangam Talks, Aug 22, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=-OlsA9KMf-0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Subhash Kak, Sanskrit and Ancient Migrations, 2021, Itihas Darpan, vol.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='in/noc21_hs12/preview 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Anuradha Chaudhary, Lecture 02: Sounds of Spoken Sanskrit: Its Alphabet, IIT Kharagpur, October 2018, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='v=UgVwzueOKRU&list=PLbRMhDVUMngfYG2GVf 2bQnIgsI0Y923g3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Shreekanth Prabhu, Annexure 1: Word Groups for Indian and European Languages, ResearchGate, January 2023, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='researchgate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='net/publication/367361269_Annexure_1_Word_Groups_for_In dian_and_European_Languages 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Shreekanth Prabhu, Annexure 2: Dravidian Theories, ResearchGate, January 2023, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='researchgate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='net/publication/367411879_Annexure_2_Dravidian_Theories 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Girdhar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Nayak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Prabhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Linguistic Classification Using Instance- Based Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' In: Saraswat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Sharma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Balachandran, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=', Bansal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' (eds) Congress on Intelligent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Lecture Notes on Data Engineering and Communications Technologies, vol 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' Springer, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} +page_content='1007/978-981-16-9113-3_63' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otFMT4oBgHgl3EQf7TEy/content/2301.12463v1.pdf'} diff --git a/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/2301.01418v1.pdf.txt b/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/2301.01418v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a8e6b62fe9949d6346ba522451687ab1b1ea0d0 --- /dev/null +++ b/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/2301.01418v1.pdf.txt @@ -0,0 +1,834 @@ +Magnetic light amplification by stimulated emission of +radiation in subwavelength systems of a dielectric cavity and +magnetic quantum emitters +Zhong-Jian Yang*, Xiao-Jing Du, Ma-Long Hu, and Jun He* +Hunan Key Laboratory of Nanophotonics and Devices, School of Physics and +Electronics, Central South University, Changsha 410083, China +*E-mail: zjyang@csu.edu.cn; junhe@csu.edu.cn + +Abstract: We propose a magnetic laser in a subwavelength system consisting of a +high-refractive-index dielectric cavity and an active medium formed by magnetic +quantum emitters. Stimulated emissions of magnetic quantum emitters induced by +their coherent interactions with quantized magnetic fields of a cavity are theoretically +considered. The condition to archive such a magnetic laser is obtained. Numerical +results show that magnetic lasers are feasible in some realistic systems, for example, a +silicon disk of high-quality whispering gallery modes with embedded emitters. +Furthermore, the competitions between the electric interaction and magnetic one in +terms of their Purcell factors are also considered in some magnetic laser achievable +systems. In a wavelength-scale silicon block of a high-order magnetic mode, the ratio +of magnetic Purcell factor to the electric one can reach more than ~103 large. Our +results open up ways to enhanced magnetic light-matter interactions. +Keywords: Magnetic, laser, dielectric cavity, magnetic quantum emitters, Purcell +factor, subwavelength. + +The interactions of the magnetic component of light and magnetic dipole (MD) +transitions at optical frequencies are usually several orders of lower than their electric +counterparts [1,2]. Hence, the light-matter interactions are generally interpreted as the +couplings of electric fields and electric dipoles. Nevertheless, the magnetic +interactions provide another dimension in glimpsing light-matter interactions [2-4]. +Furthermore, strong MD transitions of quantum emitters at optical frequencies are +indeed found in some lanthanide series ions such as Eu3+ and Er3+ [5-7]. The +interactions of these magnetic quantum emitters (MQEs) with light have been +attracting increasing research interests despite high technology requirements [4,8-12]. + Putting MQEs close to photonic structures could allow one to largely turn the +couplings between MQEs and light. The spontaneous decay rate enhancement, which +is also termed as Purcell factor [13-15], can be modified a lot [4]. Due to their high +near-field confinement, plasmonic nanostructures have been utilized to interact with +MQEs [4,16-18]. However, plasmonic systems suffer from high material losses, and +complex geometries are required to obtain effective magnetic responses. In the past +years, all-dielectric (sub)wavelength-scale structures with high-refractive indexes nr +have been found to exhibit Mie-like resonances [19,20], and they can readily support +magnetic near-field responses. Those magnetic responses provide a platform for +magnetic light-matter interactions [21-24]. The common resonant modes in the +reported subwavelength all-dielectric structures are low-order electromagnetic +multipoles such as electric/magnetic dipoles, and toroidal modes, supercavity mode +[19,20,25-28]. Most of these modes show low quality (Q) factors (~101) and low + +electromagnetic near field enhancement (~101) with a common material silicon (Si) +nr~3.5. Recently, it has been demonstrated that subwavelength dielectric resonators +(nr~3.5) can also support whispering gallery modes (WGMs) with high enough Q +factors (~105) and high electromagnetic near field enhancements (~102) [29]. These +achievements of all-dielectric magnetic cavities hold great promise to enable more +efficient magnetic photons-MQEs couplings [30,31]. +Here, we theoretically propose that a magnetic laser can be obtained in a +subwavelength system of a dielectric cavity and MQEs. MQEs are modeled as simple +two-level emitters with MD transitions. The MQEs can undergo stimulated emission +of radiation though coherent interactions with the photons in the cavity. The +stimulated emission is similar to that in common lasers or spasers [32-39] while the +couplings here are magnetic interactions instead of the electric ones. A subwavelength +dielectric disk with high-Q WGMs is numerically considered. The magnetic laser can +be obtained due to the high-Q features of the WGMs. Furthermore, we also consider +the competition between the magnetic interactions and electric ones in some magnetic +laser achievable systems. Specifically, the ratio of magnetic Purcell factor to the +electric one in a subwavelength silicon block of a high-Q magnetic mode can reaches +~103 large. This property makes such a kind of dielectric cavity a suitable platform to +carry out the magnetic light-matter interactions including a magnetic laser. +The magnetic field of a high-Q resonant mode of a dielectric cavity can be +quantized based on that of the standard harmonic oscillators [37,40-42]. A MQE is +taken as a two-level emitter with a matrix element of M +⃗⃗⃗ 10 for its MD transition. Then, + +the interaction Hamiltonian between a MQE and the quantized magnetic field under +the rotating-wave approximation can be expressed as +Hint = ћg(â𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟 )) + â+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟 ))), (1) +where 𝜔𝑛 is the frequency of the photon, and â+and â are the creation and +annihilation operators of a photon, respectively. 𝜎̂10 and 𝜎̂01 are transition operators +of the MQE. 𝜑(𝑟 ) represents the spatial phase of the magnetic field at the location 𝑟 . +g is the coupling strength g = √ +𝜇0𝜔𝑛 +2ћ𝑉𝑚 M +⃗⃗⃗ 10 ∙ +𝐵⃗ (𝑟 ) +𝐵𝑚𝑎𝑥 [42], where 𝑉𝑚 is the magnetic field +mode volume of the cavity 𝑉𝑚 = ∫ 𝜇0|𝐻⃗⃗ (𝑟 )|2𝑑3𝑟 +𝜇0𝐻𝑚𝑎𝑥 +2 +, 𝐵⃗ (𝑟 )/𝐵𝑚𝑎𝑥 (𝐻⃗⃗ (𝑟 )/𝐻𝑚𝑎𝑥) is the +normalized magnetic field of the cavity mode, ħ is reduced Planck constant, and 𝜇0 +is the permeability of vacuum. The analytical description is similar to that of electric +interactions [38,43-46], while the coupling strength g should be replaced by ge +=√ +𝜔𝑛 +2ћ𝜀0𝑉e 𝜇 10 ∙ +𝐸⃗ (𝑟 ) +𝐸𝑚𝑎𝑥 for an electric interaction. 𝜇 10 is the matrix element of the +electric dipole transition of an electric quantum emitter (EQE), 𝐸⃗ (𝑟 )/𝐸𝑚𝑎𝑥 is the +normalized electric field, 𝜀0 is the permittivity of vacuum, and 𝑉𝑒 is the +electric-field mode volume of a cavity mode. +Under Fermi's golden rule, the total emission rate into photons from a MQE can +be expressed as + Γ′ = 2π𝑔2 (𝑁𝑛 + 1) ∫ 𝐹(𝜔) +𝛾𝑛 +2 +(𝜔−𝜔𝑛)2+𝛾𝑛 +2 𝑑𝜔, (2) +where the term Nn+1 represents the contributions from the stimulated 𝛤st (Nn) and +spontaneous 𝛤sp (1) emissions. ∫ 𝐹(𝜔) +𝛾𝑛 +2 +(𝜔−𝜔𝑛)2+𝛾𝑛 +2 𝑑𝜔 is the spectral overlap factor, +where 𝐹(𝜔) is the normalized-to-1 spectrum of MD transitions, and γn is the +relaxation rate of the photon. 𝐹(𝜔) is highly related to the relaxation rate (𝛾10) of the + +MQE. For 𝛾10 ≪ γn, the overlap factor is 1. While for 𝛾10 ≫ γn, the overlap factor +becomes γn / 𝛾10. Note that we have assumed the resonant couplings between the +MQE and photons. The stimulated absorption rate 𝛤sa is equal to the stimulated +emission rate 𝛤sa = 𝛤st . For the couplings of many MQEs and photons, the +generation rate of photon number 𝑁𝑛 can be expressed as +𝑁𝑛̇ = ∫ 𝛤st𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟 + ∫ 𝛤sp 𝜌1(𝑟 )𝑑3𝑟 − 𝑁𝑛γ𝑛, (3) +where 𝜌𝑒𝑓𝑓(𝑟 ) = 𝜌1(𝑟 ) − 𝜌0(𝑟 ), 𝜌1(𝑟 ) and 𝜌0(𝑟 ) are the population densities of +MQEs in the excited and ground states, respectively. The rate equation for the +population of MQEs can be written as +∫ 𝜌̇𝑒𝑓𝑓(𝑟 )𝑑3𝑟 = ∫(𝑊01 − 𝛤sp)(𝜌1(𝑟 ) + 𝜌0(𝑟 ))𝑑3𝑟 − ∫(𝑊01 + 𝛤sp + +2𝛤st) 𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟, (4) +where 𝑊01is the pumping rate of the MQEs. +The coupling strength g is an important parameter that determines if a magnetic +laser can be realized in a system. Generally, g can be obtained by calculating the +mode volume based on numerical methods, for example, the finite difference time +domain (FDTD) simulations. Alternatively, the spontaneous decay rate enhancement +can be simulated by numerical methods. Then, g can also be obtained correspondingly. +In the FDTD simulations, the calculations are carried out with the condition of 𝛾10 ≪ +γn, and the directly simulated decay rate enhancement is 2π𝑔2/Γ1, where Γ1 is the +decay rate of a MQE in the medium of nr [31]. Γ1 = 𝑛𝑟 +3Γ0, where Γ0 is the vacuum +spontaneous decay rate of a MQE [4]. Thus, the magnetic Purcell factor is 𝛤sp/Γ0 = +2π𝑔2/Γ0 = 2π𝑔2𝑛𝑟 +3/Γ1. For simplicity, we assume that the coupling strength g of + +each MQE and cavity is the same. Based on Eq. (3), the magnetic laser can occur if +the system satisfies + 𝑁𝑒𝑓𝑓2𝜋𝑔2 > 𝛾𝑛 (5) +under the situation of 𝛾10 ≪ γn. While for the situation of 𝛾10 ≫ γn, the condition to +realize a magnetic laser becomes 𝑁𝑒𝑓𝑓2𝜋𝑔2 > 𝛾10. Here, 𝑁𝑒𝑓𝑓 = ∫ 𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟 is +the inversed total number of MQEs. For the rest of discussion, we will take the +situation of 𝛾10 ≪ γn unless specified. +A realistic dielectric cavity is considered as shown in Fig. 1(a). It is a Si disk +supporting high-Q subwavelength WGMs [29]. The radius and the height are both +630 nm. The geometry is chosen to match the wavelength region that the refractive +index of Si is around nr = 3.5. The resonance of a TE WGM of the azimuthal mode +index m = 7 occurs at λ = 1230 nm [42]. The Q-factor of this mode is 1.5 x 105 and +the corresponding relaxation rate is γn = 1.6 x 109 s-1. The magnetic Purcell factor +2π𝑔2/Γ0 can reach 1.65 x 105 [42]. We assume the maximum population +inversion +𝜌1 ≫ 𝜌0. + Thus, +𝑁𝑒𝑓𝑓 ≈ 𝑁𝑀𝑄𝐸 += +∫(𝜌1(𝑟 ) + 𝜌0(𝑟 ))𝑑3𝑟 +, +where +𝑁𝑀𝑄𝐸 represents the total number of MQEs. With the situation of low enough +temperature (𝛾10 ≪ γn) and a realistic value of Γ0 = 101 s-1 [7], the threshold number +of MQEs required to achieve a magnetic laser is 𝑁𝑀𝑄𝐸 +𝑡ℎ +≈ 𝛾𝑛/2𝜋𝑔2 ≈ 9.8 x 102. +Such a threshold value should be easily satisfied experimentally. The 𝑁𝑀𝑄𝐸 +𝑡ℎ for a +magnetic laser increases dramatically as the Q-factor of a cavity mode decreases +(𝑁𝑀𝑄𝐸 +𝑡ℎ ~1/𝑄2, Fig. 1(b)), and reaches more than ~108 for m = 3 (Q ≈ 150). For the +situation of 𝛾10 ≫ γn, the 𝑁𝑀𝑄𝐸 +𝑡ℎ becomes 𝑁𝑀𝑄𝐸 +𝑡ℎ += 𝛾10/2𝜋𝑔2 . This value is + +relatively 𝛾10/γn times larger than that under the situation of low enough temperature. + +Fig. 1. (a) Schematic of a magnetic laser system consisting of a subwavelength dielectric cavity +and active MQEs. Each MQE is a two-level emitter. (b) The 𝑁𝑀𝑄𝐸 +𝑡ℎ of a WGM-resonant Si cavity +with different mode m. The line is the fitting results with an exponential decay function. The +radius and the height are both 630 nm. (c) Normalized threshold number of MQEs 𝑁𝑀𝑄𝐸 +𝑡ℎ /𝑁𝑀𝑄𝐸 +𝑡ℎ0 +as a function of the size (diameter) ratio D/D0 (red line). Purcell factor 𝛤𝑠𝑝/ 𝛤0 as a function of +the size ratio D/D0 is shown by the gray line. D0 is a reference diameter of a disk with a threshold +number of 𝑁𝑀𝑄𝐸 +𝑡ℎ0 . +By enlarging the size of a disk cavity, it becomes relatively easier to obtain a +magnetic laser. The WGM response is a geometric resonance in a dielectric cavity. +Thus, the resonant wavelength λn of a WGM is in proportional to the disk diameter D +(λn ∝ D) [29], where the height/diameter ratio of a disk is always kept the same. The + +(a) +Subwavelength +dielectric cavity +1 +H个 +w10 +10) +k +Magnetic +emitter +(b) +10° +108 +MQE +Fitting +107 +MQE +106 +un +105 +104 +103 +102 +3 +4 +5 +6 +7 +m +(c) +1.0 +/Ntho +MQE +0.9 +sp/F。 +MQE +tho +0.8 +0.7 +0.6 +0.5 +0.4 +0 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Diameter ratio D/DQ-factor of each WGM remains the same with varying the disk size [42]. The +magnetic field distribution size increases proportionally with the disk size. Thus, the +mode volume Vm increases proportionally with the geometric volume of the disk +(Vm ∝ D3). By combining all these factors, the Purcell factor +𝛤sp +𝛤0 = +2𝜋𝑔2 +𝛤0 +∝ 𝑄𝜆𝑛 +3/𝑉𝑚 +[4] also remains unchanged with disk size (Fig. 1(c)). This is also confirmed by direct +simulations [42]. The decay rate of a WGM photon γn decreases with the disk size γn +∝1/D as the resonant frequency 𝜔𝑛 decreases with D (𝜔𝑛 ∝1/D) while the Q-factor +remains unchanged. Thus, the 𝑁𝑀𝑄𝐸 +𝑡ℎ to achieve a magnetic laser becomes relatively +smaller with disk size 𝑁𝑀𝑄𝐸 +𝑡ℎ += 𝛾𝑛/2𝜋𝑔2 ∝ 1/𝐷 (Fig. 1(c)). Here, we have assumed +a fixed 𝛤0 for simplicity. The above analysis is not restricted to a specific WGM. +Thus, the relation 𝑁𝑀𝑄𝐸 +𝑡ℎ +∝ 1/𝐷 with a fixed 𝛤0 applies for any WGM in the disk +system. Furthermore, a larger disk also provides more space to host the MQEs. This is +also an important profitable factor to realize such a system in experiment. +Now let us turn to the number of photons Nn of the magnetic laser system. At +first, Nn increases as the gain is larger than the loss. Then, Nn turns to be saturated +(denoted by 𝑁𝑛 +𝑚𝑎𝑥) when the gain equals to the loss. The 𝑁𝑛 +𝑚𝑎𝑥 can be obtained by +solving the Eqs. 3 and 4 under the steady-state conditions, namely, 𝜌̇𝑒𝑓𝑓 = 0 and +𝑁𝑛̇ = 0. The analytical expression of 𝑁𝑛 +𝑚𝑎𝑥 as a function of the system parameters is +complex [42]. There are two solutions for 𝑁𝑛 +𝑚𝑎𝑥, where the negative one is omitted. +The 𝑁𝑛 +𝑚𝑎𝑥 as a function of pumping rate for cases with different number of MQEs +𝑁𝑀𝑄𝐸 are shown in Fig. 2. When 𝑁𝑀𝑄𝐸 is more than several times larger than the +𝑁𝑀𝑄𝐸 +𝑡ℎ +, 𝑁𝑛 +𝑚𝑎𝑥 increases linearly with the pumping rate W01 (𝑁𝑛 +𝑚𝑎𝑥 ∝ (𝑁𝑀𝑄𝐸 − + +𝑁𝑀𝑄𝐸 +𝑡ℎ )W01/𝛤sp). For the case where 𝑁𝑀𝑄𝐸 is equal to the 𝑁𝑀𝑄𝐸 +𝑡ℎ , 𝑁𝑛 +𝑚𝑎𝑥shows a +square root function of the pumping rate 𝑁𝑛 +𝑚𝑎𝑥 = √ +𝑊01 +2𝛤sp + +1 +4 − +1 +4 . + +Fig. 2. The saturated number of photons 𝑁𝑛𝑚𝑎𝑥 as a function of the normalized pumping rate +W01/𝛤sp. The number of MQEs 𝑁𝑀𝑄𝐸 varies from 𝑁𝑀𝑄𝐸 +𝑡ℎ to 3𝑁𝑀𝑄𝐸 +𝑡ℎ . +We shall now investigate the coupling strengths of a magnetic interaction and an +electric interaction associated with a cavity mode. This is an important factor that +determines if a magnetic laser action can exceed an electric one. Note that Eq. (5) also +holds for the electric case while the 2𝜋𝑔2 term should be replaced by 2𝜋𝑔𝑒 +2 to +represent the electric interaction. The Purcell factor of an EQE can be expressed as +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒, where 𝛤𝑒 +𝑠𝑝 and 𝛤0 +𝑒 are the spontaneous decay rate of an EQE in a cavity +and vacuum, respectively. The ratio of the vacuum decay rate of an EQE to that of a +MQE 𝛤0 +𝑒/𝛤0 can reach several orders of magnitude for a common molecular, while it +can be much smaller for a rare-earth ion [5,6]. We assume resonant couplings for both +electric and magnetic interactions. Numerical calculations show that, the +𝛤sp/Γ0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 is ~ + +9 +xe +4 +N +2 +N += Nth +MQE +MQE +0 +12 +max +8 +N +4 +Nth +MQE +MQE +0 +40 +20 +N. +MQE +MQE +0 +90 +60 +Z +30 +N. +3Nth +MQE +MQE +0 +0 +20 +40 +60 +80 +100 +W../rsp101 for a WGM of nr = 3.5 [42]. This means that 𝛤0 +𝑒/𝛤0 should be smaller than ~101 +to make the magnetic interaction stronger than the electric one. This can be realistic +for rare-earth ions. If only the magnetic interaction is a resonant coupling. The +detuning of the nonresonant electric interaction is 𝜔 − 𝜔𝑛 = 𝑓𝛾𝑛. Based on Eq. (2), +the nonresonant decay rate of an emitter is 1/(f 2+1) times of the resonant one. This +will make the ratio +𝛤sp/Γ0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 become relatively (f 2+1) times larger. + One efficient way to further enlarge the emission ratio +𝛤sp/𝛤0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 can be considered +by putting an emitter inside a less symmetrical cavity. Here, we also assume resonant +couplings for both electric and magnetic interactions for simplicity. Figure 3(a) shows +a Si block cavity with an emitter at its center. The length, width and height are 1500, +1050 and 1050 nm, respectively. Here, the size of the cavity is also chosen to match +that nr is nr = 3.5. There is a high-order magnetic mode around λ = 1375 nm which is +around the size of the cavity (Figs. 3(b)-3(d)). This mode can be efficiently excited by +a y-polarized MQE with a magnetic Purcell factor of 𝛤sp/𝛤0 ≈ 1200. The Q factor is +about 1.5 x 103. On the other hand, if an EQE is placed at the same point. The +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 for an EQE polarized in x, y and z are only 2.5, 0.3 and 0.3, respectively. We +take an average value of 𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒≈ 1 for an EQE. The ratio +𝛤sp/Γ0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 can reaches ~103. +This means that the magnetic interaction can exceed the electric one if 𝛤0 +𝑒/𝛤0 of an +emitter is smaller than ~103. +𝛤sp/Γ0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 increases exponentially with nr and reaches ~105 +around nr = 5 (Fig. 3(e)). Based on Eq. (5), the 𝑁𝑀𝑄𝐸 +𝑡ℎ for the above magnetic mode +with nr = 3.5 is 𝑁𝑀𝑄𝐸 +𝑡ℎ ~ 106. This number is achievable in such a system. The Q +factor of the mode increases almost exponentially with nr. Thus, the 𝑁𝑀𝑄𝐸 +𝑡ℎ decreases + +almost exponentially with nr (Fig. 3(f)). It is also relatively beneficial to enlarge the +cavity size, and the discussion is the same as that in a WGM cavity (Fig. 1(c)). + +Fig. 3. Schematic of a dielectric block excited by a MQE. The origin of the coordinate system is +placed at the block center. (b) Spontaneous emission rate enhancement of a MQE (𝛤sp/𝛤0) and an +EQE (𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒). The polarization of the MQE is along y-axis. The polarization of the EQE is along +x-axis (EQE-1) or z-axis (EQE-2). The emitter is located at the block center in each case. nr = 3.5. +(c,d) The magnetic field distribution on the x-y (c) and x-z (d) plane of the MQE-excited block at +λ= 1375 nm. The arrows denote the main feature of the magnetic field directions. (e) The emission +ratio +𝛤sp/Γ0 +𝛤𝑒 +𝑠𝑝/𝛤0 +𝑒 as a function of the refractive index nr. The size of the block is the same as that in (b). +(f) Q-factor and 𝑁𝑀𝑄𝐸 +𝑡ℎ as a function of nr. +In conclusion, we have theoretically proposed that a magnetic laser can be +obtained through the stimulated emissions of MQEs in a subwavelength dielectric +cavity. The quantum treatment of such a hybrid system is carried out by considering +the interactions of quantized magnetic field and two-level MQEs. The magnetic laser +can be achieved in a subwavelength cavity based on the facts that the cavity can host +high-Q electromagnetic resonances with significant magnetic near field responses. + +(a) +(c) +ratio +2.0E5 +(e) +个y +emission +Simulation +1.5E5 +Fitting +MQE +Z +1.0E5 +5.0E4 +X +3.5 +4.0 +4.5 +5.0 +Refractive index n, +(b) +(d) +1200 +MQE +(f) +enhancer +EQE-1 +个Z +107 +EQE-2 +300 +factor +10 +106 +rate +200 +uo +Q +100 +Emissic +104 +1300 +1350 +1400 +1450 +1500 +103 +Wavelength(nm) +103 +3.5 +4.0 +4.5 +5.0 +Refractive index nThe saturated number of photons 𝑁𝑛 +𝑚𝑎𝑥 shows a linear relation with the pumping +rate when the number of MQEs is more than several times larger than its threshold +value. The competition between the the electric interaction and magnetic one in terms +of their spontaneous decay rate enhancements is also considered. 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Zhuo, R. B. Jiang, Z. G. Wang, J. F. Wang, and H. Q. Lin, Adv. Opt. Mater. 9, 12 +(2021). + + +Supplemental Material for +“Magnetic light amplification by stimulated +emission of radiation in subwavelength systems of +a dielectric cavity and magnetic quantum emitters” +Zhong-Jian Yang*, Xiao-Jing Du, Ma-Long Hu, and Jun He* +Hunan Key Laboratory of Nanophotonics and Devices, School of Physics and +Electronics, Central South University, Changsha 410083, China +*E-mail: zjyang@csu.edu.cn; junhe@csu.edu.cn + + +Part 1: The interaction Hamiltonian between a MQE and the quantized +magnetic field of a cavity. +The magnetic field of a high-Q electromagnetic mode in a cavity can be expressed as +𝐻⃗⃗ (𝑟, 𝑡) = a𝑄⃗ (𝑟)cos⁡(𝜔𝑛𝑡 + 𝜑(𝑟)) += +𝑎 +2 𝑄⃗ (𝑟)𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + +𝑎 +2 𝑄⃗ (𝑟)𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟)) (S1) +where 𝑄⃗ (𝑟) is a real function of r. The maximal magnetic field Hmax is assumed to +be Hmax = a, thus 𝑄⃗ (𝑟)= 𝐻⃗⃗ (𝑟, 𝑡) / Hmax = 𝐵⃗ (𝑟, 𝑡) / Bmax. The time averaged energy +can be expressed in terms of magnetic field as +Um = +1 +2 ∫ 𝜇0|𝐻⃗⃗ (𝑟)|2𝑑3𝑟 +=c2a2 (S2) +where c2 = +1 +2 ∫ 𝜇0|𝐻⃗⃗ (𝑟)|2𝑑3𝑟. The quantized Hamiltonian becomes the harmonic + +oscillator form [37,40,41] provided that a = +√ħ𝜔𝑛 +𝑐 +𝑎̂ and a*= +√ħ𝜔𝑛 +𝑐 +𝑎̂+. The quantized +magnetic field as a function of position and time can be written as +𝐻⃗⃗ (𝑟, 𝑡) = +√ħ𝜔𝑛 +2𝑐 𝑄⃗ (𝑟)𝑎̂𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + +√ħ𝜔𝑛 +2𝑐 𝑄⃗ (𝑟)𝑎̂+𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟)) . (S3) +The interaction Hamiltonian Hint = - 𝑀⃗⃗ ∙ 𝐵⃗ . With the second quantization and rotating +wave approximation, the Hint can be expressed as +Hint = - 𝜇0 +√ħ𝜔𝑛 +𝑐 +M +⃗⃗⃗ 10 ∙ 𝑄⃗ (𝑟)(𝑎̂𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + 𝑎̂+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟))) +=-ћg(𝑎̂⁡𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟 )) + 𝑎̂+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟 ))) (S4) +where g is the coupling strength +g = 𝜇0 +√ħ𝜔𝑛 +2𝑐 M +⃗⃗⃗ 10 ∙ 𝑄⃗ (𝑟) += 𝜇0 +√ħ𝜔𝑛 +2𝑐 M +⃗⃗⃗ 10 ∙ +𝐻⃗⃗ (𝑟) +𝐻𝑚𝑎𝑥 += 𝜇0 +√ħ𝜔𝑛 +2𝑐 M +⃗⃗⃗ 10 ∙ +𝐵⃗ (𝑟) +𝐵𝑚𝑎𝑥 (S5) +The coupling strength g can also be written in terms of the mode volume of a +magnetic mode. The mode volume of a magnetic mode can be expressed as +𝑉𝑚=∫ 𝜇0H2𝑑3𝑟 +𝜇0𝐻𝑚𝑎𝑥 +2 + +=⁡ +2c2 +𝜇0 (S6) +Thus, the coupling strength g can also be written as +g = √ +𝜇0𝜔𝑛 +2ħ𝑉𝑚 M +⃗⃗⃗ 10 ∙ +𝐻(𝑟) +𝐻𝑚𝑎𝑥 += √ +𝜇0𝜔𝑛 +2ћ𝑉𝑚 M +⃗⃗⃗ 10 ∙ +𝐵(𝑟) +𝐵𝑚𝑎𝑥 (S7) + + + + + +Part 2: Figs. S1-S3 + +FIG. S1. (a) Schematic of a dielectric disk under TE excitation. The radius and the height are both +630 nm. The refractive index is nr = 3.5. (b) The scattering spectrum under TE excitation. + + +FIG. S2. (a) The magnetic field distribution at the y-z plane of the disk. The disk is the same as +that in Figs. 1(b) and S1. The disk is excited by a MQE (denoted by the green point) located at a +place with 470 nm to the disk center. (b) The magnetic Purcell factors for different modes (red +dots). The electric Purcell factors with an EQE as the excitation are also shown (black dots). + + +(a) +(b) +m=5 +8 +Q~ 4.7x103 +m=3 +m =7 +Q~ 150 +E +Q~1.5x105 +6 +k +m = 4 +Q~745 +Scattering +2 +m=6 +Q ~ 3.2x104 +0 +1200 1400 1600 1800 2000 2200 +Wavelength (nm)(a) +(b) +EQE +105 +-MQE +Purcell factor +104 +0 +103 +102 +101 +100 +3 +4 +5 +6 +7 +m +Fig. S3. Simulated (a) Q factor and (b) magnetic Purcell factor as a function of the disk size +(radius). The height is always kept the same the radius in each case. + +Part 3: The steady-state solution for the saturated number of photons. +𝑁𝑛 +𝑚𝑎𝑥 = +1 +4 − 𝑊01 +4𝛤sp + 𝑊01𝑁𝑀𝑄𝐸 +4 𝑛 ++ [ 1 +16 + 𝑊01 +8𝛤sp + 3𝑊01𝑁𝑀𝑄𝐸 +8 𝑛 ++ ( 𝑊01 +4𝛤sp − 𝑊01𝑁𝑀𝑄𝐸 +4 𝑛 +)2]1/2 + (S8) +When 𝑁𝑀𝑄𝐸 is several times larger than 𝑁𝑀𝑄𝐸 +𝑡ℎ +( 𝑁𝑀𝑄𝐸 +𝑡ℎ += + 𝑛 +𝛤sp ), the expression +[ +1 +16 + +𝑊01 +8𝛤sp + +3𝑊01𝑁𝑀𝑄𝐸 +8 𝑛 ++ ( +𝑊01 +4𝛤sp − +𝑊01𝑁𝑀𝑄𝐸 +4 𝑛 +) +2 +] in Eq. (S8) is dominated by the +( +𝑊01 +4𝛤sp − +𝑊01𝑁𝑀𝑄𝐸 +4 𝑛 +)2term. Thus, the 𝑁𝑛 +𝑚𝑎𝑥 becomes to be 𝑁𝑛 +𝑚𝑎𝑥 ≈ +1 +4+ +(𝑁𝑀𝑄𝐸−𝑁𝑀𝑄𝐸 +𝑡ℎ +)𝑊01 +2𝛤sp +. +When 𝑁𝑀𝑄𝐸 is equal to 𝑁𝑀𝑄𝐸 +𝑡ℎ , the − +𝑊01 +4𝛤sp + +𝑊01𝑁𝑀𝑄𝐸 +4 𝑛 + term becomes 0. Thus, 𝑁𝑛 +𝑚𝑎𝑥 +becomes 𝑁𝑛 +𝑚𝑎𝑥 = √ +𝑊01 +2𝛤sp + +1 +4 − +1 +4, namely, 𝑁𝑛 +𝑚𝑎𝑥 ∝ √ +𝑊01 +𝛤sp. + +References: +[37] D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 (2003). +[40] P. W. Milonni, J. Mod. Opt. 42, 1991 (1995). +[41] R. Loudon, The Quantum Theory of Light (Oxford University Press, London, 1983). + + +2.0 +2.0 +(a) +(b) +1.8 +.8 +factor (105 +Purcell factor ( +1.6 +1.6 +Q +1.4 +1.4 +600 +700 +800 +900 +1000 +600 +700 +800 +900 +1000 +Radius of disk (nm) +Radius of disk (nm) \ No newline at end of file diff --git a/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/load_file.txt b/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa99d8579f8a500e81520094c81db0bb99865986 --- /dev/null +++ b/vNAzT4oBgHgl3EQfdPyu/content/tmp_files/load_file.txt @@ -0,0 +1,829 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf,len=828 +page_content='Magnetic light amplification by stimulated emission of radiation in subwavelength systems of a dielectric cavity and magnetic quantum emitters Zhong-Jian Yang*, Xiao-Jing Du, Ma-Long Hu, and Jun He* Hunan Key Laboratory of Nanophotonics and Devices, School of Physics and Electronics, Central South University, Changsha 410083, China E-mail: zjyang@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' junhe@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='cn Abstract: We propose a magnetic laser in a subwavelength system consisting of a high-refractive-index dielectric cavity and an active medium formed by magnetic quantum emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Stimulated emissions of magnetic quantum emitters induced by their coherent interactions with quantized magnetic fields of a cavity are theoretically considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The condition to archive such a magnetic laser is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Numerical results show that magnetic lasers are feasible in some realistic systems, for example, a silicon disk of high-quality whispering gallery modes with embedded emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Furthermore, the competitions between the electric interaction and magnetic one in terms of their Purcell factors are also considered in some magnetic laser achievable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' In a wavelength-scale silicon block of a high-order magnetic mode, the ratio of magnetic Purcell factor to the electric one can reach more than ~103 large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Our results open up ways to enhanced magnetic light-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Keywords: Magnetic, laser, dielectric cavity, magnetic quantum emitters, Purcell factor, subwavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The interactions of the magnetic component of light and magnetic dipole (MD) transitions at optical frequencies are usually several orders of lower than their electric counterparts [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Hence, the light-matter interactions are generally interpreted as the couplings of electric fields and electric dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Nevertheless, the magnetic interactions provide another dimension in glimpsing light-matter interactions [2-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Furthermore, strong MD transitions of quantum emitters at optical frequencies are indeed found in some lanthanide series ions such as Eu3+ and Er3+ [5-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The interactions of these magnetic quantum emitters (MQEs) with light have been attracting increasing research interests despite high technology requirements [4,8-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Putting MQEs close to photonic structures could allow one to largely turn the couplings between MQEs and light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The spontaneous decay rate enhancement, which is also termed as Purcell factor [13-15], can be modified a lot [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Due to their high near-field confinement, plasmonic nanostructures have been utilized to interact with MQEs [4,16-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' However, plasmonic systems suffer from high material losses, and complex geometries are required to obtain effective magnetic responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' In the past years, all-dielectric (sub)wavelength-scale structures with high-refractive indexes nr have been found to exhibit Mie-like resonances [19,20], and they can readily support magnetic near-field responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Those magnetic responses provide a platform for magnetic light-matter interactions [21-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The common resonant modes in the reported subwavelength all-dielectric structures are low-order electromagnetic multipoles such as electric/magnetic dipoles, and toroidal modes, supercavity mode [19,20,25-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Most of these modes show low quality (Q) factors (~101) and low electromagnetic near field enhancement (~101) with a common material silicon (Si) nr~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Recently, it has been demonstrated that subwavelength dielectric resonators (nr~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5) can also support whispering gallery modes (WGMs) with high enough Q factors (~105) and high electromagnetic near field enhancements (~102) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' These achievements of all-dielectric magnetic cavities hold great promise to enable more efficient magnetic photons-MQEs couplings [30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Here, we theoretically propose that a magnetic laser can be obtained in a subwavelength system of a dielectric cavity and MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' MQEs are modeled as simple two-level emitters with MD transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The MQEs can undergo stimulated emission of radiation though coherent interactions with the photons in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The stimulated emission is similar to that in common lasers or spasers [32-39] while the couplings here are magnetic interactions instead of the electric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' A subwavelength dielectric disk with high-Q WGMs is numerically considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic laser can be obtained due to the high-Q features of the WGMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Furthermore, we also consider the competition between the magnetic interactions and electric ones in some magnetic laser achievable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Specifically, the ratio of magnetic Purcell factor to the electric one in a subwavelength silicon block of a high-Q magnetic mode can reaches ~103 large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This property makes such a kind of dielectric cavity a suitable platform to carry out the magnetic light-matter interactions including a magnetic laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic field of a high-Q resonant mode of a dielectric cavity can be quantized based on that of the standard harmonic oscillators [37,40-42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' A MQE is taken as a two-level emitter with a matrix element of M ⃗⃗⃗ 10 for its MD transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Then, the interaction Hamiltonian between a MQE and the quantized magnetic field under the rotating-wave approximation can be expressed as Hint = ћg(â𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟 )) + â+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟 ))), (1) where 𝜔𝑛 is the frequency of the photon, and â+and â are the creation and annihilation operators of a photon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝜎̂10 and 𝜎̂01 are transition operators of the MQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝜑(𝑟 ) represents the spatial phase of the magnetic field at the location 𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' g is the coupling strength g = √ 𝜇0𝜔𝑛 2ћ𝑉𝑚 M ⃗⃗⃗ 10 ∙ 𝐵⃗ (𝑟 ) 𝐵𝑚𝑎𝑥 [42], where 𝑉𝑚 is the magnetic field mode volume of the cavity 𝑉𝑚 = ∫ 𝜇0|𝐻⃗⃗ (𝑟 )|2𝑑3𝑟 𝜇0𝐻𝑚𝑎𝑥 2 , 𝐵⃗ (𝑟 )/𝐵𝑚𝑎𝑥 (𝐻⃗⃗ (𝑟 )/𝐻𝑚𝑎𝑥) is the normalized magnetic field of the cavity mode, ħ is reduced Planck constant, and 𝜇0 is the permeability of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The analytical description is similar to that of electric interactions [38,43-46], while the coupling strength g should be replaced by ge =√ 𝜔𝑛 2ћ𝜀0𝑉e 𝜇 10 ∙ 𝐸⃗ (𝑟 ) 𝐸𝑚𝑎𝑥 for an electric interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝜇 10 is the matrix element of the electric dipole transition of an electric quantum emitter (EQE), 𝐸⃗ (𝑟 )/𝐸𝑚𝑎𝑥 is the normalized electric field, 𝜀0 is the permittivity of vacuum, and 𝑉𝑒 is the electric-field mode volume of a cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=" Under Fermi's golden rule, the total emission rate into photons from a MQE can be expressed as Γ′ = 2π𝑔2 (𝑁𝑛 + 1) ∫ 𝐹(𝜔) 𝛾𝑛 2 (𝜔−𝜔𝑛)2+𝛾𝑛 2 𝑑𝜔, (2) where the term Nn+1 represents the contributions from the stimulated 𝛤st (Nn) and spontaneous 𝛤sp (1) emissions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' ∫ 𝐹(𝜔) 𝛾𝑛 2 (𝜔−𝜔𝑛)2+𝛾𝑛 2 𝑑𝜔 is the spectral overlap factor, where 𝐹(𝜔) is the normalized-to-1 spectrum of MD transitions, and γn is the relaxation rate of the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝐹(𝜔) is highly related to the relaxation rate (𝛾10) of the MQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For 𝛾10 ≪ γn, the overlap factor is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' While for 𝛾10 ≫ γn, the overlap factor becomes γn / 𝛾10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Note that we have assumed the resonant couplings between the MQE and photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The stimulated absorption rate 𝛤sa is equal to the stimulated emission rate 𝛤sa = 𝛤st .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For the couplings of many MQEs and photons, the generation rate of photon number 𝑁𝑛 can be expressed as 𝑁𝑛̇ = ∫ 𝛤st𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟 + ∫ 𝛤sp 𝜌1(𝑟 )𝑑3𝑟 − 𝑁𝑛γ𝑛, (3) where 𝜌𝑒𝑓𝑓(𝑟 ) = 𝜌1(𝑟 ) − 𝜌0(𝑟 ), 𝜌1(𝑟 ) and 𝜌0(𝑟 ) are the population densities of MQEs in the excited and ground states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The rate equation for the population of MQEs can be written as ∫ 𝜌̇𝑒𝑓𝑓(𝑟 )𝑑3𝑟 = ∫(𝑊01 − 𝛤sp)(𝜌1(𝑟 ) + 𝜌0(𝑟 ))𝑑3𝑟 − ∫(𝑊01 + 𝛤sp + 2𝛤st) 𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟, (4) where 𝑊01is the pumping rate of the MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The coupling strength g is an important parameter that determines if a magnetic laser can be realized in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Generally, g can be obtained by calculating the mode volume based on numerical methods, for example, the finite difference time domain (FDTD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Alternatively, the spontaneous decay rate enhancement can be simulated by numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Then, g can also be obtained correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' In the FDTD simulations, the calculations are carried out with the condition of 𝛾10 ≪ γn, and the directly simulated decay rate enhancement is 2π𝑔2/Γ1, where Γ1 is the decay rate of a MQE in the medium of nr [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Γ1 = 𝑛𝑟 3Γ0, where Γ0 is the vacuum spontaneous decay rate of a MQE [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the magnetic Purcell factor is 𝛤sp/Γ0 = 2π𝑔2/Γ0 = 2π𝑔2𝑛𝑟 3/Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For simplicity, we assume that the coupling strength g of each MQE and cavity is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (3), the magnetic laser can occur if the system satisfies 𝑁𝑒𝑓𝑓2𝜋𝑔2 > 𝛾𝑛 (5) under the situation of 𝛾10 ≪ γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' While for the situation of 𝛾10 ≫ γn, the condition to realize a magnetic laser becomes 𝑁𝑒𝑓𝑓2𝜋𝑔2 > 𝛾10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Here, 𝑁𝑒𝑓𝑓 = ∫ 𝜌𝑒𝑓𝑓(𝑟 )𝑑3𝑟 is the inversed total number of MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For the rest of discussion, we will take the situation of 𝛾10 ≪ γn unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' A realistic dielectric cavity is considered as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' It is a Si disk supporting high-Q subwavelength WGMs [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The radius and the height are both 630 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The geometry is chosen to match the wavelength region that the refractive index of Si is around nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The resonance of a TE WGM of the azimuthal mode index m = 7 occurs at λ = 1230 nm [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The Q-factor of this mode is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 x 105 and the corresponding relaxation rate is γn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='6 x 109 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic Purcell factor 2π𝑔2/Γ0 can reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='65 x 105 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' We assume the maximum population inversion 𝜌1 ≫ 𝜌0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, 𝑁𝑒𝑓𝑓 ≈ 𝑁𝑀𝑄𝐸 = ∫(𝜌1(𝑟 ) + 𝜌0(𝑟 ))𝑑3𝑟 , where 𝑁𝑀𝑄𝐸 represents the total number of MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' With the situation of low enough temperature (𝛾10 ≪ γn) and a realistic value of Γ0 = 101 s-1 [7], the threshold number of MQEs required to achieve a magnetic laser is 𝑁𝑀𝑄𝐸 𝑡ℎ ≈ 𝛾𝑛/2𝜋𝑔2 ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='8 x 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Such a threshold value should be easily satisfied experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The 𝑁𝑀𝑄𝐸 𝑡ℎ for a magnetic laser increases dramatically as the Q-factor of a cavity mode decreases (𝑁𝑀𝑄𝐸 𝑡ℎ ~1/𝑄2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(b)), and reaches more than ~108 for m = 3 (Q ≈ 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For the situation of 𝛾10 ≫ γn, the 𝑁𝑀𝑄𝐸 𝑡ℎ becomes 𝑁𝑀𝑄𝐸 𝑡ℎ = 𝛾10/2𝜋𝑔2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This value is relatively 𝛾10/γn times larger than that under the situation of low enough temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (a) Schematic of a magnetic laser system consisting of a subwavelength dielectric cavity and active MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Each MQE is a two-level emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (b) The 𝑁𝑀𝑄𝐸 𝑡ℎ of a WGM-resonant Si cavity with different mode m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The line is the fitting results with an exponential decay function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The radius and the height are both 630 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (c) Normalized threshold number of MQEs 𝑁𝑀𝑄𝐸 𝑡ℎ /𝑁𝑀𝑄𝐸 𝑡ℎ0 as a function of the size (diameter) ratio D/D0 (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Purcell factor 𝛤𝑠𝑝/ 𝛤0 as a function of the size ratio D/D0 is shown by the gray line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' D0 is a reference diameter of a disk with a threshold number of 𝑁𝑀𝑄𝐸 𝑡ℎ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' By enlarging the size of a disk cavity, it becomes relatively easier to obtain a magnetic laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The WGM response is a geometric resonance in a dielectric cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the resonant wavelength λn of a WGM is in proportional to the disk diameter D (λn ∝ D) [29], where the height/diameter ratio of a disk is always kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The (a) Subwavelength dielectric cavity 1 H个 w10 10) k Magnetic emitter (b) 10° 108 MQE Fitting 107 MQE 106 un 105 104 103 102 3 4 5 6 7 m (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 /Ntho MQE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='9 sp/F。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' MQE tho 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='4 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 Diameter ratio D/DQ-factor of each WGM remains the same with varying the disk size [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic field distribution size increases proportionally with the disk size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the mode volume Vm increases proportionally with the geometric volume of the disk (Vm ∝ D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' By combining all these factors, the Purcell factor 𝛤sp 𝛤0 = 2𝜋𝑔2 𝛤0 ∝ 𝑄𝜆𝑛 3/𝑉𝑚 [4] also remains unchanged with disk size (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This is also confirmed by direct simulations [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The decay rate of a WGM photon γn decreases with the disk size γn ∝1/D as the resonant frequency 𝜔𝑛 decreases with D (𝜔𝑛 ∝1/D) while the Q-factor remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the 𝑁𝑀𝑄𝐸 𝑡ℎ to achieve a magnetic laser becomes relatively smaller with disk size 𝑁𝑀𝑄𝐸 𝑡ℎ = 𝛾𝑛/2𝜋𝑔2 ∝ 1/𝐷 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Here, we have assumed a fixed 𝛤0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The above analysis is not restricted to a specific WGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the relation 𝑁𝑀𝑄𝐸 𝑡ℎ ∝ 1/𝐷 with a fixed 𝛤0 applies for any WGM in the disk system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Furthermore, a larger disk also provides more space to host the MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This is also an important profitable factor to realize such a system in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Now let us turn to the number of photons Nn of the magnetic laser system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' At first, Nn increases as the gain is larger than the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Then, Nn turns to be saturated (denoted by 𝑁𝑛 𝑚𝑎𝑥) when the gain equals to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The 𝑁𝑛 𝑚𝑎𝑥 can be obtained by solving the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3 and 4 under the steady-state conditions, namely, 𝜌̇𝑒𝑓𝑓 = 0 and 𝑁𝑛̇ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The analytical expression of 𝑁𝑛 𝑚𝑎𝑥 as a function of the system parameters is complex [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' There are two solutions for 𝑁𝑛 𝑚𝑎𝑥, where the negative one is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The 𝑁𝑛 𝑚𝑎𝑥 as a function of pumping rate for cases with different number of MQEs 𝑁𝑀𝑄𝐸 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' When 𝑁𝑀𝑄𝐸 is more than several times larger than the 𝑁𝑀𝑄𝐸 𝑡ℎ , 𝑁𝑛 𝑚𝑎𝑥 increases linearly with the pumping rate W01 (𝑁𝑛 𝑚𝑎𝑥 ∝ (𝑁𝑀𝑄𝐸 − 𝑁𝑀𝑄𝐸 𝑡ℎ )W01/𝛤sp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' For the case where 𝑁𝑀𝑄𝐸 is equal to the 𝑁𝑀𝑄𝐸 𝑡ℎ , 𝑁𝑛 𝑚𝑎𝑥shows a square root function of the pumping rate 𝑁𝑛 𝑚𝑎𝑥 = √ 𝑊01 2𝛤sp + 1 4 − 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The saturated number of photons 𝑁𝑛𝑚𝑎𝑥 as a function of the normalized pumping rate W01/𝛤sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The number of MQEs 𝑁𝑀𝑄𝐸 varies from 𝑁𝑀𝑄𝐸 𝑡ℎ to 3𝑁𝑀𝑄𝐸 𝑡ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' We shall now investigate the coupling strengths of a magnetic interaction and an electric interaction associated with a cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This is an important factor that determines if a magnetic laser action can exceed an electric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (5) also holds for the electric case while the 2𝜋𝑔2 term should be replaced by 2𝜋𝑔𝑒 2 to represent the electric interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The Purcell factor of an EQE can be expressed as 𝛤𝑒 𝑠𝑝/𝛤0 𝑒, where 𝛤𝑒 𝑠𝑝 and 𝛤0 𝑒 are the spontaneous decay rate of an EQE in a cavity and vacuum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The ratio of the vacuum decay rate of an EQE to that of a MQE 𝛤0 𝑒/𝛤0 can reach several orders of magnitude for a common molecular, while it can be much smaller for a rare-earth ion [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' We assume resonant couplings for both electric and magnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Numerical calculations show that, the 𝛤sp/Γ0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 is ~ 9 xe 4 N 2 N = Nth MQE MQE 0 12 max 8 N 4 Nth MQE MQE 0 40 20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' MQE MQE 0 90 60 Z 30 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3Nth MQE MQE 0 0 20 40 60 80 100 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='./rsp101 for a WGM of nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This means that 𝛤0 𝑒/𝛤0 should be smaller than ~101 to make the magnetic interaction stronger than the electric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This can be realistic for rare-earth ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' If only the magnetic interaction is a resonant coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The detuning of the nonresonant electric interaction is 𝜔 − 𝜔𝑛 = 𝑓𝛾𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (2), the nonresonant decay rate of an emitter is 1/(f 2+1) times of the resonant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This will make the ratio 𝛤sp/Γ0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 become relatively (f 2+1) times larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' One efficient way to further enlarge the emission ratio 𝛤sp/𝛤0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 can be considered by putting an emitter inside a less symmetrical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Here, we also assume resonant couplings for both electric and magnetic interactions for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Figure 3(a) shows a Si block cavity with an emitter at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The length, width and height are 1500, 1050 and 1050 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Here, the size of the cavity is also chosen to match that nr is nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' There is a high-order magnetic mode around λ = 1375 nm which is around the size of the cavity (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3(b)-3(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This mode can be efficiently excited by a y-polarized MQE with a magnetic Purcell factor of 𝛤sp/𝛤0 ≈ 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The Q factor is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 x 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' On the other hand, if an EQE is placed at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 for an EQE polarized in x, y and z are only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' We take an average value of 𝛤𝑒 𝑠𝑝/𝛤0 𝑒≈ 1 for an EQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The ratio 𝛤sp/Γ0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 can reaches ~103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This means that the magnetic interaction can exceed the electric one if 𝛤0 𝑒/𝛤0 of an emitter is smaller than ~103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝛤sp/Γ0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 increases exponentially with nr and reaches ~105 around nr = 5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (5), the 𝑁𝑀𝑄𝐸 𝑡ℎ for the above magnetic mode with nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 is 𝑁𝑀𝑄𝐸 𝑡ℎ ~ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' This number is achievable in such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The Q factor of the mode increases almost exponentially with nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the 𝑁𝑀𝑄𝐸 𝑡ℎ decreases almost exponentially with nr (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' It is also relatively beneficial to enlarge the cavity size, and the discussion is the same as that in a WGM cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Schematic of a dielectric block excited by a MQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The origin of the coordinate system is placed at the block center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (b) Spontaneous emission rate enhancement of a MQE (𝛤sp/𝛤0) and an EQE (𝛤𝑒 𝑠𝑝/𝛤0 𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The polarization of the MQE is along y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The polarization of the EQE is along x-axis (EQE-1) or z-axis (EQE-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The emitter is located at the block center in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (c,d) The magnetic field distribution on the x-y (c) and x-z (d) plane of the MQE-excited block at λ= 1375 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The arrows denote the main feature of the magnetic field directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (e) The emission ratio 𝛤sp/Γ0 𝛤𝑒 𝑠𝑝/𝛤0 𝑒 as a function of the refractive index nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The size of the block is the same as that in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (f) Q-factor and 𝑁𝑀𝑄𝐸 𝑡ℎ as a function of nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' In conclusion, we have theoretically proposed that a magnetic laser can be obtained through the stimulated emissions of MQEs in a subwavelength dielectric cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The quantum treatment of such a hybrid system is carried out by considering the interactions of quantized magnetic field and two-level MQEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic laser can be achieved in a subwavelength cavity based on the facts that the cavity can host high-Q electromagnetic resonances with significant magnetic near field responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (a) (c) ratio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0E5 (e) 个y emission Simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5E5 Fitting MQE Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0E5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0E4 X 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 Refractive index n, (b) (d) 1200 MQE (f) enhancer EQE-1 个Z 107 EQE-2 300 factor 10 106 rate 200 uo Q 100 Emissic 104 1300 1350 1400 1450 1500 103 Wavelength(nm) 103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 Refractive index nThe saturated number of photons 𝑁𝑛 𝑚𝑎𝑥 shows a linear relation with the pumping rate when the number of MQEs is more than several times larger than its threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The competition between the the electric interaction and magnetic one in terms of their spontaneous decay rate enhancements is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' In a wavelength-scale block cavity, their Purcell factor ratio can reach more than ~103 large (nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5) due to the location dependent emission properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The widely developed fabrications of combined systems of dielectric structures and rare-earth ions may provide technical support for realizing our proposed magnetic laser in experiments [47-52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Our results will enrich the laser field and could find important applications in enhanced magnetic light-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' ACKNOWLEDGMENTS This paper was supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 11704416), the Hunan Provincial Natural Science Foundation of China (No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Noginov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Staude, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kivshar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Noginova, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Neshev, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 40, 1659 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Shu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Meng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Hong, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, Photonics Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 10, 2032 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kuznetsov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} 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3481 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [24] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' He, Photonics Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 7, 1142 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [25] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Koshelev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kruk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Melik-Gaykazyan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Choi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Bogdanov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Park, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kivshar, Science 367, 288 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Evlyukhin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Bakker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Chipouline, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kuznetsov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Luk’yanchuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Chichkov, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kivshar, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 6, 8069 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Rahmani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Neshev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Miroshnichenko, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Photonics 3, 9 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [29] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} 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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Express 14, 082004 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Du, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' He, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, Phys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, Laser Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 7, 60 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [33] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Unterhinninghofen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Wiersig, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Pflugl, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Boca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Boozer, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Kimble, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' B-At.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Dong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Shi, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Zhang, ACS Photonics 9, 2956 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Emmanuele, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Maciejczyk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Smith, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Cheng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Masson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Gosztola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Hla, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Robertson, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Ma, ACS Photonics 9, 2315 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [51] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Dong, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Barillaro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Qiu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Yang, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 121, 48 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [52] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Zhuo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Jiang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Lin, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 9, 12 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Supplemental Material for “Magnetic light amplification by stimulated emission of radiation in subwavelength systems of a dielectric cavity and magnetic quantum emitters” Zhong-Jian Yang*, Xiao-Jing Du, Ma-Long Hu, and Jun He* Hunan Key Laboratory of Nanophotonics and Devices, School of Physics and Electronics, Central South University, Changsha 410083, China E-mail: zjyang@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' junhe@csu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='cn Part 1: The interaction Hamiltonian between a MQE and the quantized magnetic field of a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The magnetic field of a high-Q electromagnetic mode in a cavity can be expressed as 𝐻⃗⃗ (𝑟, 𝑡) = a𝑄⃗ (𝑟)cos\u2061(𝜔𝑛𝑡 + 𝜑(𝑟)) = 𝑎 2 𝑄⃗ (𝑟)𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + 𝑎 2 𝑄⃗ (𝑟)𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟)) (S1) where 𝑄⃗ (𝑟) is a real function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The maximal magnetic field Hmax is assumed to be Hmax = a, thus 𝑄⃗ (𝑟)= 𝐻⃗⃗ (𝑟, 𝑡) / Hmax = 𝐵⃗ (𝑟, 𝑡) / Bmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The time averaged energy can be expressed in terms of magnetic field as Um = 1 2 ∫ 𝜇0|𝐻⃗⃗ (𝑟)|2𝑑3𝑟 =c2a2 (S2) where c2 = 1 2 ∫ 𝜇0|𝐻⃗⃗ (𝑟)|2𝑑3𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The quantized Hamiltonian becomes the harmonic oscillator form [37,40,41] provided that a = √ħ𝜔𝑛 𝑐 𝑎̂ and a*= √ħ𝜔𝑛 𝑐 𝑎̂+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The quantized magnetic field as a function of position and time can be written as 𝐻⃗⃗ (𝑟, 𝑡) = √ħ𝜔𝑛 2𝑐 𝑄⃗ (𝑟)𝑎̂𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + √ħ𝜔𝑛 2𝑐 𝑄⃗ (𝑟)𝑎̂+𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (S3) The interaction Hamiltonian Hint = - 𝑀⃗⃗ ∙ 𝐵⃗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' With the second quantization and rotating wave approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' the Hint can be expressed as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='Hint = - 𝜇0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='√ħ𝜔𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='⃗⃗⃗ 10 ∙ 𝑄⃗ (𝑟)(𝑎̂𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟)) + 𝑎̂+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='=-ћg(𝑎̂\u2061𝜎̂10𝑒−𝑖(𝜔𝑛𝑡+𝜑(𝑟 )) + 𝑎̂+𝜎̂01𝑒𝑖(𝜔𝑛𝑡+𝜑(𝑟 ))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='(S4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='where g is the coupling strength ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='g = 𝜇0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='√ħ𝜔𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='2𝑐 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='⃗⃗⃗ 10 ∙ 𝑄⃗ (𝑟) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='= 𝜇0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='√ħ𝜔𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='2𝑐 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='⃗⃗⃗ 10 ∙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='𝐻⃗⃗ (𝑟) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='𝐻𝑚𝑎𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='= 𝜇0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='√ħ𝜔𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='2𝑐 M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='⃗⃗⃗ 10 ∙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='𝐵⃗ (𝑟) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='𝐵𝑚𝑎𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='(S5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='The coupling strength g can also be written in terms of the mode volume of a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='magnetic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The mode volume of a magnetic mode can be expressed as 𝑉𝑚=∫ 𝜇0H2𝑑3𝑟 𝜇0𝐻𝑚𝑎𝑥 2 =\u2061 2c2 𝜇0 (S6) Thus, the coupling strength g can also be written as g = √ 𝜇0𝜔𝑛 2ħ𝑉𝑚 M ⃗⃗⃗ 10 ∙ 𝐻(𝑟) 𝐻𝑚𝑎𝑥 = √ 𝜇0𝜔𝑛 2ћ𝑉𝑚 M ⃗⃗⃗ 10 ∙ 𝐵(𝑟) 𝐵𝑚𝑎𝑥 (S7) Part 2: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S1-S3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (a) Schematic of a dielectric disk under TE excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The radius and the height are both 630 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The refractive index is nr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (b) The scattering spectrum under TE excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (a) The magnetic field distribution at the y-z plane of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The disk is the same as that in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 1(b) and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The disk is excited by a MQE (denoted by the green point) located at a place with 470 nm to the disk center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (b) The magnetic Purcell factors for different modes (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The electric Purcell factors with an EQE as the excitation are also shown (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (a) (b) m=5 8 Q~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='7x103 m=3 m =7 Q~ 150 E Q~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='5x105 6 k m = 4 Q~745 Scattering 2 m=6 Q ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='2x104 0 1200 1400 1600 1800 2000 2200 Wavelength (nm)(a) (b) EQE 105 MQE Purcell factor 104 0 103 102 101 100 3 4 5 6 7 m Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Simulated (a) Q factor and (b) magnetic Purcell factor as a function of the disk size (radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' The height is always kept the same the radius in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Part 3: The steady-state solution for the saturated number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 𝑁𝑛 𝑚𝑎𝑥 = 1 4 − 𝑊01 4𝛤sp + 𝑊01𝑁𝑀𝑄𝐸 4 𝑛 + [ 1 16 + 𝑊01 8𝛤sp + 3𝑊01𝑁𝑀𝑄𝐸 8 𝑛 + ( 𝑊01 4𝛤sp − 𝑊01𝑁𝑀𝑄𝐸 4 𝑛 )2]1/2 (S8) When 𝑁𝑀𝑄𝐸 is several times larger than 𝑁𝑀𝑄𝐸 𝑡ℎ ( 𝑁𝑀𝑄𝐸 𝑡ℎ = 𝑛 𝛤sp ), the expression [ 1 16 + 𝑊01 8𝛤sp + 3𝑊01𝑁𝑀𝑄𝐸 8 𝑛 + ( 𝑊01 4𝛤sp − 𝑊01𝑁𝑀𝑄𝐸 4 𝑛 ) 2 ] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' (S8) is dominated by the ( 𝑊01 4𝛤sp − 𝑊01𝑁𝑀𝑄𝐸 4 𝑛 )2term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, the 𝑁𝑛 𝑚𝑎𝑥 becomes to be 𝑁𝑛 𝑚𝑎𝑥 ≈ 1 4+ (𝑁𝑀𝑄𝐸−𝑁𝑀𝑄𝐸 𝑡ℎ )𝑊01 2𝛤sp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' When 𝑁𝑀𝑄𝐸 is equal to 𝑁𝑀𝑄𝐸 𝑡ℎ , the − 𝑊01 4𝛤sp + 𝑊01𝑁𝑀𝑄𝐸 4 𝑛 term becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Thus, 𝑁𝑛 𝑚𝑎𝑥 becomes 𝑁𝑛 𝑚𝑎𝑥 = √ 𝑊01 2𝛤sp + 1 4 − 1 4, namely, 𝑁𝑛 𝑚𝑎𝑥 ∝ √ 𝑊01 𝛤sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' References: [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Bergman and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Stockman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 90, 027402 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Milonni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 42, 1991 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' Loudon, The Quantum Theory of Light (Oxford University Press, London, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='0 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='8 factor (105 Purcell factor ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='6 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} +page_content='4 600 700 800 900 1000 600 700 800 900 1000 Radius of disk (nm) Radius of disk (nm)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAzT4oBgHgl3EQfdPyu/content/2301.01418v1.pdf'} diff --git a/xtFIT4oBgHgl3EQf0StA/content/tmp_files/2301.11368v1.pdf.txt b/xtFIT4oBgHgl3EQf0StA/content/tmp_files/2301.11368v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a3db30f3acb28ac00e5ce3d64432faf30b72953 --- /dev/null +++ b/xtFIT4oBgHgl3EQf0StA/content/tmp_files/2301.11368v1.pdf.txt @@ -0,0 +1,2382 @@ +Coincident Learning for Unsupervised Anomaly Detection +Ryan Humble 1 * Zhe Zhang 2 Finn O’Shea 2 Eric Darve 1 Daniel Ratner 2 * +Abstract +Anomaly detection is an important task for com- +plex systems (e.g., industrial facilities, manufac- +turing, large-scale science experiments), where +failures in a sub-system can lead to low yield, +faulty products, or even damage to components. +While complex systems often have a wealth of +data, labeled anomalies are typically rare (or even +nonexistent) and expensive to acquire. In this pa- +per, we introduce a new method, called CoAD, for +training anomaly detection models on unlabeled +data, based on the expectation that anomalous be- +havior in one sub-system will produce coincident +anomalies in downstream sub-systems and prod- +ucts. Given data split into two streams s and q +(i.e., subsystem diagnostics and final product qual- +ity), we define an unsupervised metric, ˆFβ, out of +analogy to the supervised classification Fβ statis- +tic, which quantifies the performance of the inde- +pendent anomaly detection algorithms on s and q +based on their coincidence rate. We demonstrate +our method in four cases: a synthetic time-series +data set, a synthetic imaging data set generated +from MNIST, a metal milling data set, and a data +set taken from a particle accelerator. +1. Introduction +The problem of anomaly detection, the task of finding ab- +normal events or data, is an important task for complex +systems, such as industrial facilities, manufacturing, and +large-scale science experiments (Sun et al., 2016; Zhao et al., +2019; Lutz et al., 2020; Edelen & Cook, 2021; Lindemann +et al., 2021; Radaideh et al., 2022). Failures in these sys- +tems can lead to low yield, faulty products, or even damage +to components, making identifying these failures a high- +priority task for system operators. However, the complexity +of these systems typically ensures that labeled data is rare +*Equal contribution 1Institute for Computational and Mathemat- +ical Engineering, Stanford University, Stanford, California 2SLAC +National Laboratory, Menlo Park, California. Correspondence to: +Ryan Humble . +0 +5 +10 +15 +20 +25 +30 +35 +40 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +s data value +0 +5 +10 +15 +20 +25 +30 +35 +40 +Event index +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +q data value +Figure 1. We consider tasks with two data streams, s and q, which +should be independent except for anomalous events which exist +across both data streams. An anomaly detection algorithm iden- +tifies points independently in each stream (stars) using example +thresholds (dashed blue line). Events found in only one stream +(black stars) are ignored, and only joint events found in both data +streams are identified as anomalous (red stars). +or nonexistent and expensive to acquire. This makes tradi- +tional supervised classification methods poorly suited to the +task of identifying these failures. Even a number of “unsu- +pervised” approaches rely on a completely normal training +set (Schlegl et al., 2017; Ruff et al., 2018), which is simi- +larly unavailable for many complex systems. We, therefore, +require an unsupervised method for detecting anomalies in +these complex systems, where the data is inherently polluted +by anomalies. +Complex systems are typically broken down into multiple +subsystems, all of which impact the overall system perfor- +mance. We consider the anomaly detection task of iden- +tifying an anomalous subsystem (s) in a larger system of +connected elements that impact overall system quality (q). +The subsystem state is considered anomalous if and only if +it impacts the larger system’s overall performance; equiva- +lently, a system issue has only occurred if both the subsys- +tem and larger system experience coincident anomalies. Our +goal is to leverage the expectation of coincident anomalies +between the two data streams, s and q, to classify normal +and anomalous examples. +arXiv:2301.11368v1 [cs.LG] 26 Jan 2023 + +Coincident Learning for Unsupervised Anomaly Detection +Our motivating example comes from a particle accelera- +tor, for which we have two data streams, one containing +data from a radio frequency (RF) station subsystem (s) and +one containing electron energy data beam-position moni- +tors (BPMs) that monitor beam quality (q). During normal +operation, the variability in the signals is independent (i.e., +random fluctuations in s and q are uncorrelated). However, +the anomalous behavior of an RF station will have an im- +pact, albeit unknown, on the BPM quality data. Even with +no ground truth labels for training, we will show that we +can exploit the coincidence of abnormalities to determine +whether abnormal RF station subsystem behavior has caused +beam performance degradation. +1.1. Contributions +In this paper, we consider a subset of anomaly detection +tasks in which the input features can be separated into two +groups (s and q), for example, drawn from different sources +or by partitioning a single source in time or space. Figure 1 +demonstrates our problem setting. We assume that the im- +pact of an anomaly is apparent in both sets of features. As a +consequence, we expect that there exists an algorithm capa- +ble of dividing each data stream into normal and anomalous +sets, and crucially the sets should match. Additionally, we +assume that s and q data are independent within either the +normal or anomalous clusters. +Our main contributions are: +1. We introduce coincident learning for anomaly detec- +tion (CoAD) and an unsupervised metric ˆFβ, in anal- +ogy to the supervised classification metric Fβ, that +exploits the coincidence between s and q to classify +normal and anomalous examples. Specifically, we use +two models—taking different data streams as inputs— +to classify examples as either normal or anomalous. +2. We also present theoretical results, including showing +that our estimate ˆFβ is a lower bound of the true Fβ +under our assumptions and deriving the form of the op- +timal models under mild conditions. We also interpret +our method as an unsupervised representation learner. +3. We show that our metric ˆFβ can be used in both a +categorical and continuous sense. If the two anomaly +detection sub-methods are predefined, ˆFβ gives a prin- +cipled way of selecting the two thresholds (demon- +strated in Section 4.1). We can also use ˆFβ to train the +anomaly detectors end-to-end, with the models parame- +terized as deep neural networks (DNNs) (demonstrated +in Sections 4.2 to 4.4). +4. We demonstrate these contributions on four data sets: a +synthetic time-series data set, a synthetic imaging data +set generated from MNIST, a publicly available metal +milling data set, and an experimental data set taken +from a particle accelerator. For the synthetic cases, +we show our unsupervised method performs nearly +as well as a supervised counterpart. For the real data +sets, we train DNNs end-to-end to achieve data-driven, +unlabeled anomaly detection. +2. Coincident Learning +Given a dataset D = {(s, q)} drawn from the respective +data streams, we consider a pair of algorithms, Aθs(s) and +Aθq(q), parameterized by θs and θq. The algorithms will +each have a scalar output, ps, pq ∈ [0, 1], which we will +interpret as the confidence that the example belongs to the +anomalous class. Let Ds = {s} and Dq = {q} be the +marginal datasets for s and q respectively. Also, we assume +that the data is generated from an unseen state variable x +(i.e., both are functions of x: s(x) and q(x)). We present a +schematic of our method in Figure 2. +x +s +q +Aθs +Aθq +ˆFβ +ps +pq +Figure 2. A schematic of CoAD showing the two data streams +s and q (generated from a hidden state x) and their respective +algorithms Aθs and Aθq, which are trained to maximize our unsu- +pervised metric ˆFβ. +2.1. CoAD Objective +To begin, we restrict ourselves to the case where ps and pq +are categorical labels in {0, 1}. We define a joint event as +one where both algorithms classify the respective data as +anomalous, i.e., ps,i = pq,i = 1. Since we lack true labels, +we cannot determine which joint events are true positives +(true anomalous events) or are false positives (normal events +flagged as anomalies). However, we show below that we can +estimate the number of false positives from the disagreement +between the two algorithms. We can then compare the +actual number of observed joint events with the estimated +number of false positive events to measure the efficacy of +the algorithms. The more joint events we observe, the more +sensitive our algorithm. The fewer points with conflicting +predictions (and thus fewer estimated false positives), the +more precise our algorithm. +Let J(θs, θq) denote the fraction of joint events found in our +data (i.e., predicted positives). Suppose α is the anomaly +fraction in our data (i.e., actual positives). If we had labels +for each of the n examples in our dataset, we would evaluate + +Coincident Learning for Unsupervised Anomaly Detection +our algorithm with the supervised metric Fβ, which can be +written as +Fβ = (1 + β2)(J − FP/n) +J + αβ2 +, +where higher values are better, FP is the number of false +positives, and β balances the weighting of precision and +recall. However, since we lack labels, we will rely on an +estimate D(θs, θq) of the fraction of false positives. (We +also show later an estimate of α suffices.) We therefore +recover an unsupervised version ˆFβ: +ˆFβ = (1 + β2)(J − D) +J + αβ2 +. +The quantity ˆFβ can now be used to compare algorithms +or select model hyperparameters in the same manner as +its supervised counterpart. As in the supervised case, the +extremes are ˆP = ˆF0 and ˆR = ˆF∞, which correspond to +precision P and recall R. We can use ˆFβ to pick a model +that strikes a balance between the number of anomalous +events found (maximizing the recall) and the confidence in +the prediction (maximizing precision). +The ˆFβ definition requires an estimate of the fraction of +false positives D(θs, θq). Our strategy is based on an ob- +servation that disagreements between Aθs and Aθq reveal +the false positive rate of the individual algorithms. Under +the assumption that s and q are independent conditioned +on knowing the true label, Theorem 2.1 shows that the dis- +agreement rates provide an upper bound on the true fraction +of false positives. (All proofs are deferred to the Appendix.) +Theorem 2.1. Assume that s and q are independent con- +ditioned on knowing the true label, and assume that the +algorithms Aθs, Aθq are no worse than random guessers. +Define D(θs, θq) = E(s,q)∈D[ps|¬pq]E(s,q)∈D[pq|¬ps] to +be our estimated fraction of false positives. Then, in the cat- +egorical case, the fraction of false positives is no more than +D and the fraction of true positives is at least J − D, imply- +ing that ˆR, ˆP, and ˆFβ are lower bounds of their supervised +counterparts R, P, and Fβ. +Corollary 2.2. +Additionally define Dnaive(θs, θq) += +Es∈Ds[ps]Eq∈Dq[pq], which is equivalent to an assumption +that s and q are completely independent. Then, D(θs, θq) ≤ +Dnaive. +Therefore, by definition of a joint event and the conditional +expectation, the fraction of joint events in the data and the +estimated fraction of false positives are +J(θs, θq) = µsq +(1) +D(θs, θq) = µs − µsq +1 − µq +µq − µsq +1 − µs +, +(2) +where µs += +Es∈Ds[ps], µq similarly, and µsq += +E(s,q)∈D[pspq]. This allows us to concretely write our un- +supervised metric as +ˆFβ = (1 + β2)µsq − µsµq +µsq + αβ2 +1 − µsq +(1 − µs)(1 − µq). +(3) +It is crucial to note that we implicitly require a majority of +the events to be labeled as 0, or equivalently require the +anomalous class to be the minority class; this is a necessary +condition since precision and recall (and our unsupervised +analogues) are not invariant under a labeling flip. Thus, we +additionally impose the constraint that anomalies exist and +are rare (0 < µsq ≤ µs, µq ≤ 0.5). Also, as a point of +optimization, it might appear that maximizing ˆFβ requires +that both α and β be defined. However, as 1 + β2 is just +a constant scalar, we need only specify the quantity αβ2. +If we used an incorrect estimate of α (since the true α is +unknown), we have merely maximized our metric for a +different value of β. Moreover, the maximizers of ˆP and ˆR +do not depend on α at all. +Lastly, when developing our metric, we assumed the cat- +egorical case (ps, pq ∈ {0, 1}). This case might naturally +arise when the two algorithms Aθs, Aθq already exists and +are parameterized by two thresholds. The metric then al- +lows a principled way of setting these thresholds, as we +demonstrate in Section 4.1. But notably, our metric natu- +rally extends to the case of continuous ps, pq ∈ [0, 1]. This +allows us to train more complex algorithms Aθs, Aθq, such +as ones parameterized as DNNs and trained with gradient- +based optimizers, thereby allowing us to cluster normal and +anomalous data without having to first build the individ- +ual anomaly detection algorithms. We present a theoretical +justification for the continuous extension in the following +section. +2.2. Properties of CoAD +Under certain simplifying assumptions, we can derive re- +sults regarding the solution to the optimization problem. +Throughout we assume that ps(s) = Aθs(s), for some +choice of parameters θs, can map Ds to any element of +[0, 1]n where n is the number of samples in the training set +(and similarly for pq(q) = Aθq(q)). +We first consider the possible forms of the maximizers p∗ +s, p∗ +q +of ˆFβ. Holding ps fixed, Theorem 2.3 shows that the optimal +p∗ +q is (nearly) categorical: p∗ +q(q) ∈ {0, ρ, 1} for some ρ ∈ +[0, 1] and all q. Moreover, Theorem 2.4 shows ρ ̸∈ {0, 1} +only occurs if the constraint µq ≤ 0.5 is tight. By applying +this twice (first for fixed ps and then again with the new pq +fixed), we need only consider (nearly) categorical solutions +for ps, pq. Thus, the continuous extension (to ps, pq ∈ [0, 1]) +is almost equivalent to the original categorical case, and our +method behaves as a (nearly) hard clustering algorithm. + +Coincident Learning for Unsupervised Anomaly Detection +Theorem 2.3. Assume θs is fixed (with µs ∈ (0, 0.5]). Let +w(q) = Es|q∈Ds|q[ps(s)]. Then, the maximum of ˆFβ can +be achieved by a (nearly) categorical solution: p∗ +q(q) = +1 {w(q) > τ} + ρ1 {w(q) = τ} for some ρ, τ ∈ [0, 1]. +Theorem 2.4. Additionally, a non-categorical solution (i.e., +with ρ ̸∈ {0, 1}) can only be uniquely optimal if the con- +straint µq ≤ 0.5 is tight. +Armed with the optimal form of ps, pq, we now derive the +solution for the illustrative scenario shown in Figure 3. We +specifically consider when the anomalous and normal sets, +respectively A and Ac, might overlap in the data streams +s and q (i.e., the data streams s and q can be noisy). The- +orem 2.5 shows that, under some mild conditions, the op- +timal solution always labels the noiseless parts of s and +q according to their true cluster labels: ps(s(A \ B)) = +pq(q(A\C)) = 1 and ps(s(Ac \B)) = pq(q(Ac \C)) = 0. +The noisy part of q (i.e., the set C) is always labeled as +anomalous, but the label for the noisy part of s (i.e., the +set B) depends on the setting of β. Choosing β = 0 (i.e., +ˆF0 = ˆP) prioritizes precision, so the optimal solution does +not assign the noisy examples in B to the anomalous class; +choosing β = ∞ (i.e., ˆF∞ = ˆR) prioritizes recall, so the +optimal solution labels B as anomalous. This tradeoff oc- +curs abruptly at a critical βcrit that depends on the noise level +in both s and q. +Theorem 2.5. Suppose the variable x exists in some prob- +ability space (Ω, F, P). As depicted in Figure 3, denote +A ∈ F and Ac its complement, where α = P(A) ≤ 0.5. +Assume (i) s(A) ∩ s(Ac) = s(B) and q(A) ∩ q(Ac) = +q(C); (ii) s and q are independent when x ∈ A (and +similarly when x ∈ Ac); (iii) P(A ∪ B), P(A ∪ C) ≤ +0.5; (iv) P(A \ B \ C) +> +0; (v) P(Ac \ B \ C) +≥ +P(Ac \ B)P(Ac \ C); and (vi) P(Ac \ B \ C)P(A \ C) ≥ +P(A \ B \ C)P(Ac \ C). Wlog let P(A \ B) ≥ P(A \ C). +Then the maximum of ˆFβ is achieved when +ps(s(A \ B)) ≡ 1, +pq(q(A \ C)) ≡ 1, +ps(s(Ac \ B))) ≡ 0, +pq(q(Ac \ C)) ≡ 0, +ps(s(B))) ≡ 1 +� +β2 ≥ β2 +crit +� +, +pq(q(C))) ≡ 1. +where βcrit depends on the sets A, Ac, B, and C. +Lastly, in the case where both algorithms Aθs, Aθq are pa- +rameterized as DNNs, we note that our method performs +both feature representation and classification, simultane- +ously and in an unsupervised manner. Suppose the final +components of each DNN are a fully-connected (FC) layer +and a sigmoid activation. The final layer and sigmoid then +amount to a logistic classifier on the latent representations +(i.e., representation fed to the final FC layer). The back- +bones of the networks are then tasked with learning how +to generate linearly-separable representations of the input +data. Theorem 2.6 shows the gradients with respect to the +Ac +A +s(Ac \ B) +s(B) +s(A \ B) +q(Ac \ C) +q(C) +q(A \ C) +Figure 3. An illustrative, noisy anomaly scenario. Setting of Theo- +rem 2.5. +logistic classifier parameters ws, bs and the latent represen- +tations zs(s), and illustrates several important qualitative +behaviors. First, the gradients depend most on the uncertain +examples ps(s) and ignore examples with high confidence, +as γs(s) = 0 for ps(s) ∈ {0, 1}, meaning the examples +on the “margin” are most important. Second, the gradients +also depend on the pseudo-target ˆyq(q), which has largest +magnitude for the most certain examples pq(q). Taken to- +gether, the ˆFβ metric encourages each individual algorithm +to adopt the other’s prediction when it is uncertain but the +other algorithm is confident. +Theorem 2.6. Suppose ps, pq are parameterized as DNNs, +whose final components of each DNN are a fully-connected +layer and a sigmoid activation. Let the latent representation +fed into the final layer be zs(s), zq(q) ∈ Rp for the two +networks respectively. Define ps(s) = σ(wT +s zs(s) + bs) +and pq(q) = σ(wT +q zq(q) + bq). Then, +∇ws ˆFβ ≡ ED[ˆyq(q)γs(s)zs(s)] +∇bs ˆFβ ≡ ED[ˆyq(q)γs(s)] +∇zs(s) ˆFβ ≡ EDq|s[ˆyq(q)]γs(s)zs(s) +where ˆyp(p) = c1pq(q)−c2 and γs(s) = ps(s) (1 − ps(s)). +The gradients w.r.t. the q network are analogous. The +constants c1, c2 depend on the current parameters θs, θq +but not individual instances (s, q) ∈ D. If the networks are +better than random guessers, then c1, c2 ≥ 0. +3. Related Work +3.1. Unsupervised anomaly detection +Our method falls into an extremely large body of work +identifying anomalies in unsupervised (or semi-supervised +“normal-only”) data. The most popular classical methods, +such as One-Class SVM (OCSVM) (Sch¨olkopf et al., 2001), +Isolation Forest (Liu et al., 2008), Local Outlier Factor (Bre- + +Coincident Learning for Unsupervised Anomaly Detection +unig et al., 2000), and Kernel Density Estimation (Parzen, +1962), typically suffer from the curse of dimensionality. +These methods struggle with high dimensional inputs pri- +marily for two reasons: (i) they rely on distances in high- +dimension; and (ii) they require feature engineering be- +cause their possible decision boundaries are limited. Since +we can leverage DNNs, our approach suffers from neither +and can represent the normal and anomalous sets using +complex decision boundaries. Many recent works, such as +DSVDD (Ruff et al., 2018), RandNet (Chen et al., 2017), +and AnoGAN (Schlegl et al., 2017), also leverage DNNs +for these reasons. However, these approaches assume ei- +ther (i) the training data is completely “normal” or (ii) the +anomalies exist in low-density regions. Our method does +not make these assumptions; in fact, we explicitly assume +our data is polluted by anomalies, which are quite possibly +densely clustered (for example, repeated failures might have +extremely similar signatures). +3.2. Common representation learning +In the absence of labels for classification, many prior works +have proposed unsupervised methods for learning feature +representations of different views (or modalities) of the data. +A key concept behind these approaches is to maximize the +alignment between the feature representations. +Perhaps the best known approach uses Canonical Correla- +tion Analysis (CCA) (Hotelling, 1936) (or a kernel general- +ization (Akaho, 2006)) to create similar feature representa- +tions for multiple views (Hardoon et al., 2004). There are +a large number of variants and applications, most notably +a deep neural network adaptation called Deep Canonical +Correlation Analysis (DCCA) (Andrew et al., 2013). In +the notation of our paper, DCCA proposes an objective +maxθs,θq Corr +� +Aθs, Aθq +� +where the algorithms are param- +eterized as DNNs and the outputs are not necessarily in [0, 1]. +Both DCCAE (Wang et al., 2015) and CorrNet (Chandar +et al., 2016) build upon this idea and add autoencoders to +the formulation. +Our method differs in several respects. First, we apply our +ˆFβ objective to a single dimensional output ps, pq from each +algorithm; this is required since we are interpreting the final +output as an anomaly probability. DCCA, DCCAE, and +CorrNet predict into a multi-dimensional latent space to +achieve higher correlation. Second, the correlation objective +is scale invariant. Suppose that p∗ +s, p∗ +q are the maximizers +of the correlation objective; then any ap∗ +s, bp∗ +q for ab > 0 is +also a maximizer. This is problematic for interpreting the +output as an anomaly label, as the scale of the ps, pq is free +and can be arbitrarily small. Lastly, the correlation objec- +tive is qualitatively similar to the special case ˆF0 = ˆP of +our method, in that they both promote maximum precision +and are agnostic to the number of anomalies found. This +behavior of ˆP (and correlation) is why the control over β +(i.e., between ˆP and ˆR) is desirable. +The other common representation learning approach uses +a covariance-based objective for learning common feature +representations. If the two views of the data are the same, +this is in fact what principal component analysis (PCA) op- +timizes for. For different views, maximum covariance anal- +ysis (MCA) (Storch & Zwiers, 1999) (also known as SVD +analysis (Bretherton et al., 1992; Wallace et al., 1992)) max- +imizes the covariance between linear projections of the two +views. Deep Maximum Covariance Analysis (DMCA) (Luo +et al., 2018) allows for deeper, nonlinear projections (pa- +rameterized by neural networks) within the same covariance +framework. In the notation of our paper, DMCA proposes an +objective maxθs,θq Cov +� +Aθs, Aθq +� +and is otherwise iden- +tical to DCCA. This covariance objective is qualitatively +similar to the special case ˆF∞ = ˆR, in that they both pro- +mote maximum recall. Specifically, we have +ˆR = 1 +αCov (ps, pq) +1 − µsq +(1 − µs)(1 − µq) +The last term is always at least 1 (since 0 < µsq ≤ µs, µq). +Thus, under the assumption that s and q are indepen- +dent given the true label (see Theorem 2.1), we have +1 +αCov (ps, pq) ≤ ˆR ≤ R where R is the true recall. This +is also a natural conclusion of Corollary 2.2, which shows +that we would have recovered covariance if we had instead +made the assumption that s and q are completely indepen- +dent. Equivalently, covariance measures the number of joint +events in excess of those that would have occurred simply by +chance, which has a connection to Cohen’s kappa (Cohen, +1960). Therefore, in our setting, covariance can only be a +worse underestimate of the recall (i.e., number of anomalies) +than our metric ˆR. We show this for a synthetic example +in Section 4.1. +3.3. Self-supervision +It may also be useful to view our method through the con- +cepts of self-supervision. In contrastive self-supervision, +an algorithm (for example by augmentation) labels exam- +ples as either similar (“positive”) or dissimilar (“negative”). +The contrastive loss function then encourages the network +to map positive examples close to each other and negative +examples far apart. Our approach is most similar to that of +SimCLR (Chen et al., 2020), which applies a constrastive- +type loss to learn representations of images; it uses different +transformations of the same image as “positive” examples +and the other examples in the training batch as “negative” +examples. We assume our data is already paired into (s, q) +“positive” pairs and similarly use the training batch as the +“negative” examples through the penalty on the estimated +number of false positives D. Our metric can therefore be +interpreting as encouraging similar labeling (through µsq) + +Coincident Learning for Unsupervised Anomaly Detection +while preventing mode collapse (through D, which sends +ˆFβ to zero if all labels are identical). Unlike SimCLR, our +approach does not require the transformations to be speci- +fied and is applicable to multi-modal data or non-Siamese +networks. +4. Experiments +We assess our method on several test cases, in both the cate- +gorical and continuous settings. In the continuous setting, +we train our DNNs using the PyTorch framework (Paszke +et al., 2019), Adam optimizer (Kingma & Ba, 2015), mini- +batches, and a sigmoid-based regularizer to enforce the +constraint µs, µq ≤ 0.5. Full details on the training settings +and architectures can be found in the Appendix. +4.1. Synthetic time-series outliers +We start by illustrating the categorical case. We create a syn- +thetic dataset of 20k normal points sampled from |N(0, 1)| +for both s and q. We then introduce anomalies by sampling +from 1 + |N(0, 1.5)| for 5% of the data points. Anoma- +lies always occur simultaneously in both data streams. We +then use a simple threshold as the anomaly detection al- +gorithm, applied independently to each data stream, and +identify as anomalous any point that exceeds the threshold +simultaneously in both s and q. This setup corresponds to +pre-specified algorithms Aθs, Aθq where the parameters are +just single thresholds on the algorithm outputs. Figure 1 +illustrates a small example. +To evaluate our approach, we first determine the estimated +ˆP- ˆR curve and the s and q thresholds that define it. Since the +assumptions of Theorem 2.1 are met for this synthetic case, +ˆR, ˆP are underestimates of recall R and precision P, re- +spectively, for any choice of thresholds. As we have ground +truth labels, we can calculate the actual precision and recall +values defined by the same unsupervised s and q thresholds; +we denote this curve the “actual unsupervised” curve. As a +reference, we also find the supervised P-R curve using the +labels (i.e., find the optimal s and q thresholds directly using +the labels). Figure 4(a) shows these three precision-recall +curves - estimated, actual, and supervised - where we plot +ˆR, ˆP for the estimated case. Although our method is unsu- +pervised, the actual precision-recall curve almost exactly +matches the supervised precision-recall curve. +We can also analyze ˆFβ and the thresholds that maximize +it, for a range of β. As above, we have the estimated ˆFβ +curve and the actual Fβ curve using the same unsupervised +thresholds. To illustrate our method for estimating the frac- +tion of false positives, we compare the definitions given +in Theorem 2.1 and Corollary 2.2. We denote the latter +definition, Dnaive, as the “naive” rate. Using the ground +truth, we also calculate the supervised Fβ curve. Figure 4(b) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall, +̂ +R +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P ecision, +̂ +P +Estimated unsupe vised (AUC: 0.65) +Actual unsupe vised (AUC: 0.78) +Supe vised (AUC: 0.78) +(a) Comparison of the supervised precision-recall curve to the +estimated ˆP- ˆR curve. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +β +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +F +β +, +̂ +F +β +Estimated unsuper ised +Actual unsuper ised +Nai e estimated unsuper ised +Nai e actual unsuper ised +Super ised +(b) Comparison of supervised Fβ to estimated ˆFβ, found using +two different estimates for the fraction of false positives. +Figure 4. Analysis of our unsupervised method on a synthetic +dataset. The “actual” curves use the optimal thresholds from the +estimated curves but evaluate using the ground truth labels. +shows the five curves—“estimated unsupervised”, “actual +unsupervised”, “naive estimated unsupervised”, “naive ac- +tual unsupervised”, and “supervised”—where we plot ˆFβ +for the estimated cases. While both estimates of the false +positive fraction lead to underestimates of ˆFβ, our defi- +nition D(θs, θq) gives a significantly better estimate than +Dnaive and the corresponding ‘s‘ and ‘q‘ thresholds achieve +near supervised-level performance when assessed with the +ground truth labels. +4.2. MNIST +As a second example, we construct an imaged-based +anomaly detection task using MNIST. Each example con- +sists of a pair of images, one given as input to each network. + +Coincident Learning for Unsupervised Anomaly Detection +Normal examples consist of a pair of images of the digit +0. Anomalous examples are pairs of images drawn from +the digits 1, 2, or 3, with the same digit given to each net- +work (i.e., there are 3 different anomaly types). We then +alter the difficulty by imposing a noisy observation model, +where with some probability the image fed to each network +is replaced by another digit, such that each anomaly class +has different amounts of noise (1 having the least noise and +3 having the most noise). The full observation model is +detailed in the Appendix. +Table 1. Fraction of MNIST digits labeled as anomalous under +a noisy observation model. A digit is labeled anomalous if the +product of the two network outputs is greater than 0.5. Results are +shown for several different values of β and rounded to the nearest +tenth of a percent. +β = 0.01 +β = 1 +β = ∞ +Digit 0 +0% +0.5% +0.5% +Digit 1 +86.3% +99.1% +100% +Digit 2 +0% +96.9% +99.2% +Digit 3 +0.2% +1.2% +99.6% +Table 1 shows the results for different values of β. In partic- +ular, we illustrate that different choices of β result in classi- +fying different sets of anomalies: small β only identifies the +least noisy anomalies, and large β identifies all 3 anomaly +classes. The full violin plots are shown in the Appendix. +We can also visualize the latent representations created by +the algorithms Aθs and Aθq. Despite being trained without +labels, our method learned to separate the digits, as shown +in Figure 5. +Figure 5. t-SNE (van der Maaten & Hinton, 2008) visualization +of the latent representations of the MNIST digits. We show the +representations of Aθs (left) and Aθq (right) for β = 1. +4.3. Milling dataset +We now demonstrate our method on a real-world +dataset. The University of California, Berkeley Milling +dataset (Agogino & Goebel, 2007) is an open dataset of +acoustic, vibration, and current measurements from a set +of metal milling cuts. A recent paper (Hahn & Mechefske, +2021) provides a detailed description of the task, analysis +code, and results from a variational autoencoder (VAE). The +dataset consists of 167 different milling cuts, corresponding +to a total of approximately 100 minutes of milling. There +are six total diagnostics: acoustics and vibrations from the +spindle, acoustics and vibration from the table, and AC +and DC current. In addition, the degree of flank wear on +the milling tool is measured after a selection of the cuts. +We follow the task as described in (Hahn & Mechefske, +2021), breaking the data into 0.25 second chunks, and try to +predict whether the milling performance in each chunk is +“healthy” or “degraded/failed,” with the label determined by +the degree of flank wear. +To apply coincident learning to the milling task, we di- +vide the diagnostics into two sets: acoustics and vibration +measurements (four data streams) and AC/DC current mea- +surements (two data streams). (Note that in the milling +dataset there is not a default separation into “subsystem” +and “quality” measurements.) Also, in part because the “de- +graded/failed” examples are the majority case, our algorithm +learns to identify the most “healthy” examples as a distinct +class. Figure 6 shows the predictions from the networks +trained under coincident learning (at β = 6) as compared to +the predictions from the VAE of (Hahn & Mechefske, 2021). +Our model has perfect predictions on failed examples while +identifying more “healthy” and “degraded” examples as +anomalous compared to the VAE. We emphasize that in +contrast to the VAE, the coincident model does not require +training only on “healthy” data. +CoAD +VAE +Figure 6. Milling data set results. (Left) Violin plot showing predic- +tions from the ˆF6 model; the prediction values are the products of +the two network outputs. (Right) Violin plot showing predictions +from the VAE in (Hahn & Mechefske, 2021). +The labels defined in (Hahn & Mechefske, 2021) are quite +simplistic, using only the degree of flank wear and ignoring +the other milling parameters: metal type, cut speed, and +cut depth. While the flank wear is indicative of milling +performance, the amount of wear that will lead to anomalous +milling may differ across these different milling settings. +Figure 7 shows the model’s anomaly confidence versus + +S Net +Digit +0 +Digit 1 +Digit 2 +Digit 3Q Net +Digit O +Digit 1 +Digit 2 +Digit 3Healthy +Degraded +Failed +Normal +Abnormal(failed) +Prediction +Prediction +-Healthy +Degraded +Failed +Normal +Abnormal (failed) +Prediction +PredictionCoincident Learning for Unsupervised Anomaly Detection +the flank wear for two different milling configurations (the +other six are shown in the Appendix). There is only a +weak correlation between our predictions and flank wear in +aggregate across all eight milling settings but a very strong +correlation for each individual configuration. We emphasize +that the models never see the flank wear measurements +or milling configurations during training, and are trained +simultaneously on all milling configurations. We infer that +the simplistic labels based on flank wear alone may be +incorrect. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Cast Iron +Steel +Figure 7. Anomaly probability versus flank wear for two different +milling configurations (cast iron (left) and steel (right) at 0.75mm +cut depth and 0.25mm/rev cut speed). Black lines are sigmoid fits. +4.4. Particle accelerator RF stations +We now return to our original motivating problem of identi- +fying the source of RF station faults. We utilize the dataset +assembled and described in (Humble et al., 2022). The +subsystem (s) data stream consists of time-series data for +a single RF station, with one data point approximately ev- +ery 5 seconds. We use a sensitive trigger to actively select +time windows with the possibility of an event to reduce data +requirements; any relative change of 0.5% will trigger a +window to be acquired. The quality (q) data stream consists +of beam-position monitor (BPM) data from seven differ- +ent BPM diagnostics in dispersive (i.e., energy-sensitive) +regions of the accelerator. Each BPM stream consists of +beam positions in the dispersive direction, recorded syn- +chronously at 120 Hz. Time windows are selected to cover +the 8 seconds prior to the end of each (asynchronous) RF +station event. +Figure 8 shows examples identified as normal and anoma- +lous. Many of the anomalous cases are apparent by eye. For +the normal cases, we specifically selected non-trivial exam- +ples that a non-expert might identify as abnormal, but which +upon close examination does not correspond to an anomaly +in the selected RF station data. For example, the final normal +case even contains an abnormal energy deviation, but the +algorithm correctly determines that the BPM abnormality is +too early compared to the RF station anomaly (as described +in (Humble et al., 2022)). We note that with significant ef- +fort, it is possible to design hand-tailored anomaly detection +algorithms for each data stream, as in (Humble et al., 2022), +through manual inspection. However, the NN approach has +zero manually set parameters and avoids the need for a hand- +designed anomaly detection algorithm. Evaluating using +the ground truth labels, our NN achieves a F1 score of 0.86, +compared to 0.9 for the hand-tailored algorithms (within +the expected error rate of the hand labels). For accelerators, +the NN approach can scale to cover thousands of potential +anomaly sources. +(a) Normal examples +(b) Anomalous examples +Figure 8. Experimental RF station data examples. Each example +shows both the RF station (left) and electron BPM (right) streams. +5. Conclusion +This work presents a new unsupervised method for detect- +ing coincident anomalies in two data streams. We use the +disagreement between two anomaly detection algorithms, +one for each set of data, to eliminate the need for labels. +We derive several theoretical properties of our metric ˆFβ, +revealing it as a type of clustering algorithm whose behavior +is configurable with β. Moreover, when the algorithms are +DNNs, we show that our method simultaneously performs +representation learning and clustering. Although we only +consider a single value of β when optimizing ˆFβ in this +work, our approach can be generalized to optimizing an +entire frontier of choices simultaneously, allowing for the +representation backbone to be reused. Lastly, we demon- +strated that our method achieves near supervised-levels of +performance on several data sets—synthetic and real, time- +series and image-based. + +Normal RF Station +Normal BPM +-50 +-40 +-30 +-20 +-10 +0 +-8 +-6 +-4 +-2 +0 +Time (s) +Time (s)Anomaly RF Station +Anomaly BPM +-50 +-40 +-30 +-20 +-10 +0 +-8 +-6 +-4 +-2 +0 +Time (s) +Time (s)Coincident Learning for Unsupervised Anomaly Detection +References +Agogino, A. and Goebel, K. Milling data set. NASA Ames +Prognostics Data Repository, NASA Ames Research Cen- +ter, Moffett Field, CA, 2007. +Akaho, S. A kernel method for canonical correlation anal- +ysis, 2006. 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Supplementary Materials for Section 2 (Coincident Learning) +A.1. CoAD Objective (Proofs) +Theorem 2.1. Assume that s and q are independent conditioned on knowing the true label, and assume that the algorithms +Aθs, Aθq are no worse than random guessers. Define D(θs, θq) = E(s,q)∈D[ps|¬pq]E(s,q)∈D[pq|¬ps] to be our estimated +fraction of false positives. Then, in the categorical case, the fraction of false positives is no more than D and the fraction of +true positives is at least J − D, implying that ˆR, ˆP, and ˆFβ are lower bounds of their supervised counterparts R, P, and +Fβ. +Proof. Let y(s, q) denote the true label (0-normal, 1-anomalous). For categorical ps(s), pq(q), the fraction of false positives +is +ED[pspq¬y] = E[ps|¬y]E[pq|¬y]P(¬y) +where we use the independence of s and q conditioned on the true label. Since we do not have the labels, we cannot directly +measure E[ps|¬y] and E[pq|¬y]. We instead will bound this expectation: +E[ps|¬y] = P(ps|¬y) = P(ps|pq, ¬y) += P(ps|¬pq) − (P(ps|y) − P(ps|¬y)) P(y|¬pq) +≤ P(ps|¬pq) = E[ps|¬pq] +where in the first line we use the independence of s and q within a single class of data, and in the final line we use +P(ps|y) ≥ P(ps|¬y) (i.e., the algorithm is no worse than random). In the second line we use the identity +P(ps|¬pq) = P(ps|¬pq, ¬y)P(¬y|¬pq) + P(ps|¬pq, y)P(y|¬pq) += P(ps|¬pq, ¬y) (1 − P(y|¬pq)) + P(ps|¬pq, y)P(y|¬pq) += P(ps|¬pq, ¬y) − (P(ps|¬pq, y) − P(ps|¬pq, ¬y)) P(y|¬pq) += P(ps|¬pq, ¬y) − [P(ps|y) − P(ps|¬y)]P(y|¬pq) +where we have again applied independence of s and q within a single class. Finally plugging in E[ps|¬y] ≤ E[ps|¬pq], +E[pq|¬y] ≤ E[pq|¬ps] (by the same logic), and P(¬y) ≤ 1, we find +ED[pspq¬y] ≤ E[ps|¬pq]E[pq|¬ps] = D(θs, θq) +Lastly, since ED[pspq¬y] ≤ D(θs, θq), the fraction of true positives is +ED[pspqy] = ED[pspq] − ED[pspqy] +≥ J(θs, θq) − D(θs, θq) +and therefore ˆFβ ≤ Fβ. +Corollary 2.2. Additionally define Dnaive(θs, θq) = Es∈Ds[ps]Eq∈Dq[pq], which is equivalent to an assumption that s and +q are completely independent. Then, D(θs, θq) ≤ Dnaive. +Proof. We have +ED[ps] = E[ps|pq]P(pq) + E[ps|¬pq]P(¬pq) += E[ps|¬pq] + P(pq) (E[ps|pq] − E[ps|¬pq]) +≥ E[ps|¬pq] +since E[ps|pq] ≥ E[ps|¬pq] if the algorithms are no worse than random. + +Coincident Learning for Unsupervised Anomaly Detection +A.2. Properties of CoAD (Proofs) +Theorem 2.3. Assume θs is fixed (with µs ∈ (0, 0.5]). Let w(q) = Es|q∈Ds|q[ps(s)]. Then, the maximum of ˆFβ can be +achieved by a (nearly) categorical solution: p∗ +q(q) = 1 {w(q) > τ} + ρ1 {w(q) = τ} for some ρ, τ ∈ [0, 1]. +Proof. We can directly optimize as follows +max +pq∈[0,1]n +µq∈[0,0.5] +ˆFβ = (1 + β2) +max +pq∈[0,1]n +µq∈[0,0.5] +µsq − µsµq +µsq + β2α +1 − µsq +(1 − µs)(1 − µq) += 1 + β2 +1 − µs +� +� +� +max +pq∈[0,1]n +µq∈[0,0.5] +µsq − µsµq +µsq + β2α +1 − µsq +1 − µq +� +� +� += 1 + β2 +1 − µs +� +� max +γ∈[0,0.5] +� +� max +pq∈[0,1]n +µq=γ +µsq − µsγ +µsq + β2α(1 − µsq) +� +� +1 +1 − γ +� +� += 1 + β2 +1 − µs +� +max +γ∈[0,0.5] +µ∗ +sq(γ) − µsγ +µ∗sq(γ) + β2α +1 − µ∗ +sq(γ) +1 − γ +� +where µ∗ +sq(γ) is the solution to the inner problem. The solution to the unconstrained inner problem is +˜µsq(γ) = arg max +pq +µsq − µsγ +µsq + β2α(1 − µsq) += +� +(1 + αβ2)(µsγ + αβ2) − αβ2 +since +∂ +∂µsq +� µsq − µsγ +µsq + β2α(1 − µsq) +� += (1 + αβ2)(µsγ + αβ2) +(µsq + αβ2)2 +− 1 +and +∂2 +∂µ2sq +� µsq − µsγ +µsq + β2α(1 − µsq) +� += −2(1 + αβ2)(µsγ + αβ2) +(µsq + αβ2)3 +≤ 0. +Now we will show that µ∗ +sq(γ) ≤ ˜µsq(γ). First, we have +µ2 +sq = E[pspq]2 +≤ E +� +p2 +s +� +E +� +p2 +q +� +by Cauchy-Schwartz +≤ E[ps]E[pq] +since ps, pq ∈ [0, 1] += µsγ +and +µsq ≤ 1 +2 ≤ 1 +2(1 + µsγ) +since µsq ≤ µq ≤ 1 +2. +We can now bound (µsq + αβ2)2: +(µsq + αβ2)2 = µ2 +sq + 2µsqαβ2 + (αβ2)2 +≤ µsγ + (1 + µsγ)αβ2 + (αβ2)2 += (1 + αβ2)(µsγ + αβ2) += ˜µsq(γ). + +Coincident Learning for Unsupervised Anomaly Detection +Therefore, we have shown µ∗ +sq(γ) ≤ ˜µsq(γ), and thus the (constrained) inner optimization problem amounts to just +maximizing µsq: +µ∗ +sq(γ) = arg max +pq∈[0,1]n +µq=γ +µsq += arg max +pq∈[0,1]n +µq=γ +Eq∈Dq +� +pq(q)Es|q∈Ds|q[ps(s)] +� += arg max +pq∈[0,1]n +µq=γ +Eq∈Dq[pq(q)w(q)]. +In order to maximize this, we first assign p∗ +q mass to those q with largest w(q) and then progressively to those with smaller +w(q). We therefore have +p∗ +q(q) = +� +� +� +� +� +1 +w(q) > τ(γ) +ρ(γ) +w(q) = τ(γ) +0 +w(q) < τ(γ) +where ρ(γ), τ(γ) are set to achieve µq = γ. Specifically, let fw(q)(w) = P(w(q) = w) and ¯Fw(q)(w) = P(w(q) > w). +If ∃τ(γ) such that ¯Fw(q)(τ(γ)) = γ, then ρ(γ) = 0. Otherwise, let τ(γ) = inf{w : ¯Fw(q)(w) ≤ γ}. By definition, this +implies that fw(q)(τ(γ)) ≥ γ − ¯Fw(q)(τ(γ)), so we can set ρ(γ) = +γ− ¯ +Fw(q)(τ(γ)) +fw(q)(τ(γ)) +∈ [0, 1]. Generally, we have +τ(γ) = inf{w : ¯Fw(q)(w) ≤ γ} +ρ(γ) = +� +0 +if ¯Fw(q)(τ(γ)) = γ +γ− ¯ +Fw(q)(τ(γ)) +fw(q)(τ(γ)) +o.w. +Corollary A.1. Define µ∗ +sq(γ) = arg maxpq∈[0,1]n,µq=γ µsq. Then, µ∗ +sq(γ) is continuous in γ and concave increasing. +Proof. From its definition in Theorem 2.3, the function µ∗ +sq(γ) is increasing and continuous. Moreover, as γ increases, τ(γ) +decreases. As a consequence, µ∗ +sq(γ) is concave. +Theorem 2.4. Additionally, a non-categorical solution (i.e., with ρ ̸∈ {0, 1}) can only be uniquely optimal if the constraint +µq ≤ 0.5 is tight. +Proof. Starting at the end of the proof of Theorem 2.3, we note that τ(γ) is the generalized inverse of the complementary +cumulative distribution function ¯Fw(q)(w). If a unique inverse exists, µ∗ +sq(γ) is differentiable on (0, 1); moreover, ρ(γ) = 0 +for all γ, and we have a categorical solution for p∗ +q(q) achieving the maximum. +If instead a unique inverse does not exist, µ∗ +sq(γ) can have piecewise linear segments, where each segment corresponds to a +different τ(γ) threshold and the endpoints of each segment correspond to different categorical solutions. We need to revisit +the outer problem (over γ) to establish the form of the solution. As shown in Theorem 2.3 and Corollary A.1, µ∗ +sq(γ) is +concave increasing and is piecewise linear (where each segment corresponds to a different τ(γ) threshold and the endpoints +of each segment correspond to different categorical solutions). We also have that +µsγ = E[psγ] +≤ +max +pq∈[0,1]n +µq=γ +E[pspq] = µ∗ +sq(γ) +≤ min +� +�E[ps1], +max +pq∈[0,1]n +µq=γ +E[1pq] +� +� = min(µs, γ) +≤ 1 +2 + +Coincident Learning for Unsupervised Anomaly Detection +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.2 +0.3 +0.4 +0.5 +γm +γM +γ +µsq(γ) +µ∗ +sq(γ) +τ(γm)γ + µ∗ +sq(γm) +Figure 9. Illustration of piecewise linear behavior of µ∗ +sq(γ). +and 0 ≤ τ(γ) ≤ 1. +We now consider where the maximum lies along each piecewise linear segment. Suppose there is a segment on [γm, γM] +with 0 ≤ γm < γM ≤ 0.5. Along this segment, let µ∗ +sq(γ) = ax + b where a = τ(γm), b = µ∗ +sq(γm), and x = γ − γm. +We depict this in Figure 9. The maximum along the segment is defined by +arg max +γ∈[γm,γM] +ˆFβ(γ) = arg max +γ∈[γm,γM] +µ∗ +sq(γ) − µsγ +µ∗sq(γ) + αβ2 +1 − µ∗ +sq(γ) +1 − γ += γm + +arg max +x∈[0,γM−γm] +(b − µsγm) + (a − µs)x +(b + αβ2) + ax +(1 − b) − ax +(1 − γm) − x += γm + +arg max +x∈[0,γM−γm] +f(x)g(x). +Since µsγ ≤ µ∗ +sq(γ) ≤ 1 +2, we immediately have ˆFβ(γ) ≥ 0 for all γ > 0, which implies the null solution, γ = 0, is either +not optimal or is not uniquely optimal, so we ignore it here-forward. +For z = αβ2(a − µs) − µs (b − aγm), we have +∂ +∂xf(x) = +z +((b + αβ2) + ax)2 +∂2 +∂x2 f(x) = − +2az +((b + αβ2) + ax)3 . +The denominators are positive, since they are non-negative and only 0 if αβ2 = γm = b = x = 0 (which is the null +solution). Therefore, f(x) is (i) concave increasing if z > 0, (ii) constant if z = 0, or (iii) convex decreasing if z < 0. For +y = (1 − b) − a(1 − γm), we similarly have +∂ +∂xg(x) = +y +((1 − γm) − x)2 +∂2 +∂x2 g(x) = +2y +((1 − γm) − x)3 . +Since the denominators are positive, g(x) is (i) convex increasing if y > 0, (ii) constant if y = 0, or (ii) concave decreasing + +Coincident Learning for Unsupervised Anomaly Detection +if y < 0. However y ≥ 0: +y = (1 − b) − a(1 − γm) += (1 − a) − (b − aγm) +≥ 0 +since a = τ(γm) ≤ 1 and b = µ∗ +sq(γm) ≥ τ(γm)γm as µ∗ +sq(γm) is concave increasing. We achieve y = 0 only if +a = 1, b = γm = 0; if y = 0, we also have z ≥ 0. +Therefore, there are only two scenarios to consider: +1. z ≥ 0: f(x) is constant or increasing, g(x) is constant or increasing → γ∗ = γM +2. z < 0: f(x) is convex decreasing, g(x) is convex increasing. As we will show, f(x)g(x) is convex → γ∗ ∈ {γm, γM}. +In the case that z < 0 (and y > 0 by the contrapositive), we have +arg max +x∈[0,γM−γm] +f(x)g(x) = +arg max +x∈[0,γM−γm] +log (f(x)g(x)) += +arg max +x∈[0,γM−γm] +log +�(b − µsγm) + (a − µs)x +(b + αβ2) + ax +(1 − b) − ax +(1 − γm) − x +� += +arg max +x∈[0,γM−γm] +log +�c1 + c2x +c3 + c4x +c5 − c6x +c7 − x +� +We consider the curvature of this: +∂ +∂x2 log +�c1 + c2x +c3 + c4x +c5 − c6x +c7 − x +� += +� +c2 +4 +(c3 + c4x)2 − +c2 +2 +(c1 + c2x)2 +� ++ +� +1 +(c7 − x)2 − +c2 +6 +(c5 − c6x)2 +� +> 0 if c1c4 > c2c3 and c5 > c6c7 +But these conditions are equivalent to z < 0 and g(x) increasing: +c1c4 − c2c3 = a(b − µsγm) − (a − µs)(b + αβ2) += µs(b − aγm) − αβ2(a − µs) += −z > 0 +c5 − c6c7 = 1 − b − a(1 − γm) += y > 0. +Since we are maximizing a convex function over a bounded domain, the maximizer is one of the domain endpoints, which +implies the endpoints of all of the segments (except for possibly γM = 0.5) correspond to categorical solutions. Therefore, +the only possible optimal non-categorical solution, where ρ > 0, occurs when the constraint µq ≤ 0.5 is tight. +Lemma A.2. Suppose two solutions with µ(1) +sq , µ(2) +sq and D(1), D(2) respectively, where D = +µs−µsq +1−µs +µq−µsq +1−µq . Assume +that µ(1) +sq +≥ D(1) and µ(2) +sq +≥ D(2). Wlog let µ(1) +sq +≥ µ(2) +sq . Then, ˆF (1) +β +> ˆF (2) +β +iff z > 0, β2 > β2 +crit, where z = +� +µ(1) +sq − µ(2) +sq +� +− +� +D(1) − D(2)� +and β2 +crit = +µ(2) +sq D(1)−µ(1) +sq D(2) +αz +. Moreover, ˆF (1) +β += ˆF (2) +β +iff µ(1) +sq = µ(2) +sq , D(1) = D(2). +Proof. Using the alternate definition of ˆFβ, we see that ˆF (1) +β +≥ ˆF (2) +β +iff +(1 + β2)µ(1) +sq − D(1) +µ(1) +sq + αβ2 ≥ (1 + β2)µ(2) +sq − D(2) +µ(2) +sq + αβ2 . +This is equivalent to +αβ2 � +µ(1) +sq − µ(2) +sq + D(2) − D(1)� +≥ µ(2) +sq D(1) − µ(1) +sq D(2). + +Coincident Learning for Unsupervised Anomaly Detection +Let z = µ(1) +sq − µ(2) +sq + D(2) − D(1), and c = µ(2) +sq D(1) − µ(1) +sq D(2). We can also write c = +� +µ(1) +sq − µ(2) +sq +� � +µ(1) +sq − D(1)� +− +µ(1) +sq z. The first term is always at least 0 by assumption. Therefore, if z ≤ 0, we must have c ≥ 0, and ˆF (1) +β +is never strictly +better; it is equal iff z = 0, µ(1) +sq = µ(2) +sq . ˆF (1) +β +is strictly better iff z > 0, αβ2 > c +z. +Theorem 2.5. Suppose the variable x exists in some probability space (Ω, F, P). As depicted in Figure 3, denote A ∈ F +and Ac its complement, where α = P(A) ≤ 0.5. Assume (i) s(A) ∩ s(Ac) = s(B) and q(A) ∩ q(Ac) = q(C); (ii) s and +q are independent when x ∈ A (and similarly when x ∈ Ac); (iii) P(A ∪ B), P(A ∪ C) ≤ 0.5; (iv) P(A \ B \ C) > 0; +(v) P(Ac \ B \ C) ≥ P(Ac \ B)P(Ac \ C); and (vi) P(Ac \ B \ C)P(A \ C) ≥ P(A \ B \ C)P(Ac \ C). Wlog let +P(A \ B) ≥ P(A \ C). Then the maximum of ˆFβ is achieved when +ps(s(A \ B)) ≡ 1, +pq(q(A \ C)) ≡ 1, +ps(s(Ac \ B))) ≡ 0, +pq(q(Ac \ C)) ≡ 0, +ps(s(B))) ≡ 1 +� +β2 ≥ β2 +crit +� +, +pq(q(C))) ≡ 1. +where βcrit depends on the sets A, Ac, B, and C. +Proof. Define +P(A \ B \ C) = c1P(A) +P(Ac \ B \ C) = c4P(Ac) +P(A \ B) = (c1 + c2)P(A) +P(Ac \ B) = (c4 + c5)P(Ac) +P(A \ C) = (c1 + c3)P(A) +P(Ac \ C) = (c4 + c6)P(Ac) +d1 = 0.5 − P(A ∪ B) +P(Ac \ B) +d2 = 0.5 − P(A ∪ C) +P(Ac \ C) +d3 = +0.5 +P(Ac \ B) +d4 = +0.5 +P(Ac \ C) +At first glance, assigning the sets to labels looks combinatorially difficult, since it is unclear in what order we should +assign labels to sets and if any set(s) should be always be labeled the same. We proceed by eliminating most possibilities. +By Theorem 2.3 and Theorem 2.4, specifically the form of p∗ +q(q) (depending on w(q) = Es|q∈Ds|q[ps(s)]) and the (near) +categorical optimal, we can determine which forms of the solution are possible. +Consider a fixed ps(s) assignment with ps(s(A \ B)) = v1, ps(s(B)) = v2, ps(s(Ac \ B)) = v3. The question is +then in what order do we label q(A \ C), q(C), q(Ac \ C). Using the independence of s and q within A and Ac (i.e., +c1 +c1+c3 = +c2 +1−(c1+c3)), we have +w(q(A \ C)) = v1c1 + v2c3 +c1 + c3 += v1c2 + v2(1 − (c1 + c2 + c3)) +1 − (c1 + c3) +w(q(Ac \ C)) = v3c4 + v2c6 +c4 + c6 += v3c5 + v2(1 − (c4 + c5 + c6)) +1 − (c4 + c6) +w(q(C)) = (v1c2 + v2(1 − c1 − c2 − c3)) P(A) + (v3c5 + v2(1 − c4 − c5 − c6)) P(Ac) +(1 − (c1 + c3))P(A) + (1 − (c4 + c6))P(Ac) +. +Note that w(q(C)) is a particular combination of w(q(A\C)) and w(q(Ac\C)) such that either w(q(A\C)) ≥ w(q(C)) ≥ +w(q(Ac \ C)) or w(q(A \ C)) ≤ w(q(C)) ≤ w(q(Ac \ C)). Therefore, for any setting of v1, v2, v3, we will label in either +the order q(A \ C), q(C), q(Ac \ C) or q(Ac \ C), q(C), q(A \ C). +By flipping the role of ps and pq and applying this logic again, we need only consider solutions in s and q that label +in this manner. Moreover, s and q will label in the same order. Concretely, suppose we labeled s in the order s(A \ +B), s(B), s(Ac \ B), with one of the labeling possibilities respecting the constraint µs ≤ 0.5: (i) v1 = 1, v2 = v3 = 0; +(ii) v1 = v2 = 1, v3 = 0; or (iii) v1 = v2 = 1, v3 = d2. Then, we also label q in the order q(A \ C), q(C), q(Ac \ C)) as +w(q(A \ C)) ≥ w(q(Ac \ C)) under each possibility. +As a result, there are only 8 possible optimal solutions, that can be split into 3 groups: + +Coincident Learning for Unsupervised Anomaly Detection +• ps(s(A\B)) = pq(q(A\C)) = 1, ps(s(B)) = ρ, pq(q(C)) = η, ps(s(Ac \B)) = pq(q(Ac \C)) = 0: Four possible +solutions with ρ∗, η∗ ∈ {0, 1}. +• ps(s(A ∪ B)) = pq(q(A ∪ C)) = 1, ps(s(Ac \ B)) = ρ, pq(q(Ac \ C)) = η: Three additional possible solutions with +(ρ∗, η∗) ∈ {(0, d2), (d1, 0), (d1, d2)}. +• ps(s(A∪B)) = pq(q(A∪C)) = 0, ps(s(Ac \B)) = ρ, pq(q(Ac \C)) = η: One only possible solution ρ∗ = d3, η∗ = +d4. +Let’s consider the first group, with ps(s(A \ B)) = pq(q(A \ C)) = 1, ps(s(Ac ∪ B)) = ρ, pq(q(Ac ∪ C)) = η, ps(s(Ac \ +B)) = pq(q(Ac \ C)) = 0. We have +µsq = [c1 + ρc3 + ηc2 + ρη(1 − c1 − c2 − c3)] P(A) + ρη(1 − c4 − c5 − c6)P(Ac) +µs = [(c1 + c2) + ρ(1 − c1 − c2)] P(A) + ρ(1 − c4 − c5)P(Ac) +µq = [(c1 + c3) + η(1 − c1 − c3)] P(A) + η(1 − c4 − c6)P(Ac) +µs − µsq +1 − µs += (1 − η) [c2 + ρ(1 − c1 − c2 − c3)] P(A) + ρ [(1 − η)(1 − c4 − c5) + ηc6] P(Ac) +(1 − ρ)(1 − c1 − c2)P(A) + [1 − ρ(1 − c4 − c5)] P(Ac) +µq − µsq +1 − µq += (1 − ρ) [c3 + η(1 − c1 − c2 − c3)] P(A) + η [(1 − ρ)(1 − c4 − c6) + ρc5] P(Ac) +(1 − η)(1 − c1 − c3)P(A) + [1 − η(1 − c4 − c6)] P(Ac) +Note that due to the assumption P(A \ B) ≥ P(A \ C), we have c2 ≥ c3 ≥ 0. To analyze the possible solutions +(ρ∗, η∗ ∈ {0, 1}), we consider the alternate definition for ˆFβ +ˆFβ = (1 + β2) µsq − D +µsq + αβ2 +D = µs − µsq +1 − µq +µq − µsq +1 − µs +under the four options: +• ρ∗ = η∗ = 0: Since µs, µq ≥ µsq, we have D ≥ 0 and +ˆFβ = (1 + β2) c1P(A) − D +c1P(A) + αβ2 +• ρ∗ = 1, η∗ = 0: Since µsq = µq, we have D = 0 and +ˆFβ = (1 + β2) +(c1 + c3)P(A) +(c1 + c3)P(A) + αβ2 +• ρ∗ = 0, η∗ = 1: Since µsq = µs, we have D = 0 and +ˆFβ = (1 + β2) +(c1 + c2)P(A) +(c1 + c2)P(A) + αβ2 +• ρ∗ = η∗ = 1 +ˆFβ = (1 + β2) P(A) + (1 − c4 − c5 − c6)P(Ac) − D +P(A) + (1 − c4 − c5 − c6)P(Ac) + αβ2 +D = +c5 +c4 + c5 +c6 +c4 + c6 +Since D(ρ∗ = 0, η∗ = 0) ≥ 0, we have +ˆFβ(ρ∗ = 1, η∗ = 0) ≥ ˆFβ(ρ∗ = 0, η∗ = 0) + +Coincident Learning for Unsupervised Anomaly Detection +where the inequality is equality iff c3 = 0. Furthermore, since c2 ≥ c3 ≥ 0, we have +ˆFβ(ρ∗ = 0, η∗ = 1) ≥ ˆFβ(ρ∗ = 1, η∗ = 0) +where the inequality is equality iff β = 0 or c2 = c3. The relationship between ˆFβ(ρ∗ = 0, η∗ = 1) and ˆFβ(ρ∗ = 1, η∗ = 1) +depends on the various probabilities and also the setting of β. Specifically, by Lemma A.2, we have +ˆFβ(ρ∗ = 1, η∗ = 1) > ˆFβ(ρ∗ = 0, η∗ = 1) +iff z > 0 and β2 > β2 +crit where +z = (µsq(ρ∗ = 1, η∗ = 1) − µsq(ρ∗ = 0, η∗ = 1)) − (D(ρ∗ = 1, η∗ = 1) − D((ρ∗ = 0, η∗ = 1)) += ((1 − c1 − c2)P(A) + (1 − c4 − c5 − c6)P(Ac)) − +c5 +c4 + c5 +c6 +c4 + c6 += P(A ∩ B) + P(Ac ∩ B ∩ C) − P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) +β2 +crit = µsq(ρ∗ = 0, η∗ = 1)D(ρ∗ = 1, η∗ = 1) − µsq(ρ∗ = 1, η∗ = 1)D(ρ∗ = 0, η∗ = 1) +αz += (c1 + c2)P(A) +αz +c5 +c4 + c5 +c6 +c4 + c6 += P(A \ B) +P(A) +P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) +1 +z +We have equality iff c5c6 = 0 (i.e., B or C is the empty set); but by assumption (P(A \ B) ≥ P(A \ C)), this must be +satisfied first by B = ∅. In this case, we don’t care about the setting of ρ. So whenever we care to label s(B), the condition +on z is always met under the assumption P(Ac \ B \ C)P(A \ C) ≥ P(A \ B \ C)P(Ac \ C): +z = P(A ∩ B) + P(Ac ∩ B ∩ C) − P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) += P(A ∩ B) + P(Ac ∩ B ∩ C) − P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) += P(A ∩ B) − (1 − P(Ac)) P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) += P(A ∩ B) − P(A)P((Ac \ B) ∩ C) +P(Ac \ B) +P((Ac \ C) ∩ B) +P(Ac \ C) += P(A) +�� +1 − P(A \ B \ C) +P(A \ C) +� +− +� +1 − P(Ac \ B \ C) +P(Ac \ B) +� � +1 − P(Ac \ B \ C) +P(Ac \ C) +�� += P(A) +�P(Ac \ B \ C) +P(Ac \ C) +− P(A \ B \ C) +P(A \ C) ++ +� +1 − P(Ac \ B \ C) +P(Ac \ C) +� P(Ac \ B \ C) +P(Ac \ B) +� +> 0 +where we used the equalities +P(Ac ∩ B ∩ C) +P(Ac ∩ B) += P((Ac \ B) ∩ C) +P(Ac \ B) +P(Ac ∩ B) +P(Ac) += P((Ac \ C) ∩ B) +P(Ac \ C) +P(A ∩ B) +P(A) += P((A \ C) ∩ B +P(A \ C) +since s and q are independent within A and Ac. + +Coincident Learning for Unsupervised Anomaly Detection +Now, let’s consider the second group, with ps(s(A ∪ B)) = pq(q(A ∪ C)) = 1, ps(s(Ac \ B)) = ρ, pq(q(Ac \ C)) = η. +We can write +µsq = P(A) + [(1 − c4 − c5 − c6) + ρc5 + ηc6 + ρηc4] P(Ac) +µs = P(A) + [(1 − c4 − c5) + ρ(c4 + c5)] P(Ac) +µq = P(A) + [(1 − c4 − c6) + η(c4 + c6)] P(Ac) +D = c6 + ρc4 +c4 + c6 +c5 + ηc4 +c4 + c5 +We again analyze the possible solutions ((ρ∗, η∗) ∈ {(0, d2), (d1, 0), (d1, d2)}), and compare to (ρ∗ = η∗ = 0) for +improvement: +• ρ∗ = 0, η∗ = d2: By Lemma A.2 (the condition on z), this solution cannot be an improvement if +µ∗ +sq(ρ∗ = 0, η∗ = d2) − µ∗ +sq(ρ∗ = 0, η∗ = 0) ≤ D(ρ∗ = 0, η∗ = d2) − D(ρ∗ = 0, η∗ = 0) +d2c6P(Ac) ≤ d2c6 +1 +c4 + c6 +c4 +c4 + c5 +P(Ac) ≤ +1 +c4 + c6 +c4 +c4 + c5 +, +which is equivalently P(Ac \ B \ C) ≥ P(Ac \ B)P(Ac \ C). This holds by assumption. The solutions are equal iff +c6 = 0. +• ρ∗ = d1, η∗ = 0: By symmetry, this is not an improvement. The solutions are equal iff c5 = 0. +• ρ∗ = d1, η∗ = d2: In fact, the same condition guarantees that this solution cannot be an improvement either, as we +again have +µ∗ +sq(ρ∗ = d1, η∗ = d2) − µ∗ +sq(ρ∗ = 0, η∗ = 0) ≤ D(ρ∗ = 0, η∗ = d2) − D(ρ∗ = 0, η∗ = 0) +[d1c5 + d2c6 + d1d2c4] P(Ac) ≤ [d1c5 + d2c6 + d1d2c4] +c4 +(c4 + c6)(c4 + c5) +P(Ac) ≤ c4 +c4 +(c4 + c6)(c4 + c5) +None of these solutions are improvements (or even equal except under corner cases) and therefore can be ignored. +Finally, let’s consider the third group, with just the one possible solution: ps(s(A ∪ B)) = pq(q(A ∪ C)) = 0, ps(s(Ac \ +B)) = d3, pq(q(Ac \ C)) = d4. We will compare this to ps(s(A ∪ B)) = pq(q(A ∪ C)) = 1, ps(s(Ac \ B)) = +d1, pq(q(Ac \ C)) = d2. We can in fact we can show that the µsq for both solutions are equal: d3d4c4P(Ac) = P(A) + +[(1 − c4 − c5 − c6) + d1c5 + d2c6 + d1d2c4] P(Ac) since 1 − d1 = d3, 1 − d2 = d4. Moreover, since both solutions +achieve µs = µq = 0.5, they are equal for all β. But since the solution we are comparing against is never optimal, this is +never optimal either! +Theorem 2.6. Suppose ps, pq are parameterized as DNNs, whose final components of each DNN are a fully-connected layer +and a sigmoid activation. Let the latent representation fed into the final layer be zs(s), zq(q) ∈ Rp for the two networks +respectively. Define ps(s) = σ(wT +s zs(s) + bs) and pq(q) = σ(wT +q zq(q) + bq). Then, +∇ws ˆFβ ≡ ED[ˆyq(q)γs(s)zs(s)] +∇bs ˆFβ ≡ ED[ˆyq(q)γs(s)] +∇zs(s) ˆFβ ≡ EDq|s[ˆyq(q)]γs(s)zs(s) +where ˆyp(p) = c1pq(q)−c2 and γs(s) = ps(s) (1 − ps(s)). The gradients w.r.t. the q network are analogous. The constants +c1, c2 depend on the current parameters θs, θq but not individual instances (s, q) ∈ D. If the networks are better than +random guessers, then c1, c2 ≥ 0. + +Coincident Learning for Unsupervised Anomaly Detection +Proof. Recall that for y = σ(aT x + b), we have ∇ay = y(1 − y)x and ∇by = y(1 − y). Let us consider the gradients w.r.t. +the parameters of the linear classifier: +∇ws ˆFβ = ∇ws +� +(1 + β2)µsq − µsµq +µsq + αβ2 +1 − µsq +(1 − µs)(1 − µq) +� += ˆFβ +�∇ws(1 − µsq) +1 − µsq ++ ∇ws(µsq − µsµq) +µsq − µsµq +� +− ˆFβ +�∇ws(µsq + αβ2) +µsq + αβ2 ++ ∇ws(1 − µs) +1 − µs +� += ˆFβ +� +1 +µsq − µsµq +− +1 +1 − µsq +− +1 +µsq + αβ2 +� +ED[pq(q)∇wsps(s)] +− ˆFβ +� +µq +µsq − µsµq +− +1 +1 − µs +� +ED[∇wsps(s)] += ED[(c1pq(q) − c2)∇wsps(s)] += ED +� +(c1pq(q) − c2)ps(s)(1 − ps(s))∇ws(wT +s zs(s) + bs) +� +≡ ED[ˆyq(q)γs(s)zs(s)] +where we use the dominating convergence theorem to exchange the derivative and expectation (note that the expectations +are finite and µsq, µs are differentiable for all choices of weights; moreover the derivatives are bounded). We have defined +γs(s) = ps(s)(1 − ps(s)) +ˆyp(p) = c1pq(q) − c2 +c1 = ˆFβ +� +1 +µsq − µsµq +− +1 +1 − µsq +− +1 +µsq + αβ2 +� +c2 = ˆFβ +� +µq +µsq − µsµq +− +1 +1 − µs +� += ˆFβ +µq − µsq +(µsq − µsµq)(1 − µs) +The gradients w.r.t. the latent representation zs(s) under the ˆFβ loss is +∇zs(s) ˆFβ = EDq|s[(c1pq(q) − c2)ps(s)(1 − ps(s))] +≡ EDq|s[ˆyq(q)]γs(s)zs(s) +Lastly, we can show that c1, c2 ≥ 0 under the assumption that if the networks are better than random guessers (i.e., +µsq ≥ µsµq ). We have that c1 ≥ 0: +c1 = ˆFβ +� +1 +µsq − µsµq +− +1 +1 − µsq +− +1 +µsq + αβ2 +� += ˆFβ +�−(µsq − µsµq)(1 − µsq) + (µsq + αβ2) (1 − 2µsq + µsµq) +(µsq − µsµq)(1 − µsq)(µsq + αβ2) +� += ˆFβ +�� +µsµq − µ2 +sq +� ++ αβ2 (1 − 2µsq + µsµq) +(µsq − µsµq)(1 − µsq)(µsq + αβ2) +� +≥ 0 +where µsµq ≥ µ2 +sq by Cauchy-Schwartz (shown in more detail in the proof of Theorem 2.3) and since µsq ≤ 0.5. Also, +since µsq ≤ µq by definition, we have c2 ≥ 0. +B. Supplementary Materials for Section 4 (Experiments) +In the continuous setting, where the algorithms are represented by DNNs, we need to enforce the constraints µs, µq ≤ +0.5. In order to use gradient-based optimizers, we impose these constraints using a “wall” regularization term Lwall = + +Coincident Learning for Unsupervised Anomaly Detection +µsσ (t(µs − 0.5)) + µqσ (t(µq − 0.5)) where t is the wall’s temperature (set to 50 for all experiments) and σ(·) is the +sigmoid function. Also, to prevent the network from quickly railing to {0, 1} (and thereby suffer from vanishing gradients), +we impose a norm penalty on the magnitude of the output logits (i.e., outputs prior to the final sigmoid converting to +probability): Lmag = E +� +logit(ps)2 + logit(pq)2� +. In total, we minimize the loss +L = − ˆFβ + λwallLwall + λmagLmag +where λwall, λmag are the regularization strengths. +B.1. MNIST +For the MNIST example, we use a noisy observation model to ensure that each anomaly class (digits 1, 2 and 3) has a +different amount of noise. In particular, we suppose that each anomaly digit has a weight wi and a “blur” probability bi. Then, +for each stream of data independently, we replace the anomalous digit with the normal digit (i.e., 0) with probability bi; we +also symmetrically replace the normal digit with the anomalous digit i with probability bi. This creates a data set with mixed +image pairs. For our experiments, we used the setting w0 = 0.85, w1 = w2 = w3 = 0.05 and b1 = 0, b2 = 0.05, b3 = 0.2. +It is worth noting that our experiment is not specific to this particular observation model: other ways of generating mixed +image pairs would also work. +To better understand the behavior of our metric in this setting, we plot ˆFβ under the simplification that all digits of the +same class are identical. This allows us to derive a closed form for ˆFβ as a function of the wi, bi, β, and the labeling choice +ps(digit i) = pq(digit i) = yi. We show the 3 obvious labeling choices in Figure 10. Each labeling choice is optimal for a +range of β, and the higher choices of β (i.e., higher recall preference) lead to labeling the noisy classes as anomalous. +Figure 10. ˆFβ for different labeling choices and values of β, under the simplified MNIST model. +We transform each MNIST by randomly cropping it to 25 × 25 and rolling between 0 and 5 pixels along each dimension +independently. We define our algorithms Aθs, Aθq as convolutional neural networks (CNNs). As shown in Figure 11, our +networks consist of four 2D convolutions with 10 out channels, kernel size 5, padding 2, and strides 1, 1, 2, 2 respectively, +followed by three FC layers of output size 8, 8, 1 respectively. All layers but the final layer are followed by a ReLU activation. +We train for 3000 epochs of 1800 image pairs, with a batch size of 760, and test by simultaneously feeding images to each +network. We use the Adam optimizer (Paszke et al., 2019) with a learning rate of 10−4. We set α = 0.15, λwall = 1/α, and +λmag = 0. +For completeness, Figure 12 shows the violin plots corresponding to the results in Table 1. +B.2. Milling dataset +We define our algorithms Aθs, Aθq as CNNs. As shown in Figure 13, our networks consist of two 1D convolutions with 10 +out channels, kernel size 5, padding 2, and stride 1, followed by three FC layers of output size 8, 8, 1 respectively. All layers + +LD +y=[0 1 0 0] +y=[0 1 1 0] +60 +y=[0 1 1 1] +0.B +0.7 +0.6 +0.5 +t0 +0.D +50 +LD +15 +2D +25 +3.D +3.5 +4.DCoincident Learning for Unsupervised Anomaly Detection +Input +2D Convolution + ReLU +2D Convolution + ReLU +2D Convolution + ReLU +2D Convolution + ReLU +Fully-connected + ReLU +Fully-connected + ReLU +Fully-connected + Sigmoid +Anomaly Probability +(1,25,25) +(10,25,25) +(10,25,25) +(10,13,13) +490 +8 +8 +1 +Figure 11. Model architecture for each MNIST DNN, showing the input size to each layer. +β = 0.01 +β = 1 +β = ∞ +Figure 12. Violin plot showing predictions from the ˆFβ models on the MNIST digits. Prediction values are the products of the two +network outputs. From left to right, β = 0.01, 1, ∞. +but the final layer are followed by a ReLU activation. The s net has 2 input channels; the q net has 4 input channels. We +train for 50 epochs using a batch size of 1024 and the train-test splits of (Hahn & Mechefske, 2021). We use the Adam +optimizer (Paszke et al., 2019) with a learning rate of 3 × 10−3. We set α = 0.438, according the flank wear labels. We set +λwall = 1 and λmag = 2e − 4. +As discussed in Section 4.3, the labels defined in (Hahn & Mechefske, 2021) are quite simplistic, since they ignore the +milling parameters: metal type (iron or steel), cut speed (0.25mm/rev or 0.5mm/rev), and cut depth (0.75mm or 1.5mm). +Figure 14 shows the model’s anomaly confidence versus the flank wear for the 8 different milling configurations (using +β = 6). We see there is, at best, a weak correlation between our predictions and flank wear in aggregate but a very strong +correlation for each individual configuration. In fact, our anomaly confidence is physically interpretable: as the feed rate and +depth increase (i.e., more material volume cut per revolution), the degree of flank wear corresponding to anomalous milling +decreases. + +FbetaLoss:beta=0.0l(S*Q) +Digit 0 +Digit 1 +C +. +C +. +Digit 2 +Digit 3 +Normal +Abnormal +Prediction +Predictionβ=1 +O +O +Digit 0 +Digit 1 +Q +0 +O +C +Digit 2 +O +0 +O +O +O +Digit 3 +O +O +O +Normal +Abnormal +Prediction +PredictionF_beta Loss: beta = Inf (S * Q) +Digit 0 +. +Digit 1 +Digit 2 +Digit 3 +Normal +Abnormal +Prediction +PredictionCoincident Learning for Unsupervised Anomaly Detection +Input +1D Convolution + ReLU +1D Convolution + ReLU +Fully-connected + ReLU +Fully-connected + ReLU +Fully-connected + Sigmoid +Anomaly Probability +(2 or 4,64) +(10,64) +640 +8 +8 +1 +Figure 13. Model architecture for the milling DNNs, showing the input size to each layer. +B.3. Particle accelerator RF stations +We use the dataset constructed in (Humble et al., 2022) with additional transformations: (i) normalize the data, (ii) add +0.1 × N (0, 1) noise to each input, (iii) roll the BPM data between 0 and 10 points randomly, and (iv) select only the +last 50 points in the RF data and the last 1000 in the BPM data. We define our algorithms Aθs, Aθq as CNNs. As shown +in Figure 15(a), our s network consists of two 1D convolutions with 5 out channels, kernel size 10, padding 0, and stride 1, +followed by three FC layers of output size 6, 6, 1 respectively. As shown in Figure 15(b), our q network is very similar, using +15 out channels, a kernel size 20, and a stride of 5 instead. All layers but the final layer are followed by a ReLU activation. +We train for 2000 epochs using a batch size of 400 and a train-test split of 85% − 15%. We use the Adam optimizer (Paszke +et al., 2019) with a learning rate of 10−4. We set α = 0.2, which is the approximate anomaly rate based on the labels. We +set λwall = 1/α and λmag = 0. + +Coincident Learning for Unsupervised Anomaly Detection +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Cast iron, feed=0.25mm/rev, depth=0.75mm +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Cast iron, feed=0.5mm/rev, depth=0.75mm +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Cast iron, feed=0.25mm/rev, depth=1.5mm +0.0 +0.2 +0.4 +0.6 +0.8 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Cast iron, feed=0.5mm/rev, depth=1.5mm +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Steel, feed=0.25mm/rev, depth=0.75mm +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Steel, feed=0.5mm/rev, depth=0.75mm +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Steel, feed=0.25mm/rev, depth=1.5mm +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Flank wear +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Confidence in anomaly +Steel, feed=0.5mm/rev, depth=1.5mm +Figure 14. Anomaly probability versus flank wear for different milling configurations. Black line is a sigmoid fit. + +Coincident Learning for Unsupervised Anomaly Detection +Input +1D Convolution + ReLU +1D Convolution + ReLU +Fully-connected + ReLU +Fully-connected + ReLU +Fully-connected + Sigmoid +Anomaly Probability +(5,50) +(5,41) +160 +6 +6 +1 +(a) Architecture for RF station data. +Input +1D Convolution + ReLU +1D Convolution + ReLU +Fully-connected + ReLU +Fully-connected + ReLU +Fully-connected + Sigmoid +Anomaly Probability +(14,1000) +(15,197) +540 +6 +6 +1 +(b) Architecture for BPM data. +Figure 15. Model architecture for the particle accelerator DNNs, showing the input size to each layer. + diff --git a/xtFIT4oBgHgl3EQf0StA/content/tmp_files/load_file.txt b/xtFIT4oBgHgl3EQf0StA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a18f2c565bd6fa79ef5d2fd88ddc29ed3ae770e1 --- /dev/null +++ b/xtFIT4oBgHgl3EQf0StA/content/tmp_files/load_file.txt @@ -0,0 +1,1549 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf,len=1548 +page_content='Coincident Learning for Unsupervised Anomaly Detection Ryan Humble 1 * Zhe Zhang 2 Finn O’Shea 2 Eric Darve 1 Daniel Ratner 2 * Abstract Anomaly detection is an important task for com- plex systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', industrial facilities, manufac- turing, large-scale science experiments), where failures in a sub-system can lead to low yield, faulty products, or even damage to components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' While complex systems often have a wealth of data, labeled anomalies are typically rare (or even nonexistent) and expensive to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' In this pa- per, we introduce a new method, called CoAD, for training anomaly detection models on unlabeled data, based on the expectation that anomalous be- havior in one sub-system will produce coincident anomalies in downstream sub-systems and prod- ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Given data split into two streams s and q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', subsystem diagnostics and final product qual- ity), we define an unsupervised metric, ˆFβ, out of analogy to the supervised classification Fβ statis- tic, which quantifies the performance of the inde- pendent anomaly detection algorithms on s and q based on their coincidence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' We demonstrate our method in four cases: a synthetic time-series data set, a synthetic imaging data set generated from MNIST, a metal milling data set, and a data set taken from a particle accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Introduction The problem of anomaly detection, the task of finding ab- normal events or data, is an important task for complex systems, such as industrial facilities, manufacturing, and large-scale science experiments (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Edelen & Cook, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Lindemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Radaideh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Failures in these sys- tems can lead to low yield, faulty products, or even damage to components, making identifying these failures a high- priority task for system operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' However, the complexity of these systems typically ensures that labeled data is rare Equal contribution 1Institute for Computational and Mathemat- ical Engineering, Stanford University, Stanford, California 2SLAC National Laboratory, Menlo Park, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFIT4oBgHgl3EQf0StA/content/2301.11368v1.pdf'} +page_content=' Correspondence to: Ryan Humble + +1.055 +45 +40 +00 +Gravity +1.05 +35 +000 +30 +Standard +1.045 +25 +M +20 +o +15 +1.04 +o +10 +000 +5 +0 +5 +10 +15 +20 +25 +30 +Trial number0.20 +0.15 +R 0.10 +0.05 +0.00 +1500 2000 2500 3000 +ReInstability Transitions in a Low Viscosity Jet +9 +1600, the transition M increases as a function of Re; below Re=1600, the instability +was weak, and it was difficult to distinguish the nature of the mode. For higher Re, the +transition value of M increases with Re, and at Re=2800 (not shown in Fig. 7, the first +trial showed a helical mode before switching into an axisymmetric mode, suggesting the +critical viscosity required is higher. + +10 +X. Tan et al. +(a) +(b) +Figure 5: (Top) Growth of axisymmetric instabilities for M=1 and multiple Re: (a) +Re=428 (b) Re=1036 (c) Re=1545 (d) Re=2009 (e) Re=2539 (f) Re = 3009 (Bottom) +Helical modes observed at M=45, Re = 1332, 1676, 2013, 2339. + +a +dInstability Transitions in a Low Viscosity Jet +11 +Figure 6: Images showing the transition from helical to axisymmetric modes as M is +decreased +Figure 7: Transition boundary in (M,Re) space from helical to axisymmetric modes at a +fixed Reynolds number of 2013. From left to right, the values of M are 45, 33, 28, 23 and +20. Here M=28 appears to be closest to the transition between modes. + +50 +45 +- +40 +35 +30 +M +25 +20 +15 +10 +Transition +Helical +5 +Axisymmetric +Combined +0 +1200 +1400 +1600 +1800 +2000 +2200 +2400 +Re12 +X. Tan et al. +[hb] +Figure 8: Evolution of velocity spectra in the downstream direction for M=39, Re= 2013 +on the jet axis and in the shear layer. Top: centerline variation for Re=2013. Bottom: +spectra in the shear layer. +3.2. Hot Film Anemometry +Hot film anemometry was used to characterize the flow for values of M greater than +unity. With the mixing of the two fluids and the change in Prandtl number, the calibration +for the hot film could no longer be used, and the voltage response is presented. Here we +are interested in the spectral content of the velocity fluctuations at different downstream +distances, as well as the rate of growth of the disturbance relative to the constant property +jet. Figure 8(a) and (b) show the evolution of the spectrum along the centerline and in the +shear layer for M= * and Re=1682. A distinct frequency peak is visible at all locations +in the near field. The strength of voltage fluctuations (Fig. 9) for M=1 shows a relatively +gentle increase downstream; for large M the strength shows a sharp increase within one +jet diameter, appearing to saturate within a few diameters. +The dependence of the dominant frequency on the viscosity ratio, as detected by hot +film anemometry, is plotted in Fig. 10. Following the experimental sequence and moving +from high values of M to low values, one sees an increasing trend while the helical mode +remains dominant. . +3.3. Image Analysis +The hot film measurements strongly indicate the existence of a single dominant mode +that saturates in intensity in the first few diameters downstream of the jet exit. However, +the increased conductivity of the liquid due to the dissolved salts resulted in increased +contamination, pickup of electrical line noise despite probe shielding, and the occasional +air bubbles introduced into the tank due to mixing that would stick to the hot film, +together resulted in a very low rate at which meaningful data were acquired. As a result, +LID data were chosen as a means of investigating the growth of unstable modes. The +orange filter on the camera lens ensured that the jet could be strongly distinguished +against the background, by isolating the emission from the Rhodamine dye under blue +illumination. Applying a threshold intensity to the grayscale images allows determination +of the jet boundary; the diameter of the jet as determined from the result was very weakly +sensitive to the threshold value, but we are primarily interested in the frequency, which + +×10-3 +Re =2013 at z/D =0, centerline +×10~3 +Re = 2013 at z/D = 1, centerline +×10-3 +Re = 2013 at z/D = 2, centerline +×10-3 +Re = 2013 at z/D = 3, centerline +3. +3 +3 +2.5 +2.5 +2.5 +2.5 +M2 +2 +2 +.5 +1.5 +1.5 +0.5 +0.5 +0.5 +0.5 +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +50 +60 +70 +80 +Freq (Hz) +Freq (Hz) +Freq (Hz) +Freq (Hz) +Re = 2013 at z/D = 0, shear layer +Re = 2013 at z/D = 1, shear layer +Re = 2013 at z/D = 2, shear layer +Re = 2013 at z/D = 3, shear layer +0.01 +0.01 +0.01 +0.01 +0.008 +008 +0.008 +0.008 +(V) +006 +0.006 +0.006 +004 +0.004 +0.004 +A +0.002 +002 +0.002 +0.002 +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +60 +70 +80 +20 +30 +40 +50 +60 +70 +80 +20 +30 +40 +60 +70 +80 +Freg (Hz) +Fre (Hz) +Freg (Hz) +Fre (Hz)Instability Transitions in a Low Viscosity Jet +13 +Figure 9: Root mean square value of voltage fluctuations along the centerline and shear +layer for M=1 and M=45 at Re=2000 +Figure 10: Stanton numbers corresponding to the dominant frequency, as identified by +hot film anemometry, as a function of M for Re=1682. +is unaffected by the choice of threshold. To study the spatial evolution of the oscillations +of the interface, we examine the jet width at 4 locations downstream of the jet exit, as +plotted in Fig. 11. The amplitude of oscillations shows a non-linear increase, and in Fig. 12 +we further examine the amplitude and frequency of these oscillations. Figure 12 shows the +sharp peak in the power spectrum that might be surmised from the oscillatory waveform +in Fig. 11. Figure 12(b) shows the variation of the amplitude of the dominant frequency in +the downstream direction. Again, as with the anemometry measurements, the oscillations +show an exponential increase in the disturbance amplitude, before saturating at z/D=* +. +To ascertain the nature of this instability, that develops much faster than the axisym- +metric instability of the constant property jet, we verified that the frequencies at the +different downstream stations shown in Fig. 11 are identical. Another way of assessing +the spatially invariant ‘global’ nature of this frequency is to examine the intensity records +of at single pixels in the shear layer. Figure 13 shows the frequency spectrum of 4 pixels at +two different downstream locations, on either side of the jet. The frequencies are identical, + +15 +eM=39.Centerline +M=39,ShearLayer +M=1,Centerline +M=1,ShearLayer +10 +(%) +VI V. +5 +04 +0 +2 +3 +4 +5 +z/D0.8 +Re=1682 +Re=842,Trial 1 +0.7 +Re=842, Trial 2 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +20 +30 +40 +50 +M14 +X. Tan et al. +[b] +Figure 11: (a) Time variation of the jet width in pixels at different downstream distance +(b) Square of the amplitude of the coefficients of the Fourier transform, A2(f) +(a) +(b) +Figure 12: (a) Power spectrum of oscillation of interface at z/D= for M=38, Re=2400. +(b) Growth of the square of the amplitude of the Fourier coefficient of the dominant +mode in the downstream direction +and provide further circumstantial evidence that the instability observed at large M is a +global mode, corresponding to absolute instability of the near-field profiles. This putative +global mode has a frequency that depends on the parameters that define the flow, such +as the inlet Reynolds number, viscosity ratio M, and the inlet velocity profile, specified +by the momentum thickness θ. In the experiment, the values of Re and θ are conjoined +through the specific geometry of the nozzle. Further, it is experimentally difficult to +conduct trials at constant M, while changing Re. Therefore, we present the global mode +frequency at constant Re (and θ) as a function of M. Figure 14(a) shows that the waves +developing on the interface have a frequency that decreases as the ambient viscosity is +reduced from a starting value of M≈ 45. For this data set taken at Re= 2000, there is +a sharp increase in frequency near the observed transition from helical to axisymmetric +mode. We interpret this as further evidence of the helical mode being driven by an +absolute instability of near-field profiles. After the transition to the axisymmetric mode, +there is a further decrease in frequency, before the curve displays an asymptotic behavior +as M is further reduced. + +9mm +6mm +3mm +0.5mm +0.0 +0.5 +10 +15 +2.0 +2.5 +3.0 +3.5 +Time(s)0.4 +E0 +0.2 +0.1 +0.0 +20 +40 +60 +801.0 +0.8 +0.6 +A^2 +0.4 +0.2 +0.Q +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +z/DInstability Transitions in a Low Viscosity Jet +15 +Figure 13: Spectrum of pixel intensity fluctuation in shear layer at four locations in the +near-field of the jet. +The corresponding wavelengths, as determined from inspection of the images, are +shown in Figure 14(b). Knowledge of the wavelength and frequency allows us to calculate +the phase velocity of the dominant mode; this is plotted in Figure 14(c). +4. Summary and Discussion +We have presented experiments characterizing the flow behavior for the specific con- +figuration of a low-viscosity jet with a laminar boundary layer emerging into an ambient +medium of relatively higher viscosity. The fluids, while perfectly miscible, have a low +binary diffusion coefficient, with the result that the diffusion region is expected to +be extremely thin, approaching a sharp interface with zero surface tension. The flow +visualization results clearly indicate the onset of a helical mode at or close to the nozzle +exit plane, when the viscosity ratio M is increased substantially beyond unity, with +enhanced mixing relative to the M=1 case. For that baseline case, it is apparent that the +jet can stay coherent and nearly parallel for a significant distance downstream indicating +a slowly evolving spatial instability that saturates in the far-field. For latge M, hot film +anemometry confirms the presence of a self-sustained oscillation with a discrete peak in +the spectrum, with the disturbance energy rapidly growing and saturating within the first +few diameters downstream. Further, we have characterized this dominant frequency as a +function of M and Re over a limited range, showing that this frequency is an increasing +function of Re and a decreasing function of M. However, over the range of M studied, the +anemometry does not display any sharp jumps in the frequency, leaving open the question +of the convective/absolute nature of the instability. To resolve issues with anemometry +in conducting fluids, a larger range of Reynolds numbers were examined using image +analysis. With this technique,the initial linear growth, followed by an exponential rise in +the amplitude of oscillations is more clearly evident. We are able to confirm the frequency +trends observed with anemometry, a discrete jump in frequency at a certain value of M, +as well as the spatial invariance of this frequency. +Turning to the convective/absolute nature of the instability, at first glance, one cannot + +50 +40 +40 +30 +30 +20 +20 +10 +10 +5 +10 +15 +20 +25 +30 +35 +5 +10 +15 +20 +25 +30 +35 +17.5 +50 +15.0 +40 +12.5 +10.0 +30 +7.5 +20 +5.0 +2.5 +10 +0.0 +0 +5 +10 +15 +20 +25 +30 +35 +Ln +10 +15 +20 +25 +30 +3516 +X. Tan et al. +(a) +(b) +(c) +Figure 14: Variation of instability characteristics along the constant (Re,θ) curve, as M +decreases dueing the experiment. (a) Frequency (b) Wavelength (c) Phase velocity +rule out the possibility that these observations reflect a rapidly growing convective +instability that happens to be clearly visible against a low-noise background established +by pump-less flow. However, there are several features of the flow, as discussed above, +that suggest otherwise. It is generally accepted that for an instability to be considered a +global mode that arises from a linear, absolutely unstable mechanism (as understood +in the context of spatio-temporal linear stability analysis), it should display certain +hallmarks. These are: the presence of a single self-sustained oscillation that can be +detected everywhere in the domain; the sharp onset of a regime of enhanced mixing +in which this instability appears, as determined by the appropriate control parameters; +and the insensitivity to external forcing. The first criterion is found to be amply satisfied +in the present experiment. The image analysis does show a sharp jump in frequency +and wavelength as the viscosity ratio is decreased, and the transition value of M is +reasonably consistent with visual observations. Further evidence for the sharp onset is +afforded by an examination of the singular values obtained through Principal Component +Analysis, which suggest that near the critical value of M corresponding to the frequency +jump, there is always only one dominant mode, implying no mode competition close to +the transition point. We also note that in line with the theoretical understanding that + +1.0 +0.9 +number +0.8 +Strouhal +0.7 +0.6 +0.5 +0.4, +10 +20 +30 +40 +50 +60 +Viscosity2.0 +1.8 +1.6 +入/D +1.4 +1.2 +1.0 +10 +20 +30 +40 +50 +60 +Viscosity1.1 +1.0 +0.9 +/ +0.8 +0.7 +0.6 +10 +20 +30 +40 +50 +60 +ViscosityInstability Transitions in a Low Viscosity Jet +17 +that the development length for the global mode is, in principle, infinite at the transition +boundary and onset of the global mode (in terms of the controlling parameters) (Couairon +& Chomaz 1997) but shortens away from the transition boundary as one moves into the +regime of absolute instability, the disappearance of a region of parallel flow for large M, +compared to M=1 is an indicator of the instability being controlled by inlet conditions. +The question of sensitivity (or lack thereof) to external noise remains to be explored. +Since the flow is gravity-fed, pump-driven oscillations such as those used by dOl (2009) +are difficult o implement and current work focuses on applying vibrations to the diffuser +section upstream of the nozzle. A strong response by the system, as in terms of enhanced +amplitude of oscillations, to forcing frequencies near the natural frequency of the insta- +bility would provide further strong support for the idea that the observed helical mode +correspodnds to the absolutely unstable mode found in the companion computational +paper by Yang et al. (2021). However, it should be noted that this may not necessarily +constitute clinching evidence; indeed, Hallberg & Strykowski (2008) have shown that the +global mode is a low-density jet can be overwhelmed by sufficiently strong forcing. We +also note that the decrease in the Strouhal number with increasing M as exemplified in +Figs. 10 and 14(a) is closely aligned with behavior of the absolutely unstable frequency, +as predicted by spatio-temporal linear stability analysis for a specific tanh-type profile +(see Fig. of citetYang2022). +Lastly, it should be noted that global mode characteristics are closely linked to inlet +profiles, and therefore the influence of the boundary layer thickness needs to be explicitly +addressed. In the present study, this thickness is directly linked to the Reynolds number +through the nozzle contraction profile, and future studies will require disentangling the +relative contributions of these two parameters to the evolution of the flow. +Acknowledgement We are grateful for useful discussions with David Forliti and Paul +Strykowski during the preparation of this manuscript. 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Journal of fluid mechanics 255, 323–347. + diff --git a/ztFRT4oBgHgl3EQfkDcP/content/tmp_files/load_file.txt b/ztFRT4oBgHgl3EQfkDcP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a12abb5eaa585f66f9254c7668f34a2da3dc7e4 --- /dev/null +++ b/ztFRT4oBgHgl3EQfkDcP/content/tmp_files/load_file.txt @@ -0,0 +1,547 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf,len=546 +page_content='This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Global Instabilities and Mode Transitions in a Low Viscosity Jet Emerging Into a High Viscosity Medium Vinod Srinivasan1, Xijun Tan1, Ian Wright1 and Akash Dhotre1† 1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' accepted xx) The effect of viscosity contrast between a jet and its surroundings is experimentally investigated, using density-matched fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A gravity-driven flow is established, with a jet of saltwater emerging into an ambient medium composed of high-viscosity propylene glycol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Jet Reynolds numbers Re ranging from 1600 to 3400 were studied, for an ambient- to-jet viscosity ratio M between 1 and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Visualization suggests that at low values of the viscosity ratio, the jet breakdown mode is axisymmetric, while helical modes develop at high values of viscosity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The transition between these two modes is attempted to be delineated using a variety of diagnostic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Hot film anemometry measurements indicate that the onset of the helical mode was accompanied by the observation of a discrete peak in the frequency spectrum of velocity fluctuations, which exhibited little spatial variation for the first several diameters in the downstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Dye injection accompanied by Laser-Induced Fluorescence was used to identify the jet boundary against the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' An analysis of high-speed images acquired using the LIF technique enables identification of the spatial growth rate of waves on the jet boundary, as well as the frequency of oscillation of the weakly diffusive interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Temporal fluctuations in of fluorescence intensity are found to be spatially invariant in the jet near-field, further attesting to behavior consistent with that of a self-sustained oscillation whose frequency depends on the viscosity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Singular Value Decomposition was used to analyze the images and identify the various spatial modes, and suggests the existence of a single dominant mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Together, thee observations provide strong circumstantial evidence for the ecidence of a glonal mode that arises solely due to viscosity variation in a jet flow, without any additional effects due to density variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Key words: Jets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' shear layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' shear-flow instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' absolute/convective instability, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Introduction Shear layers with spatially variable fluid physical properties occur in a variety of industrial and natural systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The variations in density and/or viscosity may occur due to temperature gradients, as in the case of a plasma torch (Duan & Heberlein 2002), static mixers (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2003), reacting flows (Pathikonda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2021), or due to species concentration gradients, such as salinity gradients set up when an estuary enters an ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' While most situations will feature both density and viscosity effects, the fluid dynamics † Email address for correspondence: vinods@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='13593v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='flu-dyn] 31 Jan 2023 2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' of variable density jets have been extensively studied, both theoretically (Huerre & Monkewitz 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yu & Monkewitz 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Lesshafft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2006) and experimentally (Kyle & Sreenivasan 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yu & Monkewitz 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Hallberg & Strykowski 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In particular, low-density jets have been shown to be a member of a class of globally unstable flows (Huerre & Monkewitz 1990), which are characterized by enhanced by the sudden onset of a regime with enhanced mixing, self-sustained oscillations and insensitivity to external forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The frequency of the global modes in the near-field of low density jets has been linked to the existence of local profiles over a finite streamwise extent that are absolutely unstable in the framework of local spatio-temporal linear stability analysis (Chomaz 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Pier & Huerre 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' While the primary mechanism of breakdown of the flow is inviscid, arising from the baroclinic torque established by gradients in density and pressure, there are some indications (Hallberg & Strykowski 2006) that viscosity does modify the onset of global modes, as well as their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Hallberg & Strykowski (2006) conducted experiments with multiple nozzle geometries, thereby independently studying the effects of shear layer thickness, density ratio and jet Reynolds number, and found a weak but perceptible effect of jet Reynolds number on the global mode frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The linear stability calculations of Lesshafft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2006) and Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2010) also suggest that the frequency and transition boundary between convective and absolute instability are affected by the viscosity in the form of the Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It is therefore natural to inquire into the effects of variations in viscosity between the jet and ambient, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Strong gradients in viscosity are unlikely to be established in gas flows, and we look to other situations where the role of viscosity gradients has been more extensively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In fact, in contrast to free shear flow, an extensive body of literature on variable viscosity flows addresses pressure-driven internal flows of high Schmidt number fluids (Govindarajan & Sahu 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' While viscosity is instinctively assumed to have a stabilizing influence on the growth of disturbances, it is responsible for altering the base state of a flow, often creating sharp velocity gradients through the no-slip condition and therefore serving as a source of disturbance kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It has long been known (Yih 1967) that a jump in viscosity across a sharp interface can lead to long-wave at any Reynolds number or short-wave instabilities at low Re (Hooper & Boyd 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Here we focus on flows of miscible fluids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' the immiscible situation is covered in reviews by Joseph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (1997) and more recently, Govindarajan & Sahu (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Mention should also be made of the extensive work done in the context of liquid atomization, on planar shear layers with gas-;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='iquid streams (Matas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Otto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Fuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Bozonnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Together, these studies have shown that viscous stability calculations are required to match theory with experiments on gas-liquid shear layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' that absolute instability of co-flowing gas/liquid streams is supported when velocity defects immediately downstream of a splitter plate are considered, and match experimentally observed frequencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' and that confinement and the finite thickness of the gas stream play an important role in destabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, viscosity ratios of the two streams, when considered, were always extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' and the effects of this ratio were rarely isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Ern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2003) showed the destabilizing effects of a finite thickness interface marked by gradients in velocity and viscosity, and demonstrated that for certain parameter ranges, the instability could be stronger than that of the corresponding sharp-interface configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Sharp gradients in velocity profile, combined with variations in velocity profile, lead to additional source terms in the equation for disturbance kinetic energy, which drive the growth of instabilities near the diffuse interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Ranganathan & Govin- darajan (2001) performed a temporal stability analysis of the effects of diffusion in Instability Transitions in a Low Viscosity Jet 3 channel flow of two fluids in a three-layer configuration, and found that when the critical layer (the region where the wave speed matched the mean velocity profile), the flow was strongly stabilized or destabilized when the more viscous fluid was adjacent to the wall or in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For the pipe geometry, Selvam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (Selvam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2007, 2009) performed a linear stability analysis that predicted the onset of absolute instability for low Reynolds numbers when the viscosity contrast is sufficiently high, and the diffuse interface is located in a certain range of radial locations with respect to the pipe radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' They find that when the core fluid is more viscous, the flow can be at best unstable over a certain Reynolds number range, with the axisymmetric mode being dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' When the less viscous liquid is in the core, helical modes are favored, and can lead to absolute instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' These calculations were supported by Direct Numerical Simulation and a global linear stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' These calculations partially reproduced the experimental observations of D’Olce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2008), who reported pearl- and mushroom-shaped instabilities in the Reynolds number range of 2-60, with no sharp transitions in either wavelength or frequency that would provide strong evidence of a global mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The helical modes observed by Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2003) for injection of a low-viscosity fluid into a static mixer are in line with the theoretical results discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Virtually no experimental data are available for miscible free shear layer flows with strong viscosity gradients, in either planar or cylindrical geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A notable situation that features shear layers with significant viscosity variation, accompanied by density variation is that of the buoyant jet (Subbarao & Cantwell 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Bharadwaj & Das 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This has been confirmed to be an absolute instability (Chakravarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2018) when realistic viscosity profiles are included and either the density ratio between the ambient and the jet is greater than 2, or when the Richardson number is greater than 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' however viscosity ratios are not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It is difficult to translate insights from confined flows to unconfined shear layers such as jet flows, due to the fundamental differences in velocity profiles, characterized by inflection points in the case of jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Any insights from pressure-driven flow studies has to be interpreted with caution, since confinement is known to play both stabilizing and destabilizing roles in other situations involving absolute instability of single phase (Juniper 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Healey 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2021) or two-phase flows (Bozonnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Further, the seminal works of Rayleigh [] and Tomotika Tomotika (1935) considered capillary flows of liquid filaments in another viscous medium in the limit of negligible Reynolds number and are not relevant to the present work which is focused on large Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, it is interesting to note that the helical mode is found to arise from an inviscid mechanism based on the Rayleigh criterion for the base profiles used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This would be expected to further favor the establishment of such modes in free shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Sahu & Govindarajan (2014) considered a planar shear layer configuration, and the emergence of an overlap mode when the gradients in velocity and viscosity occur in the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Destabilization was enhanced when these layers overlapped, and with decreasing thickness of either of the gradient regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In line with inviscid theory, the configuration was found to be absolutely unstable when countercurrent velocity profiles were used for the base state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' More recently, Yang & Srinivasan (2022) carried out a linear stability analysis of base profiles corresponding to the near-field of a jet emerging into an ambient with a different viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Their base profiles reflected modifications to the standard tanh- profile typically used in the analysis of jet flows (Mattingly & Chang 1974), such as an inward radial shift due to the decelerating effects of a more viscous ambient, and concentration gradient regions that were much thinner than the momentum thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' As is typical of jet flows, the axisymmetric and helical m odes had nearly equal temporal growth rates over a wide range of conditions specified by 4 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' the jet Reynolds number, ambient-to-jet viscosity ratio, momentum and concentration layer thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, beyond a critical value of viscosity ratio that was Reynolds number=dependent, absolute instability of the flow was supported, with the helical mode being strongly favored over the axisymmetric mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A more systematic study of the transition boundary between convective and absolute instability as a function of the operating parameters is currently underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' With the above preliminary results in mind, we carry out a study that seeks to isolate the effects of large viscosity contrast between a jet and its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The goals of the present study are to characterize the near-field of a low-viscosity jet at moderate Reynolds numbers (1500 < Re < 3500) for ambient-to-jet viscosity ratios ranging from 1 to 45, and to examine the flow field for any evidence of global modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This article is structured as fellows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Section 2 describes the experimental facility used to achieve a neutrally buoyant jet with high viscosity contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Section 3 describes the flow visualization and the observation of disturbance modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Section 5 describes identification of the dominant modes using a Proper Orthogonal Decomposition (POD)-based technique applied to the images from visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Section 6 provides a summary and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Experiments The study of the effects of viscosity gradients alone on the development of instabilities in the near-field of a jet requires the elimination of density effects, as well as a quiet facility with a minimum amount of external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Accordingly, experiments were carried out in a jet facility shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 1 that utilizes gravity to attain the required flow rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The experiments were performed in the vertical configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A large overhead reservoir delivered fluid to a nozzle located in a test section of square cross-section through a flow meter, a diffuser section and a flow straightener composed of laminar flow elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The nozzle has a fifth-order polynomial profile with zero slope and curvature at its inlet and exit planes, and imposes an area contraction of 87 on the entering flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The nozzle exit diameter D is 6 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Jet Reynolds numbers are defined based on the nozzle exit diameter D and the average velocity ¯U of the flow, as inferred from measurement of the volumetric flow rate from the flowmeter (Dwyer ****, accuracy of 2%), Re = ρ ¯UD µj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1) where µj is the viscosity of the injected fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The requirement of having a wide range of viscosity contrast defined in terms of the ambient-to-jet viscosity ratio requires the use of liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Salt water of nominal density 1042 kg/m3) is chosen as the test fluid, in order to facilitate density-matching as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Reynolds numbers up to 4000 can be attained using this reservoir/nozzle combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The jet exhausts into the test section, which is made of transparent polycarbonate and has a square cross-section with inner dimensions 240 × 240mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Overflow ports near the top of the tank enable maintenance of a constant fluid height in the test section during operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The top of the tank is open to allow direct mounting of a hot-film anemometry system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The fluid in the tank creates the desired viscosity ratio, which is defined as M = µ∞ µj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2) where the subscript ∞ refers to test section conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For this study, propylene glycol Instability Transitions in a Low Viscosity Jet 5 Figure 1: Sketch of the jet facility used to produce a low-viscosity jet sing gravity-driven flow and salt water were used as the two fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Propylene glycol in its pure form has a viscosity of 42 mPas, approximately 45 times that of water, and has a density of 1036 kg/m3, which is only a few percent above the density of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Industrial-grade propylene glycol used in this work was often found to have even higher viscosity values, and therefore each batch of glycol was measured for its density and dynamic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A salt water solution was then prepared in order to match the density to within a tenth of a percent ( ∆ρ ρ = | ρj−ρ∞ ρj | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' These fluids are Newtonian over the range of strain rates imposed, are very miscible with each other, eliminating surface tension as a relevant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Nevertheless, as we shall see, the interface thickness has no time to develop diffusively and essentially remains a sharp interface in the near -field of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Constant Property Jet Profiles Hot-film anemometry was used to first characterize the jet facility to establish the base flow for a constant property jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Fpr a water jet issuing to a water ambient, anemometry was used to characterize the mean velocity profiles and background noise level, as well the shear layer thickness of the jet at the nozzle exit plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 2(a) shows velocity profiles emerging from the jet for multiple Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The profiles are mostly top-hat, characterized by a steep decrease in magnitude in the shear layer towards the quiescent ambient fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A two-dimensional trace of voltage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2b) at the exit plane (z/D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1) confirms the axisymmetric nature of these profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Momentum thickness of the shear layer were evaluated as a function of Reynolds number by integrating radially from the centerline to a location where the velocity decreased to 10% of the centerline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' further radial measurements were avoided as the hot film responds unreliably to the low velocities in the entrained flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The momentum thickness is evaluated as θθ = � ∞ 0 U(r) − U∞ Uc − U∞ [1 − U(r) − U∞ Uc − U∞ ]dr (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='3) AnemometerProbe 3-Axis Traverse AmbientTank NozzleSystem Fluid Supply from Reservoir6 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 2: (a) Velocity profiles at multiple Reynolds numbers (b) Two-dimensional trace of velocity at z/D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1, showing symmetry of profile about the axis (c) Shear layer thickness as a function of Reynolds number The laminar nature of the jet boundary layer at the exit plane is checked (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2(c)) by observing a linear relationship between D/θ and √ Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The constants in the fit are unique to the nozzle geometry, reflecting the acceleration imposed by the area contraction and the resultant thinning of the boundary layer entering the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Profiles at multiple downstream locations within the first half-diameter can be well-represented by an equation of the form used by Mattingly and Chang Mattingly & Chang (1974): u Uc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5(1 + tanh( D 8θ(1 r − r)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4) We now turn to the fluctuations in the jet velocity at the exit plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The turbulence intensity normalized by the average velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The profiles are shown in figure 3(a) alongside profiles measured by Todde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2009) in their work on low Reynolds number free jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It can be seen that the turbulence intensity profile has a comparable trend, although the current work has much lower centerline turbulence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This speaks to the benefit of having a gravity-fed jet, free from any disturbances downstream from pumps or fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Lastly, we examine the spectral content of the flow at the exit plane in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 3, and find no discrete peaks in the frequency spectrum, assuring that the jet is a low-turbulence system with little ambient noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Test Procedure For experiments with propylene glycol as the ambient fluid, each set of experiments is conducted at a constant Reynolds number governed by the flow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Over the course of several test runs, the viscosity of the tank (and hence the value of M) decreases due to eRe=428 Re=725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 4Re=1036 Re=1330 BRe=1676 xew 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 率—Re=2009 n/n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 1 r/DJetVelocity-Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 00 5 10 5 10 xposition 0 yposition24 Experiment 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='20VRe 22 20 D/0 18 16 14 12 1 30 40 50 /ReInstability Transitions in a Low Viscosity Jet 7 Figure 3: (a) Radial profile of turbulence intensity at Re = 1688, M=1 (b) Frequency spectrum of voltage fluctuations mixing with the injected salt water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This makes the test runs inherently quasi-transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Therefore, a careful procedure was followed to minimize the effects of variation of M during each test run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We first estimate the variation of M during a typical test run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' At a salt water-glycol interface, diffusion acts to thicken the interface to yield a concentration thickness given by √γt where γ is the binary diffusion coefficient of propylene glycol into water (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 × 10−9 m2/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For a one-minute long trial run, this yields a diffusion length of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='01D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In practice, test runs were much shorter, typically lasting 20-30s after the initial starting vortex had passed out of the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' During this period, high-speed images were acquired digital camera operating at 500 fps and at 1024 × 1024 px resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' After the image acquisition was completed, the flow was turned off, the tank was stirred with a mixer and allowed to settle and become quiescent again, before the next trial (typically 30 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A sample of tank fluid was taken for subsequent viscosity and density measurement for determining the value of M for each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In this way, at each Reynolds numbers, values of M starting from 50 and descending down to 15 were attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It is reasonable to expect that since the salt-water and propylene glycol were initially density matched to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1% before the start of the experiments, that the density would remain the same through the experiments, even as the bulk viscosity in the tank decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This expectation was somewhat belied — aqueous solutions of propylene glycol undergo a slight contraction in volume that is concentration-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 4(a) shows the variation of density and viscosity in the test section after a series of trial runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It is apparent that the specific gravity varies between the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='036 of pure propylene glycol to a maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='051, or about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The injected salt water jet continues to have a specific gravity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='036, raising the prospect of confounding buoyancy effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The importance of buoyant forces relative to jet inertia is assessed by evaluating the Richardson number, Ri = g∆ρD ρmU 2 j (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5) This is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 4(b) against the Reynolds number, assuming an average density difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='7% over all runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For Reynolds numbers greater than 1500, Ri is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 e-Measuredturbulent intensity Todde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' turbulent intensity (%) max "n/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 r/D×10~5 8 Amplitude (V) 2 0 10 20 30 40 50 60 70 80 Freq (Hz)8 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 4: (a) Variation of density and viscosity in test section during a sequence of runs (b) Richardson number as a function of Re 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Flow Visualization Preliminary images for M=1 (water jet into water ambient) were acquired with an 18MP camera whose lens was equipped with an orange filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The jet fluid was dyed with Rhodamine 6G, and the tank volume was illuminated with a blue LED light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The emission by rhodamine in the orange part of the spectrum was captured and shows the breakdown of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 5 shows the axisymmetric nature of the instabilities dominating the breakdown process, after developing from an initial nearly parallel near-field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' As Re increases, this distance becomes palpably shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Unlike the observations of Mattingly & Chang (1974), no evidence of an eventual competition between the axisymmetric mode and a growing helical mode in the far-field is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' On the other hand, when the jet emerges into an ambient medium of propylene glycol (M=45), helical instabilities are observed over a range of Re from approximately 1600 to 2600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 5(b) shows a sequence of images taken at M=1 and a Reynolds number of 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Of note is the disappearance of the parallel flow region in the near-field, with the helical mode almost instantaneously developing at the exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Some discrete bright spots visible in every image are artifacts due to bubbles being introduced in the test section during the stirring process, which remain suspended due to the high fluid viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We also note that the wavelength of the disturbances seems substantially lower than that of the axisymmetric instability at M=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' These two sets of images suggest that there must exist a transition value (or range) of M for every fixed value of Re, where the dominant mode changes from axisymmetric to helical, and experiments were conducted to elucidate the transition behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 6 shows a sequence of images captured for Re= 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The transition of the dominant instability from helical to axisymmetric, as M decreases from * to * is clearly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Nevertheless, it is difficult to assign a precise value for the transition value of M with confidence in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Inspection of multiple images allows to assign a transition value of M close to ** in this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, in some cases, the images appear to show both axisymmetric and helical features, with two distinct frequency peaks in the hot film spectrum (discussed subsequently), and no clear transition is evident, especially at lower Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Due to the nature of the experiment, involving discrete steps in M, a fine-grained transition value could not be determined in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Nevertheless, observations clearly indicate that this transition value of M is Reynolds number-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 7 shows our estimate for the transition value of M as a function of Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For Re> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='055 45 40 00 Gravity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='05 35 000 30 Standard 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='045 25 M 20 o 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='04 o 10 000 5 0 5 10 15 20 25 30 Trial number0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='15 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='00 1500 2000 2500 3000 ReInstability Transitions in a Low Viscosity Jet 9 1600, the transition M increases as a function of Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' below Re=1600, the instability was weak, and it was difficult to distinguish the nature of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For higher Re, the transition value of M increases with Re, and at Re=2800 (not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 7, the first trial showed a helical mode before switching into an axisymmetric mode, suggesting the critical viscosity required is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 10 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (a) (b) Figure 5: (Top) Growth of axisymmetric instabilities for M=1 and multiple Re: (a) Re=428 (b) Re=1036 (c) Re=1545 (d) Re=2009 (e) Re=2539 (f) Re = 3009 (Bottom) Helical modes observed at M=45, Re = 1332, 1676, 2013, 2339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' a dInstability Transitions in a Low Viscosity Jet 11 Figure 6: Images showing the transition from helical to axisymmetric modes as M is decreased Figure 7: Transition boundary in (M,Re) space from helical to axisymmetric modes at a fixed Reynolds number of 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' From left to right, the values of M are 45, 33, 28, 23 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Here M=28 appears to be closest to the transition between modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 50 45 40 35 30 M 25 20 15 10 Transition Helical 5 Axisymmetric Combined 0 1200 1400 1600 1800 2000 2200 2400 Re12 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' [hb] Figure 8: Evolution of velocity spectra in the downstream direction for M=39, Re= 2013 on the jet axis and in the shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Top: centerline variation for Re=2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Bottom: spectra in the shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Hot Film Anemometry Hot film anemometry was used to characterize the flow for values of M greater than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' With the mixing of the two fluids and the change in Prandtl number, the calibration for the hot film could no longer be used, and the voltage response is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Here we are interested in the spectral content of the velocity fluctuations at different downstream distances, as well as the rate of growth of the disturbance relative to the constant property jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 8(a) and (b) show the evolution of the spectrum along the centerline and in the shear layer for M= * and Re=1682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A distinct frequency peak is visible at all locations in the near field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The strength of voltage fluctuations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 9) for M=1 shows a relatively gentle increase downstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' for large M the strength shows a sharp increase within one jet diameter, appearing to saturate within a few diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The dependence of the dominant frequency on the viscosity ratio, as detected by hot film anemometry, is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Following the experimental sequence and moving from high values of M to low values, one sees an increasing trend while the helical mode remains dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Image Analysis The hot film measurements strongly indicate the existence of a single dominant mode that saturates in intensity in the first few diameters downstream of the jet exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, the increased conductivity of the liquid due to the dissolved salts resulted in increased contamination, pickup of electrical line noise despite probe shielding, and the occasional air bubbles introduced into the tank due to mixing that would stick to the hot film, together resulted in a very low rate at which meaningful data were acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' As a result, LID data were chosen as a means of investigating the growth of unstable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The orange filter on the camera lens ensured that the jet could be strongly distinguished against the background, by isolating the emission from the Rhodamine dye under blue illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Applying a threshold intensity to the grayscale images allows determination of the jet boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' the diameter of the jet as determined from the result was very weakly sensitive to the threshold value, but we are primarily interested in the frequency, which ×10-3 Re =2013 at z/D =0, centerline ×10~3 Re = 2013 at z/D = 1, centerline ×10-3 Re = 2013 at z/D = 2, centerline ×10-3 Re = 2013 at z/D = 3, centerline 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 M2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 20 30 40 50 60 70 80 20 30 40 50 60 70 80 20 30 40 50 60 70 80 20 30 40 50 60 70 80 Freq (Hz) Freq (Hz) Freq (Hz) Freq (Hz) Re = 2013 at z/D = 0, shear layer Re = 2013 at z/D = 1, shear layer Re = 2013 at z/D = 2, shear layer Re = 2013 at z/D = 3, shear layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='008 008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='008 (V) 006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='006 004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='004 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='002 002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='002 20 30 40 50 60 70 80 20 30 40 60 70 80 20 30 40 50 60 70 80 20 30 40 60 70 80 Freg (Hz) Fre (Hz) Freg (Hz) Fre (Hz)Instability Transitions in a Low Viscosity Jet 13 Figure 9: Root mean square value of voltage fluctuations along the centerline and shear layer for M=1 and M=45 at Re=2000 Figure 10: Stanton numbers corresponding to the dominant frequency, as identified by hot film anemometry, as a function of M for Re=1682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' is unaffected by the choice of threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' To study the spatial evolution of the oscillations of the interface, we examine the jet width at 4 locations downstream of the jet exit, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The amplitude of oscillations shows a non-linear increase, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 12 we further examine the amplitude and frequency of these oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 12 shows the sharp peak in the power spectrum that might be surmised from the oscillatory waveform in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 12(b) shows the variation of the amplitude of the dominant frequency in the downstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Again, as with the anemometry measurements, the oscillations show an exponential increase in the disturbance amplitude, before saturating at z/D=* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' To ascertain the nature of this instability, that develops much faster than the axisym- metric instability of the constant property jet, we verified that the frequencies at the different downstream stations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 11 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Another way of assessing the spatially invariant ‘global’ nature of this frequency is to examine the intensity records of at single pixels in the shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 13 shows the frequency spectrum of 4 pixels at two different downstream locations, on either side of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The frequencies are identical, 15 eM=39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='Centerline M=39,ShearLayer M=1,Centerline M=1,ShearLayer 10 (%) VI V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 5 04 0 2 3 4 5 z/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 Re=1682 Re=842,Trial 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='7 Re=842, Trial 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 20 30 40 50 M14 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' [b] Figure 11: (a) Time variation of the jet width in pixels at different downstream distance (b) Square of the amplitude of the coefficients of the Fourier transform, A2(f) (a) (b) Figure 12: (a) Power spectrum of oscillation of interface at z/D= for M=38, Re=2400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (b) Growth of the square of the amplitude of the Fourier coefficient of the dominant mode in the downstream direction and provide further circumstantial evidence that the instability observed at large M is a global mode, corresponding to absolute instability of the near-field profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' This putative global mode has a frequency that depends on the parameters that define the flow, such as the inlet Reynolds number, viscosity ratio M, and the inlet velocity profile, specified by the momentum thickness θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In the experiment, the values of Re and θ are conjoined through the specific geometry of the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Further, it is experimentally difficult to conduct trials at constant M, while changing Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Therefore, we present the global mode frequency at constant Re (and θ) as a function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Figure 14(a) shows that the waves developing on the interface have a frequency that decreases as the ambient viscosity is reduced from a starting value of M≈ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For this data set taken at Re= 2000, there is a sharp increase in frequency near the observed transition from helical to axisymmetric mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We interpret this as further evidence of the helical mode being driven by an absolute instability of near-field profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' After the transition to the axisymmetric mode, there is a further decrease in frequency, before the curve displays an asymptotic behavior as M is further reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 9mm 6mm 3mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 10 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 Time(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 E0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 20 40 60 801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 A^2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 z/DInstability Transitions in a Low Viscosity Jet 15 Figure 13: Spectrum of pixel intensity fluctuation in shear layer at four locations in the near-field of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The corresponding wavelengths, as determined from inspection of the images, are shown in Figure 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Knowledge of the wavelength and frequency allows us to calculate the phase velocity of the dominant mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' this is plotted in Figure 14(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Summary and Discussion We have presented experiments characterizing the flow behavior for the specific con- figuration of a low-viscosity jet with a laminar boundary layer emerging into an ambient medium of relatively higher viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The fluids, while perfectly miscible, have a low binary diffusion coefficient, with the result that the diffusion region is expected to be extremely thin, approaching a sharp interface with zero surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The flow visualization results clearly indicate the onset of a helical mode at or close to the nozzle exit plane, when the viscosity ratio M is increased substantially beyond unity, with enhanced mixing relative to the M=1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For that baseline case, it is apparent that the jet can stay coherent and nearly parallel for a significant distance downstream indicating a slowly evolving spatial instability that saturates in the far-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' For latge M, hot film anemometry confirms the presence of a self-sustained oscillation with a discrete peak in the spectrum, with the disturbance energy rapidly growing and saturating within the first few diameters downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Further, we have characterized this dominant frequency as a function of M and Re over a limited range, showing that this frequency is an increasing function of Re and a decreasing function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, over the range of M studied, the anemometry does not display any sharp jumps in the frequency, leaving open the question of the convective/absolute nature of the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' To resolve issues with anemometry in conducting fluids, a larger range of Reynolds numbers were examined using image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' With this technique,the initial linear growth, followed by an exponential rise in the amplitude of oscillations is more clearly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We are able to confirm the frequency trends observed with anemometry, a discrete jump in frequency at a certain value of M, as well as the spatial invariance of this frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Turning to the convective/absolute nature of the instability, at first glance, one cannot 50 40 40 30 30 20 20 10 10 5 10 15 20 25 30 35 5 10 15 20 25 30 35 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 50 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 40 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0 5 10 15 20 25 30 35 Ln 10 15 20 25 30 3516 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (a) (b) (c) Figure 14: Variation of instability characteristics along the constant (Re,θ) curve, as M decreases dueing the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (a) Frequency (b) Wavelength (c) Phase velocity rule out the possibility that these observations reflect a rapidly growing convective instability that happens to be clearly visible against a low-noise background established by pump-less flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, there are several features of the flow, as discussed above, that suggest otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' It is generally accepted that for an instability to be considered a global mode that arises from a linear, absolutely unstable mechanism (as understood in the context of spatio-temporal linear stability analysis), it should display certain hallmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' These are: the presence of a single self-sustained oscillation that can be detected everywhere in the domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' the sharp onset of a regime of enhanced mixing in which this instability appears, as determined by the appropriate control parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' and the insensitivity to external forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The first criterion is found to be amply satisfied in the present experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The image analysis does show a sharp jump in frequency and wavelength as the viscosity ratio is decreased, and the transition value of M is reasonably consistent with visual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Further evidence for the sharp onset is afforded by an examination of the singular values obtained through Principal Component Analysis, which suggest that near the critical value of M corresponding to the frequency jump, there is always only one dominant mode, implying no mode competition close to the transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We also note that in line with the theoretical understanding that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='9 number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 Strouhal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4, 10 20 30 40 50 60 Viscosity2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 入/D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 10 20 30 40 50 60 Viscosity1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='9 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content='6 10 20 30 40 50 60 ViscosityInstability Transitions in a Low Viscosity Jet 17 that the development length for the global mode is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' in principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' infinite at the transition boundary and onset of the global mode (in terms of the controlling parameters) (Couairon & Chomaz 1997) but shortens away from the transition boundary as one moves into the regime of absolute instability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' the disappearance of a region of parallel flow for large M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' compared to M=1 is an indicator of the instability being controlled by inlet conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' The question of sensitivity (or lack thereof) to external noise remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Since the flow is gravity-fed, pump-driven oscillations such as those used by dOl (2009) are difficult o implement and current work focuses on applying vibrations to the diffuser section upstream of the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' A strong response by the system, as in terms of enhanced amplitude of oscillations, to forcing frequencies near the natural frequency of the insta- bility would provide further strong support for the idea that the observed helical mode correspodnds to the absolutely unstable mode found in the companion computational paper by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' However, it should be noted that this may not necessarily constitute clinching evidence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' indeed, Hallberg & Strykowski (2008) have shown that the global mode is a low-density jet can be overwhelmed by sufficiently strong forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We also note that the decrease in the Strouhal number with increasing M as exemplified in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 10 and 14(a) is closely aligned with behavior of the absolutely unstable frequency, as predicted by spatio-temporal linear stability analysis for a specific tanh-type profile (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' of citetYang2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Lastly, it should be noted that global mode characteristics are closely linked to inlet profiles, and therefore the influence of the boundary layer thickness needs to be explicitly addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' In the present study, this thickness is directly linked to the Reynolds number through the nozzle contraction profile, and future studies will require disentangling the relative contributions of these two parameters to the evolution of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Acknowledgement We are grateful for useful discussions with David Forliti and Paul Strykowski during the preparation of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' We also acknowledge the assistance from Justin Chen in acquiring some of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' numerical simulations and nonlinear global modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Journal of Fluid Mechanics 618, 323–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Srinivasan, V, Hallberg, M P & Strykowski, P J 2010 Viscous linear stability of axisymmetric low-density jets: Parameters influencing absolute instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Physics of Fluids 22 (2), 24103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Subbarao, ER & Cantwell, BJ 1992 Investigation of a 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' & Srinivasan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' 2022 Linear stability analysis of a round jet emerging into an ambient medium of different viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Submitted to PRF or JFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yang, Jinwei, Anderson, Matt J, Strykowski, Paul J & Srinivasan, Vinod 2021 Effects of confinement on absolute and convective instabilities for momentum-driven countercurrent shear layers.' 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1175–1181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Yu, Ming-Huei & Monkewitz, Peter A 1993 Oscillations in the near field of a heated two- dimensional jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'} +page_content=' Journal of fluid mechanics 255, 323–347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztFRT4oBgHgl3EQfkDcP/content/2301.13593v1.pdf'}